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BEGIN:VEVENT
SUMMARY:Rick Jardine (University of Western Ontario.)
DTSTART:20200916T230000Z
DTEND:20200917T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/1
DESCRIPTION:Title: Posets\, metric spaces\, and topological data analysis.
\nby Rick Jardine (University of Western Ontario.) as part of New York Cit
y Category Theory Seminar\n\n\nAbstract\nTraditional TDA is the analysis o
f homotopy invariants of systems of spaces V(X) that arise from finite met
ric spaces X\, via distance measures. These spaces can be expressed in ter
ms of posets\, which are barycentric subdivisions of the usual Vietoris-Ri
ps complexes V(X). The proofs of stability theorems in TDA are sharpened c
onsiderably by direct use of poset techniques.\n\nExpanding the domain of
definition to extended pseudo metric spaces enables the construction of a
realization functor on diagrams of spaces\, which has a right adjoint Y |-
-> S(Y)\, called the singular functor. The realization of the Vietoris-Rip
s system V(X) for an ep-metric space X is the space itself. The counit of
the adjunction defines a map \\eta: V(X) --> S(X)\, which is a sectionwise
weak equivalence - the proof uses simplicial approximation techniques.\n\
nThis is the context for the Healy-McInnes UMAP construction\, which will
be discussed if time permits. UMAP is non-traditional: clusters for UMAP a
re defined by paths through sequences of neighbour pairs\, which can be a
highly efficient process in practice.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Ellerman (University of Ljubljana)
DTSTART:20200930T230000Z
DTEND:20201001T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/2
DESCRIPTION:Title: The Logical Theory of Canonical Maps: The Elements & Distin
ctions Analysis of the Morphisms\, Duality\, Canonicity\, and Universal Co
nstructions in Sets.\nby David Ellerman (University of Ljubljana) as p
art of New York City Category Theory Seminar\n\n\nAbstract\nAbstract: Cate
gory theory gives a mathematical characterization of naturality but not of
canonicity. The purpose of this paper is to develop the logical theory of
canonical maps based on the broader demonstration that the dual notions o
f elements & distinctions are the basic analytical concepts needed to unpa
ck and analyze morphisms\, duality\, canonicity\, and universal constructi
ons in Sets\, the category of sets and functions. The analysis extends dir
ectly to other Sets-based concrete categories (groups\, rings\, vector spa
ces\, etc.). Elements and distinctions are the building blocks of the two
dual logics\, the Boolean logic of subsets and the logic of partitions. Th
e partial orders (inclusion and refinement) in the lattices for the dual l
ogics define morphisms. The thesis is that the maps that are canonical in
Sets are the ones that are defined (given the data of the situation) by th
ese two logical partial orders and by the compositions of those maps.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathon Funk (Queensborough CUNY)
DTSTART:20201014T220000Z
DTEND:20201014T233000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/3
DESCRIPTION:Title: Pseudogroup Torsors.\nby Jonathon Funk (Queensborough C
UNY) as part of New York City Category Theory Seminar\n\n\nAbstract\nAbstr
act: We use sheaf theory to analyze the topos of etale actions on the germ
groupoid of a pseudogroup in the sense that we present a site for this to
pos\, which we call the classifying topos of the pseudogroup. Our analysis
carries us further into how pseudogroup morphisms and geometric morphisms
are related. Ultimately\, we shall see that the classifying topos classif
ies what we call a pseudogroup torsor. In hindsight\, we see that pseudogr
oups form a bicategory of `flat' bimodules.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrei V. Rodin (Saint Petersburg State University.)
DTSTART:20201021T230000Z
DTEND:20201022T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/4
DESCRIPTION:Title: Vladimir Voevodsky’s Unachieved Project\nby Andrei V.
Rodin (Saint Petersburg State University.) as part of New York City Categ
ory Theory Seminar\n\n\nAbstract\nSoon after receiving the Fields Medal fo
r his proof of Milnor Conjecture and the related work in the Motivic Theor
y\, Vladimir delivered a series of two public lectures in the Wuhan Univer
sity (China) titled “What is most important for mathematics in the near
future?” where he described the most urgent tasks as follows: 1) to buil
d a computerised version of Bourbaki’s ‘Elements’ and 2) to bridge p
ure and applied mathematics. The first project resulted into the Univalent
foundations of mathematics. The second project remained unachieved in spi
te of significant time and efforts that Vladimir spent for its realisation
. More specifically\, during 2007-2009 Vladimir worked on a mathematical t
heory of Population Dynamics but then abandoned this project and focused o
n the Univalent Foundations until the sudden end of his life in 2017. \n
\nUsing extensive unpublished materials available via Vladimir Voevo
dsky’s memorial webpage (h
ttps://www.math.ias.edu/Voevodsky/)\n I reconstruct Vladimir’s visio
n of mathematics and its role in science incuding his original strategy of
bridging the gap between the pure and applied mathematics. Finally\, I sh
ow a relevance of Univalent Foundations to Vladimir’s unachieved project
and speculate about a possible role of Univalent Foundations in science.\
n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Larry Moss (Indiana University)
DTSTART:20201028T230000Z
DTEND:20201029T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/5
DESCRIPTION:Title: Coalgebra in Continuous Mathematics.\nby Larry Moss (In
diana University) as part of New York City Category Theory Seminar\n\n\nAb
stract\nAbstract: A slogan from coalgebra in the 1990's holds that\n\n'dis
crete mathematics : algebra :: continuous mathematics : coalgebra'\n\nThe
idea is that objects in continuous math\, like real numbers\, are often un
derstood via their approximations\, and coalgebra gives tools for understa
nding and working with those objects. Some examples of this are Pavlovic a
nd Escardo's relation of ordinary differential equations with coinduction\
, and also Freyd's formulation of the unit interval as a final coalgebra.
My talk will be an organized survey of several results in this area\, incl
uding (1) a new proof of Freyd's Theorem\, with extensions to fractal sets
\; (2) other presentations of sets of reals as corecursive algebras and fi
nal coalgebras\; (3) a coinductive proof of the correctness of policy iter
ation from Markov decision processes\; and (4) final coalgebra presentatio
ns of universal Harsanyi type spaces from economics.\n\nThis talk reports
on joint work with several groups in the past 5-10 years\, and also some o
ngoing work.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Scoccola (Michigan State University)
DTSTART:20201105T000000Z
DTEND:20201105T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/6
DESCRIPTION:Title: Locally persistent categories and approximate homotopy theo
ry.\nby Luis Scoccola (Michigan State University) as part of New York
City Category Theory Seminar\n\n\nAbstract\nAbstract: In applied homotopy
theory and topological data analysis\, procedures use homotopy invariants
of spaces to study and classify discrete data\, such as finite metric spac
es. To show that such a procedure is robust to noise\, one endows the coll
ection of possible inputs and the collection of outputs with metrics\, and
shows that the procedure is continuous with respect to these metrics\, so
one is interested in doing some kind of approximate homotopy theory. I wi
ll show that a certain type of enriched categories\, which I call locally
persistent categories\, provide a natural framework for the study of appro
ximate categorical structures\, and in particular\, for the study of metri
cs relevant to applied homotopy theory and metric geometry.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noah Chrein (University of Maryland)
DTSTART:20201112T000000Z
DTEND:20201112T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/7
DESCRIPTION:Title: Yoneda ontologies.\nby Noah Chrein (University of Maryl
and) as part of New York City Category Theory Seminar\n\n\nAbstract\nAbstr
act: We will discuss a 2-categorical model of ontology\, and how to view c
ertain higher categories as ontologies in this language. We can translate
the various Yoneda lemmas associated to higher categories into the languag
e of ontology\, and in turn\, discuss what it means for a generic ontology
to have a yoneda lemma. These will be the "Yoneda Ontologies".\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Enrico Ghiorzi (Appalachian State University)
DTSTART:20201119T000000Z
DTEND:20201119T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/8
DESCRIPTION:Title: Internal enriched categories.\nby Enrico Ghiorzi (Appal
achian State University) as part of New York City Category Theory Seminar\
n\n\nAbstract\nAbstract: Internal categories feature a notion of completen
ess which is remarkably well behaved. For example\, the internal adjoint f
unctor theorem requires no solution set condition. Indeed\, internal categ
ories are intrinsically small\, and thus immune from the size issues commo
nly afflicting standard category theory. Unfortuntely\, they are not quite
as expressive as we would like: for example\, there is no internal Yoneda
lemma. To increase the expressivity of internal category theory\, we defi
ne a notion of internal enrichment over an internal monoidal category and
develop its theory of completeness. The resulting theory unites the good p
roperties of internal categories with the expressivity of enriched categor
y theory\, thus providing a powerful framework to work with.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dan Shiebler (Oxford University)
DTSTART:20201210T000000Z
DTEND:20201210T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/9
DESCRIPTION:Title: Functorial Manifold Learning and Overlapping Clustering.\nby Dan Shiebler (Oxford University) as part of New York City Category T
heory Seminar\n\n\nAbstract\nAbstract: We adapt previous research on funct
orial clustering and topological unsupervised learning to develop a functo
rial perspective on manifold learning algorithms. Our framework characteri
zes a manifold learning algorithm in terms of the loss function that it op
timizes\, which allows us to focus on the algorithm's objective rather tha
n the details of the learning process. We demonstrate that we can express
several state of the art manifold learning algorithms\, including Laplacia
n Eigenmaps\, Metric Multidimensional Scaling\, and UMAP\, as functors in
this framework. This functorial perspective allows us to reason about the
invariances that these algorithms preserve and prove refinement bounds on
the kinds of loss functions that any such functor can produce. Finally\, w
e experimentally demonstrate how this perspective enables us to derive and
analyze novel manifold learning algorithms.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Winkler
DTSTART:20201203T000000Z
DTEND:20201203T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/10
DESCRIPTION:Title: Functors as homomorphisms of quivered algebras.\nby An
drew Winkler as part of New York City Category Theory Seminar\n\n\nAbstrac
t\nAbstract: A quiver induces a minimalist algebraic structure which is\,
nonetheless\, balanced\, associative\, elementwise strongly irreducible\,
and both left and right quivered\, in a functorial way\; a homomorphism of
quivers induces a homomorphism of algebras. Q balanced\, quivered algebra
possesses a quiver structure\, but it is not true in general that a homom
orphism for the algebra is also a homomorphism for the quiver. It will be
precisely when it is also a homomorphism for the algebra structure induced
by the quiver structure it induces. Such a bihomorphism\, in the special
case of categories\, (where the associativity property and a composition-i
nducing property hold)\, is precisely a functor. This facet of categories\
, as possessing two compatible composition structures\, explains in some s
ense a bifurcation in the structure of monads.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arthur Parzygnat (IHES)
DTSTART:20201216T180000Z
DTEND:20201216T193000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/11
DESCRIPTION:Title: A functorial characterization of classical and quantum ent
ropies.\nby Arthur Parzygnat (IHES) as part of New York City Category
Theory Seminar\n\n\nAbstract\nAbstract: Entropy appears as a useful concep
t in a wide variety of academic disciplines. As such\, one would suspect t
hat category theory would provide a suitable language to encompass all or
most of these definitions. The Shannon entropy has recently been given a c
haracterization as a certain affine functor by Baez\, Fritz\, and Leinster
. This characterization is the only characterization I know of that uses l
inear assumptions (as opposed to additive\, exponential\, logarithmic\, et
c). Here\, we extend that characterization to include the von Neumann entr
opy as well as highlight the new categorical structures that arise when tr
ying to do so. In particular\, we introduce Grothendieck fibrations of con
vex categories\, and we review the notion of a disintegration\, which is a
key part of conditional probability and Bayesian statistics and plays a c
rucial role in our characterization theorem. The characterization of Baez\
, Fritz\, and Leinster interprets Shannon entropy in terms of the informat
ion loss associated to a deterministic process\, which is possible since t
he entropy difference associated to such a process is always non-negative.
This fails for quantum entropy\, and has important physical consequences.
\n
\nReferences:
\n
Paper (and references therein)
\n Paper (original paper of Baez\, Fritz\, and Leinster)\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jason Parker (Brandon University)
DTSTART:20210204T000000Z
DTEND:20210204T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/12
DESCRIPTION:Title: Isotropy Groups of Quasi-Equational Theories.\nby Jaso
n Parker (Brandon University) as part of New York City Category Theory Sem
inar\n\n\nAbstract\nAbstract: In [2]\, my PhD supervisors (Pieter Hofstra
and Philip Scott) and I studied the new topos-theoretic phenomenon of isot
ropy (as introduced in [1]) in the context of single-sorted algebraic theo
ries\, and we gave a logical/syntactic characterization of the isotropy gr
oup of any such theory\, thereby showing that it encodes a notion of inner
automorphism or conjugation for the theory. In the present talk\, I will
summarize the results of my recent PhD thesis\, in which I build on this e
arlier work by studying the isotropy groups of (multi-sorted) quasi-equati
onal theories (also known as essentially algebraic\, cartesian\, or finite
limit theories). In particular\, I will show how to give a logical/syntac
tic characterization of the isotropy group of any such theory\, and that i
t encodes a notion of inner automorphism or conjugation for the theory. I
will also describe how I have used this characterization to exactly charac
terize the ‘inner automorphisms’ for several different examples of qua
si-equational theories\, most notably the theory of strict monoidal catego
ries and the theory of presheaves valued in a category of models. In parti
cular\, the latter example provides a characterization of the (covariant)
isotropy group of a category of set-valued presheaves\, which had been an
open question in the theory of categorical isotropy.\n\n[1] J. Funk\, P. H
ofstra\, B. Steinberg. Isotropy and crossed toposes. Theory and Applicatio
ns of Categories 26\, 660-709\, 2012.\n\n[2] P. Hofstra\, J. Parker\, P.J.
Scott. Isotropy of algebraic theories. Electronic Notes in Theoretical Co
mputer Science 341\, 201-217\, 2018.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Hines (University of York)
DTSTART:20210211T000000Z
DTEND:20210211T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/13
DESCRIPTION:Title: Shuffling cards as an operad.\nby Peter Hines (Univers
ity of York) as part of New York City Category Theory Seminar\n\n\nAbstrac
t\nThe theory of how two packs of cards may be shuffled together to form a
single pack has been remarkably well-studied in combinatorics\, group the
ory\, statistics\, and other areas of mathematics. This talk aims to study
natural extensions where 1/ We may have infinitely many cards in a deck\,
2/ We may take the result of a previous shuffle as one of our decks of ca
rds (i.e. shuffles are hierarchical)\, and 3/ There may even be an infinit
e number of decks of cards.\n\nFar from being 'generalisation for generali
sation's sake'\, the original motivation came from theoretical & practical
computer science. The mathematics of card shuffles is commonly used to de
scribe processing in multi-threaded computations. Moving to the infinite c
ase gives a language in which one may talk about potentially non-terminati
ng processes\, or servers with an unbounded number of clients\, etc.\n\nHo
wever\, this talk is entirely about algebra & category theory -- just as i
n the finite case\, the mathematics is of interest in its own right\, and
should be studied as such.\n\nWe model shuffles using operads. The intuiti
on behind them of allowing for arbitrary n-ary operations that compose in
a hierarchical manner makes them a natural\, inevitable choice for describ
ing such processes such as merging multiple packs of cards.\n\nWe use very
concrete examples\, based on endomorphism operads in groupoids of arithme
tic operations. The resulting structures are at the same time both simple
(i.e. elementary arithmetic operations)\, and related to deep structures i
n mathematics and category theory (topologies\, tensors\, coherence\, asso
ciahedra\, etc.)\n\nWe treat this as a feature\, not a bug\, and use it to
describe complex structures in elementary terms. We also aim to give prev
iously unobserved connections between distinct areas of mathematics.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Richard Blute (University of Ottawa)
DTSTART:20210218T000000Z
DTEND:20210218T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/14
DESCRIPTION:Title: Finiteness Spaces\, Generalized Polynomial Rings and Topol
ogical Groupoids.\nby Richard Blute (University of Ottawa) as part of
New York City Category Theory Seminar\n\n\nAbstract\nAbstract: The categor
y of finiteness spaces was introduced by Thomas Ehrhard as a model of clas
sical linear logic\, where a set is equipped with a class of subsets to be
thought of as finitary. Morphisms are relations preserving the finitary s
tructure. The notion of finitary subset allows for a sharp analysis of com
putational structure.\n\nWorking with finiteness spaces forces the number
of summands in certain calculations to be finite and thus avoid convergenc
e questions. An excellent example of this is how Ribenboim’s theory of g
eneralized power series rings can be naturally interpreted by assigning fi
niteness monoid structure to his partially ordered monoids. After Ehrhard
’s linearization construction is applied\, the resulting structures are
the rings of Ribenboim’s construction.\n\nThere are several possible cho
ices of morphism between finiteness spaces. If one takes structure-preserv
ing partial functions\, the resulting category is complete\, cocomplete an
d symmetric monoidal closed. Using partial functions\, we are able to mode
l topological groupoids\, when we consider composition as a partial functi
on. We can associate to any hemicompact etale Hausdorff groupoid a complet
e convolution ring. This is in particular the case for the infinite paths
groupoid associated to any countable row-finite directed graph.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joshua Sussan (Medgar Evers)
DTSTART:20210304T000000Z
DTEND:20210304T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/15
DESCRIPTION:Title: Categorification and quantum topology.\nby Joshua Suss
an (Medgar Evers) as part of New York City Category Theory Seminar\n\n\nAb
stract\nAbstract: The Jones polynomial of a link could be defined through
the representation theory of quantum sl(2). It leads to a 3-manifold invar
iant and 2+1 dimensional TQFT. In the mid 1990s\, Crane and Frenkel outlin
ed the categorification program with the aim of constructing a 3+1 dimensi
onal TQFT by upgrading the representation theory of quantum sl(2) to some
categorical structures. We will review these ideas and give examples of va
rious categorifications of quantum sl(2) constructions.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tobias Fritz (University of Innsbruck)
DTSTART:20210317T230000Z
DTEND:20210318T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/16
DESCRIPTION:Title: Categorical Probability and the de Finetti Theorem\nby
Tobias Fritz (University of Innsbruck) as part of New York City Category
Theory Seminar\n\n\nAbstract\nI will give an introduction to categorical p
robability in terms of Markov categories\, followed by a discussion of the
classical de Finetti theorem within that framework. Depending on whether
current ideas work out or not\, I may (or may not) also present a sketch o
f a purely categorical proof of the de Finetti theorem based on the law of
large numbers. Joint work with Tomáš Gonda\, Paolo Perrone and Eigil Fj
eldgren Rischel.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ross Street (Macquarie University)
DTSTART:20210414T230000Z
DTEND:20210415T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/17
DESCRIPTION:Title: Absolute colimits for differential graded categories.\
nby Ross Street (Macquarie University) as part of New York City Category T
heory Seminar\n\n\nAbstract\nA little enriched category theory will be rev
iewed\, in particular\, absolute colimits and Cauchy completion. Then the
focus will be on the monoidal category DGAb of chain complexes of abelian
groups which is at the heart of homological and homotopical algebra. Categ
ories enriched in DGAb are called differential graded categories (DG-categ
ories). Recent joint work with Branko Nikolic and Giacomo Tendas on the ab
solute colimit completion of a DG-category will be described. The talk is
dedicated to the memory of two great New Yorkers\, Sammy Eilenberg and Ale
x Heller.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Juan Orendain (University of Mexico\, UNAM.)
DTSTART:20210505T230000Z
DTEND:20210506T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/18
DESCRIPTION:Title: How long does it take to frame a bicategory?\nby Juan
Orendain (University of Mexico\, UNAM.) as part of New York City Category
Theory Seminar\n\n\nAbstract\nAbstract: Framed bicategories are double cat
egories having all companions and conjoints. Many structures naturally org
anize into framed bicategories\, e.g. open Petri nets\, polynomials functo
rs\, polynomial comonoids\, structured cospans\, algebras\, etc. Symmetric
monoidal structures on framed bicategories descend to symmetric monoidal
structures on the corresponding horizontal bicategories. The axioms defini
ng symmetric monoidal double categories are much more tractible than those
defining symmetric monoidal bicategories. It is thus convenient to study
ways of lifting a given bicategory into a framed bicategory along an appro
priate category of vertical morphisms. Solutions to the problem of lifting
bicategories to double categories have classically being useful in expres
sing Kelly and Street's mates correspondence and in proving the higher dim
ensional Seifert-van Kampen theorem of Brown et. al.\, amongst many other
applications. We consider lifting problems in their full generality.\n\nGl
obularly generated double categories are minimal solutions to lifting prob
lems of bicategories into double categories along given categories of vert
ical arrows. Globularly generated double categories form a coreflective su
b-2-category of general double categories. This\, together with an analysi
s of the internal structure of globularly generated double categories yiel
ds a numerical invariant on general double categories. We call this invari
ant the vertical length. The vertical length of a double category C measur
es the complexity of mixed compositions of globular and horizontal identit
y squares of C and thus provides a measure of complexity for lifting probl
ems on the horizontal bicategory HC of C. I will explain recent results on
the theory of globularly generated double categories and the vertical len
gth invariant. The ultimate goal of the talk is to present conjectures on
the vertical length of framed bicategories and possible applications.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tobias Fritz (University of Innsbruck)
DTSTART:20210324T230000Z
DTEND:20210325T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/19
DESCRIPTION:Title: Categorical Probability and the de Finetti Theorem\nby
Tobias Fritz (University of Innsbruck) as part of New York City Category
Theory Seminar\n\n\nAbstract\nI will give an introduction to categorical p
robability in terms of Markov categories\, followed by a discussion of the
classical de Finetti theorem within that framework. Depending on whether
current ideas work out or not\, I may (or may not) also present a sketch o
f a purely categorical proof of the de Finetti theorem based on the law of
large numbers. Joint work with Tomáš Gonda\, Paolo Perrone and Eigil Fj
eldgren Rischel.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gemma De las Cuevas (University of Innsbruck)
DTSTART:20211006T230000Z
DTEND:20211007T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/20
DESCRIPTION:Title: From simplicity to universality and undecidability\nby
Gemma De las Cuevas (University of Innsbruck) as part of New York City Ca
tegory Theory Seminar\n\n\nAbstract\nWhy is it so easy to generate complex
ity? I will suggest that this is due to the phenomenon of universality —
essentially every non-trivial system is universal\, and thus able to expl
ore all complexity in its domain. We understand universality in spin model
s\, automata and neural networks. I will present the first step toward rig
orously linking the first two\, where we cast classical spin Hamiltonians
as formal languages and classify the latter in the Chomsky hierarchy. We p
rove that the language of (effectively) zero-dimensional spin Hamiltonians
is regular\, one-dimensional spin Hamiltonians is deterministic context-f
ree\, and higher-dimensional and all-to-all spin Hamiltonians is context-s
ensitive. I will also talk about the other side of the coin of universalit
y\, namely undecidability\, and will raise the question of whether univers
ality is "visible" in Lawvere’s Theorem.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dan Shiebler (Oxford University)
DTSTART:20211020T230000Z
DTEND:20211021T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/21
DESCRIPTION:Title: Out of Sample Generalization with Kan Extensions\nby D
an Shiebler (Oxford University) as part of New York City Category Theory S
eminar\n\n\nAbstract\nA common problem in data science is "use this functi
on defined over this small set to generate predictions over that larger se
t." Extrapolation\, interpolation\, statistical inference and forecasting
all reduce to this problem. The Kan extension is a powerful tool in catego
ry theory that generalizes this notion. In this work we explore several ap
plications of Kan extensions to data science.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dusko Pavlovic (University of Hawai‘i at Mānoa)
DTSTART:20211103T230000Z
DTEND:20211104T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/22
DESCRIPTION:Title: Geometry of computation and string-diagram programming in
monoidal computer\nby Dusko Pavlovic (University of Hawai‘i at Māno
a) as part of New York City Category Theory Seminar\n\n\nAbstract\nA monoi
dal computer is a monoidal category with a distinguished type carrying the
structure of a single-instruction programming language. The instruction w
ould be written as "run"\, but it is usually drawn as a string diagram. Eq
uivalently\, the monoidal computer structure can be viewed as a typed lamb
da-calculus without lambda abstraction\, even implicit. Any Turing complet
e programming language\, including Turing machines and partial recursive f
unctions\, gives rise to a monoidal computer. We have thus added yet anoth
er item to the Church-Turing list of models of computation. It differs fro
m other models by its categoricity. While the other Church-Turing models c
an be programmed to simulate each other in many different ways\, and each
interprets even itself in infinitely many non-isomorphic ways\, the struct
ure of a monoidal computer is unique up to isomorphism. A monoidal categor
y can be a monoidal computer in at most one way\, just like it can be clos
ed in at most one way\, up to isomorphism. In other words\, being a monoid
al computer is a property\, not structure. Computability is thus a categor
ical property\, like completeness. This opens an alley towards an abstract
treatment of parametrized complexity\, one-way and trapdoor functions on
one hand\, and of algorithmic learning in the other.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marco Schorlemmer (Spanish National Research Council)
DTSTART:20211118T000000Z
DTEND:20211118T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/23
DESCRIPTION:Title: A Uniform Model of Computational Conceptual Blending\n
by Marco Schorlemmer (Spanish National Research Council) as part of New Yo
rk City Category Theory Seminar\n\n\nAbstract\nWe present a mathematical m
odel for the cognitive operation of conceptual blending that aims at being
uniform across different representation formalisms\, while capturing the
relevant structure of this operation. The model takes its inspiration from
amalgams as applied in case-based reasoning\, but lifts them into categor
y theory so as to follow Joseph Goguen’s intuition for a mathematically
precise characterisation of conceptual blending at a representation-indepe
ndent level of abstraction. We prove that our amalgam-based category-theor
etical model of conceptual blending is essentially equivalent to the pusho
ut model in the ordered category of partial maps as put forward by Goguen.
But unlike Goguen’s approach\, our model is more suitable to capture co
mputational realisations of conceptual blending\, and we exemplify this by
concretising our model to computational conceptual blends for various rep
resentation formalisms and application domains.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Geroch (University of Chicago)
DTSTART:20211202T000000Z
DTEND:20211202T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/24
DESCRIPTION:Title: An Alien's Perspective on Mathematics (and Physics).\n
by Robert Geroch (University of Chicago) as part of New York City Category
Theory Seminar\n\n\nAbstract\nAbstract: We describe what might be called
a "point of view" toward mathematics. This view touches on such issues as
how Godel's theorem might be interpreted\, the relevance to physics of mat
hematical axioms such as the axiom of choice\, and the possibility of usin
g physics to "solve" unsolvable mathematical problems.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samantha Jarvis (The CUNY Graduate Center)
DTSTART:20211216T000000Z
DTEND:20211216T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/25
DESCRIPTION:Title: Language as an Enriched Category.\nby Samantha Jarvis
(The CUNY Graduate Center) as part of New York City Category Theory Semina
r\n\n\nAbstract\nWe review enriched category theory\, with particular focu
s on enriching over posets such as [0\,1]. We then apply this to natural l
anguage\, making a language category into an enriched language category as
in Bradley-Vlassopoulos-Terilla (our advisor!) [2106.07890.pdf (arxiv.org
)]. The statements of enriched category theory have concrete (and interest
ing!) interpretations when applied to this enriched language category.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Todd Trimble (Western Connecticut State University)
DTSTART:20211223T000000Z
DTEND:20211223T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/26
DESCRIPTION:Title: Categorifying negatives: roadblocks and detours.\nby T
odd Trimble (Western Connecticut State University) as part of New York Cit
y Category Theory Seminar\n\n\nAbstract\nThe challenge of finding meaningf
ul categorified interpretations of "reciprocals" of objects and "negatives
" of objects poses some intriguing problems. In this talk\, we consider a
few responses to this challenge\, with particular attention to extending t
he substitution product on species to "negative species" and "virtual spec
ies".\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ralph Wojtowicz (Shenandoah University)
DTSTART:20220203T000000Z
DTEND:20220203T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/27
DESCRIPTION:Title: On Logic-Based Artificial Intelligence and Categorical Log
ic.\nby Ralph Wojtowicz (Shenandoah University) as part of New York Ci
ty Category Theory Seminar\n\n\nAbstract\nThe objective of this talk is to
reformulate the logic-based artificial intelligence algorithms and exampl
es from the text of Russell and Norvig using the syntax and categorical se
mantics of Johnstone’s Sketches of an Elephant in order to: (1) identify
the fragments of first-order logic required\; (2) enable symbolic reasoni
ng about richly-structured semantic objects (e.g.\, graphs\, dynamic syste
ms and objects in categories other than Set)\; (3) clarify the separation
between syntax and semantics\; and (4) support use of other category-theor
etic infrastructure such as Morita equivalence and transformations between
theories and sketches.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emilio Minichiello (CUNY Graduate Center)
DTSTART:20220217T000000Z
DTEND:20220217T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/28
DESCRIPTION:Title: Category Theory ∩ Differential Geometry.\nby Emilio
Minichiello (CUNY Graduate Center) as part of New York City Category Theor
y Seminar\n\n\nAbstract\nIn this talk we will take a tour through some are
as of math at the intersection of category theory and differential geometr
y. We will talk about how the use of category theory works towards solving
2 problems: 1) to give rigorous definitions and techniques to study incre
asingly complicated objects in differential geometry that are coming from
physics\, like orbifolds and bundle gerbes\, and 2) to find good categorie
s in which to embed the category of finite dimensional smooth manifolds\,
without losing too much geometric intuition. This involves the study of Li
e groupoids\, sheaves\, diffeological spaces\, stacks\, and infinity stack
s. I will try to motivate the use of these mathematical objects and how th
ey help mathematicians understand differential geometry and expand its sco
pe.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jens Hemelaer (University of Antwerp)
DTSTART:20211209T000000Z
DTEND:20211209T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/29
DESCRIPTION:by Jens Hemelaer (University of Antwerp) as part of New York C
ity Category Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Roberts
DTSTART:20220224T000000Z
DTEND:20220224T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/30
DESCRIPTION:Title: Do you have what it takes to use the diagonal argument?\nby David Roberts as part of New York City Category Theory Seminar\n\n\n
Abstract\nLawvere's reformulation of the diagonal argument captured many i
nstances from the literature in an elegant and abstract category-theoretic
treatment. The original version used cartesian closed categories\, but ga
ve a nod to how the statement of the argument could be adjusted so as to m
ake fewer demands on the category. In fact the argument\, and the fixed-po
int theorem that Lawvere provided as the positive version of the argument\
, both require much less than Lawvere stated. This talk will give an outli
ne of Lawvere's version of the diagonal argument\, his corresponding fixed
-point theorem\, and then cover a few versions obtained recently that drop
assumptions on the properties/structure of the category at hand.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Morgan Rogers (Universit`a degli Studi dell’Insubria.)
DTSTART:20220330T230000Z
DTEND:20220331T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/32
DESCRIPTION:Title: Toposes of Topological Monoid Actions.\nby Morgan Roge
rs (Universit`a degli Studi dell’Insubria.) as part of New York City Cat
egory Theory Seminar\n\n\nAbstract\nAnyone encountering topos theory for t
he first time will be familiar with the fact that the category of actions
of a monoid on sets is a special case of a presheaf topos. It turns out th
at if we equip the monoid with a topology and consider the subcategory of
continuous actions\, the result is still a Grothendieck topos. It is possi
ble to characterize such toposes in terms of their points\, and along the
way extract canonical representing topological monoids\, the complete mono
ids. I'll sketch the trajectory of this story\, present some positive and
negative results about Morita-equivalence of topological monoids\, and exp
lain how one can extract a semi-Galois theory from this set-up.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jin-Cheng Guu (Stony Brook University)
DTSTART:20220316T230000Z
DTEND:20220317T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/33
DESCRIPTION:Title: Topological Quantum Field Theories from Monoidal Categorie
s\nby Jin-Cheng Guu (Stony Brook University) as part of New York City
Category Theory Seminar\n\n\nAbstract\nAbstract: We will introduce the not
ion of a topological quantum field theory (tqft) and a monoidal category.
We will then construct a few (extended) tqfts from monoidal categories\, a
nd show how quantum invariants of knots and 3-manifolds were obtained. If
time permits\, I will discuss (higher) values in (higher) codimensions bas
ed on my recent work on categorical center of higher genera (joint with A.
Kirillov and Y. H. Tham).\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph Dimos
DTSTART:20220323T230000Z
DTEND:20220324T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/34
DESCRIPTION:Title: Introduction to Fusion Categories and Some Applications.\nby Joseph Dimos as part of New York City Category Theory Seminar\n\n\n
Abstract\nAbstract: Tensor categories and multi-tensor categories have str
ong alignment with module categories. We can use the multi-tensor categori
es C in conjunction with classifying tensor algebras wrt C. From here\, we
can illustrate some examples of tensor categories: the category Vec of k-
vector spaces that gives us a fusion category. This is defined as a catego
ry Rep(G) of some finite dimensional k-representations of a group G. From
here\, I will walk through the correspondence of tensor categories (Etingo
f) and fusion categories. Throughout\, I will indicate a few unitary and n
on-unitary cases of fusion categories. Those unitary fusion categories are
those that admit a uniquely monoidal structure. For example\, this draws
upon [Jones 1983] for finite index and finite depth that bridges a subfact
or A-bimodule B to provide a full subcategory of some category A by its mo
dule structure. I will discuss some of these components throughout and exp
lain the Morita equivalence of fusion categories.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jason Parker (Brandon University in Manitoba.)
DTSTART:20220406T230000Z
DTEND:20220407T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/35
DESCRIPTION:Title: Enriched structure-semantics adjunctions and monad-theory
equivalences for subcategories of arities.\nby Jason Parker (Brandon U
niversity in Manitoba.) as part of New York City Category Theory Seminar\n
\n\nAbstract\nSeveral structure-semantics adjunctions and monad-theory equ
ivalences have been established in category theory. Lawvere (1963) develop
ed a structure-semantics adjunction between Lawvere theories and tractable
Set-valued functors\, which was subsequently generalized by Linton (1969)
\, while Dubuc (1970) established a structure-semantics adjunction between
V-theories and tractable V-valued V-functors for a symmetric monoidal clo
sed category V. It is also well known (and due to Linton) that there is an
equivalence between Lawvere theories and finitary monads on Set. Generali
zing this result\, Lucyshyn-Wright (2016) established a monad-theory equiv
alence for eleutheric systems of arities in arbitrary closed categories. B
uilding on earlier work by Nishizawa and Power\, Bourke and Garner (2019)
subsequently proved a general monad-theory equivalence for arbitrary small
subcategories of arities in locally presentable enriched categories. Howe
ver\, neither of these equivalences generalizes the other\, and there has
not yet been a general treatment of enriched structure-semantics adjunctio
ns that specializes to those established by Lawvere\, Linton\, and Dubuc.\
n\nMotivated by these considerations\, we develop a general axiomatic fram
ework for studying enriched structure-semantics adjunctions and monad-theo
ry equivalences for subcategories of arities\, which generalizes all of th
e aforementioned results and also provides substantial new examples of rel
evance for topology and differential geometry. For a subcategory of aritie
s J in a V-category C over a symmetric monoidal closed category V\, Linton
’s notion of clone generalizes to provide enriched notions of J-theory a
nd J-pretheory\, which were also employed by Bourke and Garner (2019). We
say that J is amenable if every J-theory admits free algebras\, and is str
ongly amenable if every J-pretheory admits free algebras. If J is amenable
\, then we obtain an idempotent structure-semantics adjunction between cer
tain J-pretheories and J-tractable V-categories over C\, which yields an e
quivalence between J-theories and J-nervous V-monads on C. If J is strongl
y amenable\, then we also obtain a rich theory of presentations for J-theo
ries and J-nervous V-monads. We show that many previously studied subcateg
ories of arities are (strongly) amenable\, from which we recover the afore
mentioned structure-semantics adjunctions and monad-theory equivalences. W
e conclude with the result that any small subcategory of arities in a loca
lly bounded closed category is strongly amenable\, from which we obtain st
ructure-semantics adjunctions and monad-theory equivalences in (e.g.) many
convenient categories of spaces.\n\nJoint work with Rory Lucyshyn-Wright.
\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Martsinkovsky (Northeastern University)
DTSTART:20220413T230000Z
DTEND:20220414T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/36
DESCRIPTION:Title: A Reflector in Search of a Category.\nby Alex Martsink
ovsky (Northeastern University) as part of New York City Category Theory S
eminar\n\n\nAbstract\nThe last several months have seen an explosive growt
h of activities centered around the defect of a finitely presented functor
. This notion made its first appearance in M. Auslander's fundamental work
on coherent functors in the mid-1960s\, although at that time it was most
ly used just as a technical tool. A phenomenological study of that concept
was initiated by Jeremy Russell in 2016. What transpired in the recent mo
nths is the ubiquitous nature of the defect\, explained in part by the fac
t that it is adjoint to the Yoneda embedding. Thus any branch of mathemati
cs\, computer science\, physics\, or any applied science that references t
he Yoneda embedding automatically becomes a candidate for applications of
the defect.\n\nIn this expository talk I will first give a streamlined int
roduction to the original notion of defect of a finitely presented functor
defined on a module category and then show how to generalize it to arbitr
ary additive functors. Along the way I will give a dozen or so examples il
lustrating various use cases for the defect. The ultimate goal of this lec
ture is to provide a background for the upcoming talk of Alex Sorokin\, wh
o will report on his vast generalization of the defect to arbitrary profun
ctors enriched in a cosmos.\n\nThis presentation is based on joint work in
progress with Jeremy Russell.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Sorokin (Northeastern University)
DTSTART:20220427T230000Z
DTEND:20220428T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/37
DESCRIPTION:Title: The defect of a profunctor.\nby Alex Sorokin (Northeas
tern University) as part of New York City Category Theory Seminar\n\n\nAbs
tract\nIn the mid 1960s Auslander introduced a notion of the defect of a f
initely presented functor on a module category. In 2021 Martsinkovsky gene
ralized it to arbitrary additive functors. In this talk I will show how to
define a defect of any enriched functor with a codomain a cosmos. Under m
ild assumptions\, the covariant (contravariant) defect functor turns out t
o be a left covariant (right contravariant) adjoint to the covariant (cont
ravariant) Yoneda embedding. Both defects can be defined for any profuncto
r enriched in a cosmos V. They happen to be adjoints to the embeddings of
V-Cat in V-Prof. Moreover\, the Isbell duals of a profunctor are completel
y determined by the profunctor's covariant and contravariant defects. Thes
e results are based on applications of the Tensor-Hom-Cotensor adjunctions
and the (co)end calculus.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gershom Bazerman (Arista Networks.)
DTSTART:20220504T230000Z
DTEND:20220505T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/38
DESCRIPTION:Title: Classes of Closed Monoidal Functors which Admit Infinite T
raversals.\nby Gershom Bazerman (Arista Networks.) as part of New York
City Category Theory Seminar\n\n\nAbstract\nIn functional programming\, f
unctors that are equipped with a traverse\noperation can be thought of as
data structures which permit an\nin-order traversal of their elements. Thi
s has been made precise by\nthe correspondence between traversable functor
s and finitary\ncontainers (aka polynomial functors). This correspondence
was\nestablished in the context of total\, necessarily terminating\,\nfunc
tions. However\, the Haskell language is non-strict and permits\nfunctions
that do not terminate. It has long been observed that\ntraversals can at
times\, in practice\, operate over infinite lists\, for\nexample in distri
buting the Reader applicative. We present work in\nprogress that character
izes when this situation occurs\, making use of\nthe toolkit of guarded re
cursion.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergei Burkin (University of Tokyo)
DTSTART:20220907T230000Z
DTEND:20220908T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/39
DESCRIPTION:Title: Segal conditions and twisted arrow categories of operads\nby Sergei Burkin (University of Tokyo) as part of New York City Catego
ry Theory Seminar\n\n\nAbstract\nSeveral categories\, including the simple
x category Delta and Moerdijk-Weiss dendroidal category Omega\, allow to e
ncode structures (in this case categories and operads reprectively) as Seg
al presheaves. There are other examples of such categories\, which were de
fined intuitively\, by analogy with Delta. We will describe a general cons
truction of categories from operads that produces categories that admit Se
gal presheaves. This construction explains why these categories appear in
homotopy theory\, why these allow to encode homotopy coherent structures a
s simplicial presheaves that satisfy weak Segal condition. Further general
ization of this construction to clones shows that these categories are not
as canonical as one might have hoped.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Prakash Panangaden (McGill University)
DTSTART:20220914T230000Z
DTEND:20220915T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/40
DESCRIPTION:Title: Quantitative Equational Logic\nby Prakash Panangaden (
McGill University) as part of New York City Category Theory Seminar\n\nAbs
tract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Ellerman (University of Ljubljana)
DTSTART:20221019T230000Z
DTEND:20221020T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/41
DESCRIPTION:Title: To Interpret Quantum Mechanics:``Follow the Math'': The ma
th of QM as the linearization of the math of partitions\nby David Elle
rman (University of Ljubljana) as part of New York City Category Theory Se
minar\n\n\nAbstract\nAbstract: Set partitions are dual to subsets\, so the
re is a logic of partitions dual to the Boolean logic of subsets. Partitio
ns are the mathematical tool to describe definiteness and indefiniteness\,
distinctions and distinctions\, as well as distinguishability and indisti
nguishability. There is a semi-algorithmic process or ``Yoga'' of lineariz
ation to transform the concepts of partition math into the corresponding v
ector space concepts. Then it is seen that those vector space concepts\, p
articularly in Hilbert spaces\, are the mathematical framework of quantum
mechanics. (QM). This shows that those concepts\, e.g.\, distinguishabilit
y versus indistinguishability\, are the central organizing concepts in QM
to describe an underlying reality of objective indefiniteness--as opposed
to the classical physics and common sense view of reality as ``definite al
l the way down'' This approach thus supports what Abner Shimony called the
``Literal Interpretation'' of QM which interprets the formalism literally
as describing objective indefiniteness and objective probabilities--as we
ll as being complete in contrast to the other realistic interpretations su
ch as the Bohmian\, spontaneous localization\, and many world interpretati
ons which embody other variables\, other equations\, or other worldly idea
s.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrei Rodin (University of Lorraine (Nancy\, France))
DTSTART:20221110T000000Z
DTEND:20221110T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/42
DESCRIPTION:Title: Kolmogorov's Calculus of Problems and Homotopy Type theory
\nby Andrei Rodin (University of Lorraine (Nancy\, France)) as part of
New York City Category Theory Seminar\n\n\nAbstract\nA. N. Kolmogorov in
1932 proposed an original version of mathematical intuitionism where the c
oncept of problem plays a central role\, and which differs in its content
from the versions of intuitionism developed by A. Heyting and other follow
ers of L. Brouwer. The popular BHK-semantics of Intuitionistic logic follo
ws Heyting's line and conceals the original features of Kolmogorov's logic
al ideas. Homotopy Type theory (HoTT) implies a formal distinction between
sentences and higher-order constructions and thus provides a mathematical
argument in favour of Kolmogorov's approach and against Heyting's approac
h. At the same time HoTT does not support the constructive notion of negat
ion applicable to general problems\, which is informally discussed by Kolm
ogorov in the same context. Formalisation of Kolmogorov-style constructive
negation remains an interesting open problem.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Saeed Salehi (University of Tabriz)
DTSTART:20221124T000000Z
DTEND:20221124T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/43
DESCRIPTION:Title: Self-Reference and Diagonalization: their difference and a
short history.\nby Saeed Salehi (University of Tabriz) as part of New
York City Category Theory Seminar\n\n\nAbstract\nWhat is now called the D
iagonal (or the Self-Reference) Lemma\, is the statement that for every f
ormula F(x)
\, with the only free variable x\, there exists a sentence &sigm
a\; such that &sigma\; is equivalent to the F of the Gö\;del code of &sigma\;\, i.e.\, &sigma\;
&equiv\; F(#&sigma\;)\; and this equivalence is provable in certai
n weak arithmetics. This lemma is credited to Gö\;del (1931)\, in the special case when F i
s the unprovability predicate\, and to Carnap (1934) in the more general case.\n
\nIn th
is talk\, we will argue that Gö
\;del-Carnap's origin
al Diagonal Lemma is not the modern formulation and was more similar to\,
but not exactly identical with\, the Strong Diagonal (or Direct Self-Refer
ence) Lemma. This lemma\, so-called recently\, says that for every formula
F(x)\, in
a sufficiently expressive language\, there exists a sentence &sigma\; such that &sigma\; is equal to the F of the Gö\;del code of &sigma\;\, i.e.\, &sigma\; =<
/b> F(#&
sigma\;)\; and this equality is provable in sufficiently strong theori
es. We will attempt at tracking down the first appearance of the modern fo
rmulation of the Diagonal Lemma in the equivalent form\, also in the stron
g direct form of equality.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Pare (Dalhousie University)
DTSTART:20221208T000000Z
DTEND:20221208T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/45
DESCRIPTION:Title: The horizontal/vertical synergy of double categories\n
by Robert Pare (Dalhousie University) as part of New York City Category Th
eory Seminar\n\n\nAbstract\nA double category is a category with two types
of arrows\, horizontal and vertical\, related by double cells. Think of s
ets with functions and relations as arrows and implications as double cell
s. The theory is 2-dimensional just like for 2-categories. In fact 2-categ
ories were originally defined as double categories in which all vertical a
rrows were identities. Most of the theory of 2-categories extends to doubl
e categories resulting in a deeper understanding. This is one aspect of do
uble categories: they’re “new and improved” 2-categories.\n\nFrom a
purely formal point of view\, a double category is a category object in CA
T. Once a familiarity with double categories has developed\, it is amusing
and instructive to see how the various constructs of formal category theo
ry play out in this setting.\n\nBut these two aspects of double categories
\, fancy 2-categories or internal categories\, are only part of the pictur
e. Perhaps the most important thing is the interplay between the horizonta
l and the vertical.\n\nI will start with some examples of double categorie
s to give a feeling for the objects I will be discussing\, and then look a
t several concepts indicative of the rich interplay between the horizontal
and the vertical.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Torre
DTSTART:20220928T230000Z
DTEND:20220929T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/46
DESCRIPTION:Title: Diagonalization\, and the Limits of Limitative Theorems\nby James Torre as part of New York City Category Theory Seminar\n\nAbst
ract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Astra Kolomatskaia (Stony Brook)
DTSTART:20221102T230000Z
DTEND:20221103T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/47
DESCRIPTION:Title: The Objective Metatheory of Simply Typed Lambda Calculus\nby Astra Kolomatskaia (Stony Brook) as part of New York City Category
Theory Seminar\n\n\nAbstract\nLambda calculus is the language of functions
. One reduces the application of a function to an argument by substituting
the argument for the function's formal parameter inside of the function's
body. The result of such a reduction may have further instances of functi
on application. We can write down expressions\, such as ((λ f. f f) (λ f
. f f))\, in which this process does not terminate. In the presence of typ
es\, however\, one has a normalisation theorem\, which effectively states
that "programs can be run". One proof of this theorem\, which only works f
or the most elementary of type theories\, is to assign some monotone well-
founded invariant to a given reduction algorithm. A much more surprising p
roof proceeds by constructing the normal form of a term by structural recu
rsion on the term's syntactic representation\, without ever performing red
uction. Such normalisation algorithms fall under the class of Normalisatio
n by Evaluation. Since the accidental discovery of the first such algorith
m\, it was clear that NbE had some underlying categorical content\, and\,
in 1995\, Altenkirch\, Hofmann\, and Streicher published the first categor
ical normalisation proof. Discovering this content requires first asking t
he question “What is STLC?”\, perhaps preceded by the question “What
is a type theory?”. In this talk we will lay out the details of Altenki
rch's seminal paper and explore conceptual refinements discovered in the p
rocess of its formalisation in Cubical Agda.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ross Street (Macquarie University)
DTSTART:20221026T230000Z
DTEND:20221027T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/48
DESCRIPTION:Title: The core groupoid can suffice\nby Ross Street (Macquar
ie University) as part of New York City Category Theory Seminar\n\n\nAbstr
act\nAbstract: Let V be the monoidal category of modules over a commuative
ring R. I am interested in categories A for which there is a groupoid G s
uch that the functor categories [A\,V] and [G\,V] are equivalent. In parti
cular\, G could be the core groupoid of A\; that is\, the subcategory with
the same objects and with only the invertible morphisms. Every category A
can be regarded as a V-category (that is\, an R-linear category)\, denote
d RA\, with the same objects and with hom R-module RA(a\,b) free on the ho
mset A(a\,b). Indeed\, RA is the free V-category on A so that the V-functo
r category [RA\,V] is the ordinary functor category [A\,V] with the pointw
ise R-linear structure. In these terms\, we are interested in when RA and
RG are Morita equivalent V-categories. In my joint work with Steve Lack on
Dold-Kan-type equivalences\, we had many examples of this phenomenon. How
ever\, the example of Nick Kuhn\, where A is the category of finite vector
spaces over a fixed finite field F with all F-linear functions and G is t
he general linear groupoid over F\, does not fit our theory. Yet the ``ker
nel'' of the equivalence is of the same type. The present work shows that
the category theory behind the Kuhn result also covers our Dold-Kan-type s
etting. I plan to start with a baby example which highlights the ideas.\n\
nI am grateful to Nick Kuhn and Ben Steinberg for their patient email corr
espondence with me on this topic.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Baković (University of Osijek\, Croatia.)
DTSTART:20230202T000000Z
DTEND:20230202T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/49
DESCRIPTION:Title: Enhanced 2-adjunctions.\nby Igor Baković (University
of Osijek\, Croatia.) as part of New York City Category Theory Seminar\n\n
\nAbstract\nWhenever one has a class of objects possessing certain structu
re and a hierarchy of morphisms that preserve structure more or less tight
ly\, we are in an enhanced context. Enhanced 2-categories were introduced
by Lack and Shulman in 2012 with a paradigmatic example of an enhanced 2-c
ategory T-alg of strict algebras for a 2-monad and whose tight and loose 1
-cells are pseudo- and lax morphisms of algebras\, respectively. They can
be defined in two equivalent ways: either as 2-functors\, which are the id
entity on objects\, faithful\, and locally fully faithful\, or as categori
es enriched over the cartesian closed category F\, whose objects are funct
ors that are fully faithful and injective on objects. Lack and Shulman cal
led objects of F full embeddings\, but we will call them "enhanced categor
ies" because they are nothing else but categories with a distinguished cla
ss of objects\, which we call tight.The 2-category F has a much richer str
ucture besides being cartesian closed\; there are additional closed (but n
ot monoidal) structures\, and we show how 2-categories with a right ideal
of 1-cells as in 2-categories with Yoneda structure on them can be present
ed as categories enriched in F in the sense of Eilenberg and Kelly. Since
Lack and Shulman were mainly motivated by limits in enhanced 2-categories\
, they didn't further develop the theory of enhanced (co)lax functors and
their enhanced lax adjunctions. The purpose of this talk is to lay the fou
ndations of the theory of enhanced 2-adjunctions and give their examples t
hroughout mathematics and theoretical computer science.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikhail Khovanov (Columbia University.)
DTSTART:20230209T000000Z
DTEND:20230209T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/50
DESCRIPTION:Title: Universal construction and its applications.\nby Mikha
il Khovanov (Columbia University.) as part of New York City Category Theor
y Seminar\n\n\nAbstract\nUniversal construction starts with an evaluation
of closed n-manifolds and builds a topological theory (a lax TQFT) for n-c
obordisms. A version of it has been used for years as an intermediate step
in constructing link homology theories\, by evaluating foams embedded in
3-space. More recently\, universal construction in low dimensions has been
used to find interesting structures related to Deligne categories\, forma
l languages and automata. In the talk we will describe the universal const
ruction and review these developments.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mee Seong Im (United States Naval Academy\, Annapolis)
DTSTART:20230216T000000Z
DTEND:20230216T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/51
DESCRIPTION:Title: Automata and topological theories.\nby Mee Seong Im (U
nited States Naval Academy\, Annapolis) as part of New York City Category
Theory Seminar\n\n\nAbstract\nTheory of regular languages and finite state
automata is part of the foundations of computer science. Topological quan
tum field theories (TQFT) are a key structure in modern mathematical physi
cs. We will interpret a nondeterministic automaton as a Boolean-valued one
-dimensional TQFT with defects labelled by letters of the alphabet for the
automaton. We will also describe how a pair of a regular language and a c
ircular regular language gives rise to a lax one-dimensional TQFT.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joshua Sussan (CUNY)
DTSTART:20230223T000000Z
DTEND:20230223T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/52
DESCRIPTION:Title: Non-semisimple Hermitian TQFTs.\nby Joshua Sussan (CUN
Y) as part of New York City Category Theory Seminar\n\n\nAbstract\nTopolog
ical quantum field theories coming from semisimple categories build upon i
nteresting structures in representation theory and have important applicat
ions in low dimensional topology and physics. The construction of non-semi
simple TQFTs is more recent and they shed new light on questions that seem
to be inaccessible using their semisimple relatives. In order to have pot
ential applications to physics\, these non-semisimple categories and TQFTs
should possess Hermitian structures. We will define these structures and
give some applications.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jens Hemelaer (University of Antwerp.)
DTSTART:20230315T230000Z
DTEND:20230316T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/53
DESCRIPTION:Title: EILC toposes.\nby Jens Hemelaer (University of Antwerp
.) as part of New York City Category Theory Seminar\n\n\nAbstract\nIn topo
s theory\, local connectedness of a geometric morphism is a very geometric
property\, in the sense that it is stable under base change\, can be chec
ked locally\, and so on. In some situations however\, the weaker property
of being essential is easier to verify. In this talk\, we will discuss EIL
C toposes: toposes E such that any essential geometric morphism with codom
ain E is automatically locally connected. It turns out that many toposes o
f interest are EILC\, including toposes of sheaves on Hausdorff spaces and
classifying toposes of compact groups.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jim Otto
DTSTART:20230329T230000Z
DTEND:20230330T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/54
DESCRIPTION:Title: P Time\, A Bounded Numeric Arrow Category\, and Entailment
s.\nby Jim Otto as part of New York City Category Theory Seminar\n\n\n
Abstract\nWe revisit the characterization of the P Time functions from our
McGill thesis.\n\n1. We build on work of L. Roman (89) on primitive recur
sion and of A. Cobham (65) and Bellantoni-Cook(92) on P Time.\n\n2. We use
base 2 numbers with the digits 1 & 2. Let N be the set of these numbers.
We split the tapes of a multi-tape Turing machine each into 2 stacks of di
gits 1 & 2. These are (modulo allowing an odd numberof stacks) the multi-s
tack machines we use to study P Time.\n\n3. Let Num be the category with o
bjects the finite products of N and arrows the functions between these. Fr
om its arrow category Num^2 we abstract the doctrine (here a category of s
mall categories with chosen structure) PTime of categories with with finit
e products\, base 2 numbers\, 2-comprehensions\, flat recursion\, & safe r
ecursion. Since PTime is a locally finitely presentable category\, it has
an initial category I. Our characterization is that the bottom of the imag
e of I in Num^2 consists of the P Time functions.\n\n4. We can use I (thin
king of its arrows as programs) to run multi-stack machines long enough to
get P Time.This is the completeness of the characterization.\n\n5. We cut
down the numeric arrow category Num^2\, using Bellantoni-Cook growth & ti
me bounds on the functions\, to get a bounded numeric arrow category B. B
is in the doctrine PTime. This yields the soundness of the characterizatio
n.\n\n6. For example\, the doctrine of toposes with base 1 numbers\, choic
e\, & precisely 2 truth values (which captures much of ZC set theory) like
ly lacks an initial category\, much as there is an initial ring\, but no i
nitial field.\n\n7. On the other hand\, the L. Roman doctrine PR of catego
ries with finite products\, base 1 numbers\, & recursion (that is\, produc
t stable natural numbers objects) does have an initial category as it cons
ists of the strong models of a finite set of entailments. And is thus loca
lly finitely presentable. We sketch the signature graph for these entailme
nts. And some of these entailments. Similarly (but with more complexity) t
here are entaiments for the doctrine PTime.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Walter Tholen (York University)
DTSTART:20230419T230000Z
DTEND:20230420T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/55
DESCRIPTION:Title: What does “smallness” mean in categories of topologica
l spaces?\nby Walter Tholen (York University) as part of New York City
Category Theory Seminar\n\n\nAbstract\nQuillen’s notion of small object
and the Gabriel-Ulmer notion of finitely presentable or generated object
are fundamental in homotopy theory and categorical algebra. Do these notio
ns always lead to rather uninteresting classes of objects in categories of
topological spaces\, such as the class of finite discrete spaces\, or jus
t the empty space \, as the examples and remarks in the existing literatur
e may suggest?\n\nIn this talk we will demonstrate that the establishment
of full characterizations of these notions (and some natural variations th
ereof) in many familiar categories of spaces\, such as those of T_i-spaces
(i= 0\, 1\, 2)\, can be quite challenging and may lead to unexpected surp
rises. In fact\, we will show that there are significant differences in th
is regard even amongst the categories defined by the standard separation c
onditions\, with the T1-separation condition standing out. The findings ab
out these specific categories lead us to insights also when considering ra
ther arbitrary full reflective subcategories of Top.\n\n(Based on joint wo
rk with J. Adamek\, M. Husek\, and J. Rosicky.)\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dusko Pavlovic (University of Hawai‘i at Mānoa)
DTSTART:20230426T230000Z
DTEND:20230427T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/56
DESCRIPTION:Title: Program-closed categories.\nby Dusko Pavlovic (Univers
ity of Hawai‘i at Mānoa) as part of New York City Category Theory Semin
ar\n\n\nAbstract\nLet CC be a symmetric monoidal category with a comonoid
on every object. Let CC* be the cartesian subcategory with the same object
s and just the comonoid homomorphisms. A *programming language* is a well-
ordered object P with a *program closure*: a family of X-natural surjectio
ns\n\nCC(XA\,B) <<--run_X-- CC*(X\,P)\n\none for every pair A\,B. In this
talk\, I will sketch a proof that program closure is a property: Any two p
rogramming languages are isomorphic along run-preserving morphisms. The re
sult counters Kleene's interpretation of the Church-Turing Thesis\, which
has been formalized categorically as the suggestion that computability is
a structure\, like a group presentation\, and not a property\, like comple
teness. We prove that it is like completeness. The draft of a book on cate
gorical computability is available from the web site dusko.org.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arthur Parzygnat (Nagoya University.)
DTSTART:20230517T230000Z
DTEND:20230518T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/58
DESCRIPTION:Title: Inferring the past and using category theory to define ret
rodiction.\nby Arthur Parzygnat (Nagoya University.) as part of New Yo
rk City Category Theory Seminar\n\n\nAbstract\nClassical retrodiction is t
he act of inferring the past based on knowledge of the present. The primar
y example is given by Bayes' rule P(y|x) P(x) = P(x|y) P(y)\, where we use
prior information\, conditional probabilities\, and new evidence to updat
e our belief of the state of some system. The question of how to extend th
is idea to quantum systems has been debated for many years. In this talk\,
I will lay down precise axioms for (classical and quantum) retrodiction u
sing category theory. Among a variety of proposals for quantum retrodictio
n used in settings such as thermodynamics and the black hole information p
aradox\, only one satisfies these categorical axioms. Towards the end of m
y talk\, I will state what I believe is the main open question for retrodi
ction\, formalized precisely for the first time. This work is based on the
preprint https://arxiv.org/abs/2210.13531 and is joint work with Francesc
o Buscemi.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tomáš Gonda (University of Innsbruck)
DTSTART:20230927T230000Z
DTEND:20230928T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/59
DESCRIPTION:Title: A Framework for Universality in Physics\, Computer Science
\, and Beyond.\nby Tomáš Gonda (University of Innsbruck) as part of
New York City Category Theory Seminar\n\n\nAbstract\nTuring machines and s
pin models share a notion of universality according to which some simulate
all others. We set up a categorical framework for universality which incl
udes as instances universal Turing machines\, universal spin models\, NP c
ompleteness\, top of a preorder\, denseness of a subset\, and others. By i
dentifying necessary conditions for universality\, we show that universal
spin models cannot be finite. We also characterize when universality can b
e distinguished from a trivial one and use it to show that universal Turin
g machines are non-trivial in this sense. We leverage a Fixed Point Theore
m inspired by a result of Lawvere to establish that universality and negat
ion give rise to unreachability (such as uncomputability). As such\, this
work sets the basis for a unified approach to universality and invites the
study of further examples within the framework.\n\nTALK AT 5PM. Not 7PM.
NOTE SPECIAL TIME!!!\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thiago Alexandre (University of São Paulo (Brazil))
DTSTART:20231011T230000Z
DTEND:20231012T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/60
DESCRIPTION:Title: Internal homotopy theories\nby Thiago Alexandre (Unive
rsity of São Paulo (Brazil)) as part of New York City Category Theory Sem
inar\n\n\nAbstract\nThe idea of 'Homotopy theories' was introduced by Hell
er in his seminal paper from 1988. Two years later\, Grothendieck discover
ed the theory of derivators (1990)\, exposed in his late manuscript Les D
érivateurs\, and developed further by several authors. Essentially\, ther
e are no significant differences between Heller's homotopy theories and Gr
othendieck's derivators. They are tautologically the same 2-categorical yo
ga. However\, they come from distinct motivations. For Heller\, derivators
should be a definitive answer to the question "What is a homotopy theory?
"\, while for Grothendieck\, who was strongly inspired by topos cohomology
\, the first main motivation for derivators was to surpass some technical
deficiencies that appeared in the theory of triangulated categories. Indee
d\, Grothendieck designed the axioms of derivators in light of a certain 2
-functorial construction\, which associates the corresponding (abelian) de
rived category to each topos\, and more importantly\, inverse and direct c
ohomological images to each geometric morphism. It was from this 2-functor
ial construction\, from where topos cohomology arises\, that Grothendieck
discovered the axioms of derivators\, which are surprisingly the same as H
eller's homotopy theories. Nowadays\, it is commonly accepted that a homot
opy theory is a quasi-category\, and they can all be presented by a locali
zer (M\,W)\, i.e.\, a couple composed by a category M and a class of arrow
s in W. This point of view is not so far from Heller\, since pre-derivator
s\, quasi-categories\, and localizers\, are essentially equivalent as an a
nswer to the question "What is a homotopy theory?". In my talk\, I will ex
pose these subjects in more detail\, and I am also going to explore how to
internalize a homotopy theory in an arbitrary (Grothendieck) topos\, a pr
oblem which strongly relates formal logic and homotopical algebra.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Shulman (University of San Diego)
DTSTART:20231018T230000Z
DTEND:20231019T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/61
DESCRIPTION:Title: The derivator of setoids\nby Michael Shulman (Universi
ty of San Diego) as part of New York City Category Theory Seminar\n\n\nAbs
tract\nThe question of "what is a homotopy theory" or "what is a higher ca
tegory" is already interesting in classical mathematics\, but in construct
ive mathematics (such as the internal logic of a topos) it becomes even mo
re subtle. In particular\, existing constructive attempts to formulate a
homotopy theory of spaces (infinity-groupoids) have the curious property t
hat their "0-truncated objects" are more general than ordinary sets\, bein
g instead some kind of "free exact completion" of the category of sets (a.
k.a. "setoids"). It is at present unclear whether this is a necessary fea
ture of a constructive homotopy theory or whether it can be avoided someho
w. One way to find some evidence about this question is to use the "deriv
ators" of Heller\, Franke\, and Grothendieck\, as they give us access to h
igher homotopical structure without depending on a preconcieved notion of
what such a thing should be. It turns out that constructively\, the free
exact completion of the category of sets naturally forms a derivator that
has a universal property analogous to the classical category of sets and t
o the classical homotopy theory of spaces: it is the "free cocompletion of
a point" in a certain universe. This suggests that either setoids are an
unavoidable aspect of constructive homotopy theory\, or more radical modi
fications to the notion of homotopy theory are needed.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emilio Minichiello (CUNY Graduate Center)
DTSTART:20231025T230000Z
DTEND:20231026T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/62
DESCRIPTION:Title: A Mathematical Model of Package Management Systems.\nb
y Emilio Minichiello (CUNY Graduate Center) as part of New York City Categ
ory Theory Seminar\n\n\nAbstract\nAbstract: In this talk\, I will review s
ome recent joint work with Gershom Bazerman and Raymond Puzio. The motivat
ion is simple: provide a mathematical model of package management systems\
, such as the Hackage package respository for Haskell\, or Homebrew for Ma
c users. We introduce Dependency Structures with Choice (DSC) which are se
ts equipped with a collection of possible dependency sets for every elemen
t and satisfying some simple conditions motivated from real life use cases
. We define a notion of morphism of DSCs\, and prove that the resulting ca
tegory of DSCs is equivalent to the category of antimatroids\, which are m
athematical structures found in combinatorics and computer science. We ana
lyze this category\, proving that it is finitely complete\, has coproducts
and an initial object\, but does not have all coequalizers. Further\, we
construct a functor from a category of DSCs equipped with a certain subcla
ss of morphisms to the opposite of the category of finite distributive lat
tices\, making use of a simple finite characterization of the Bruns-Lakser
completion.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Larry Moss (Indiana University\, Bloomington)
DTSTART:20231109T000000Z
DTEND:20231109T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/63
DESCRIPTION:Title: On Kripke\, Vietoris\, and Hausdorff Polynomial Functors.<
/a>\nby Larry Moss (Indiana University\, Bloomington) as part of New York
City Category Theory Seminar\n\n\nAbstract\nThe Vietoris space of compact
subsets of a given Hausdorff space yields an endofunctor V on the category
of Hausdorff spaces. Vietoris polynomial endofunctors on that category ar
e built from V\, the identity and constant functors by forming products\,
coproducts and compositions. These functors are known to have terminal coa
lgebras and we deduce that they also have initial algebras. We present an
analogous class of endofunctors on the category of extended metric spaces\
, using in lieu of V the Hausdorff functor H. We prove that the ensuing Ha
usdorff polynomial functors have terminal coalgebras and initial algebras.
Whereas the canonical constructions of terminal coalgebras for Vietoris p
olynomial functors takes omega steps\, one needs \\omega + \\omega steps i
n general for Hausdorff ones. We also give a new proof that the closed set
functor on metric spaces has no fixed points.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pedro Sota (CANCELLED)
DTSTART:20231123T000000Z
DTEND:20231123T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/64
DESCRIPTION:Title: CANCELLED\nby Pedro Sota (CANCELLED) as part of New Yo
rk City Category Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Charlotte Aten (University of Denver)
DTSTART:20231130T000000Z
DTEND:20231130T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/65
DESCRIPTION:Title: A categorical semantics for neural networks\nby Charlo
tte Aten (University of Denver) as part of New York City Category Theory S
eminar\n\n\nAbstract\nIn recent work on discrete neural networks\, I consi
dered such networks whose activation functions are polymorphisms of finite
\, discrete relational structures. The general framework I provided was no
t entirely categorical in nature but did provide a steppingstone to a cate
gorical treatment of neural nets which are definitionally incapable of ove
rfitting. In this talk I will outline how to view neural nets as categorie
s of functors from certain multicategories to a target multicategory. More
over\, I will show that the results of my PhD thesis allow one to systemat
ically define polymorphic learning algorithms for such neural nets in a ma
nner applicable to any reasonable (read: functorial) finite data structure
.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Juan Orendain (Case Western Univeristy)
DTSTART:20240508T230000Z
DTEND:20240509T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/66
DESCRIPTION:Title: Canonical squares in fully faithful and absolutely dense e
quipments.\nby Juan Orendain (Case Western Univeristy) as part of New
York City Category Theory Seminar\n\n\nAbstract\nZOOM TALK. \nAbstract: Eq
uipments are categorical structures of dimension 2 having two separate typ
es of 1-arrows -vertical and horizontal- and supporting restriction and ex
tension of horizontal arrows along vertical ones. Equipments were defined
by Wood in [W] as 2-functors satisfying certain conditions\, but can also
be understood as double categories satisfying a fibrancy condition as in [
Sh]. In the zoo of 2-dimensional categorical structures\, equipments nicel
y fit in between 2-categories and double categories\, and are generally co
nsidered as the 2-dimensional categorical structures where synthetic categ
ory theory is done\, and in some cases\, where monoidal bicategories are m
ore naturally defined.\n\nIn a previous talk in the seminar\, I discussed
the problem of lifting a 2-category into a double category along a given c
ategory of vertical arrows\, and how this problem allows us to define a no
tion of length on double categories. The length of a double category is a
number that roughly measures the amount of work one needs to do to reconst
ruct the double category from a bicategory along its set of vertical arrow
s.\n\nIn this talk I will review the length of double categories\, and I w
ill discuss two recent developments in the theory: In the paper [OM] a met
hod for constructing different double categories from a given bicategory i
s presented. I will explain how this construction works. One of the main i
ngredients of the construction are so-called canonical squares. In the pre
print [O] it is proven that in certain classes of equipments -fully faithf
ul and absolutely dense- every square that can be canonical is indeed cano
nical. I will explain how from this\, it can be concluded that fully faith
ful and absolutely dense equipments are of length 1\, and so they can be '
easily' reconstructed from their horizontal bicategories.\n\nReferences:\n
[O] Length of fully faithful framed bicategories. arXiv:2402.16296.\n\n[OM
] J. Orendain\, R. Maldonado-Herrera\, Internalizations of decorated bicat
egories via π-indexings. To appear in Applied Categorical Structures. arX
iv:2310.18673.\n\n[W] R. K. Wood\, Abstract Proarrows I\, Cahiers de topol
ogie et géométrie différentielle 23 3 (1982) 279-290.\n\n[Sh] M. Shulma
n\, Framed bicategories and monoidal fibrations. Theory and Applications o
f Categories\, Vol. 20\, No. 18\, 2008\, pp. 650–738.\n\nZoom Talk.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raymond Puzio
DTSTART:20240515T230000Z
DTEND:20240516T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/67
DESCRIPTION:Title: Uniqueness of Classical Retrodiction\nby Raymond Puzio
as part of New York City Category Theory Seminar\n\n\nAbstract\nIN PERSON
TALK.\nAbstract: In previous talks at this Category seminar and at the To
pology\, Geometry and Physics seminar\, Arthur Parzygnat showed how Bayesi
an inversion and its generalization to quantum mechanics may be interprete
d as a functor on a suitable category of states which satisfies certain ax
ioms. Such a functor is called a retrodiction and Parzygnat and collaborat
ors conjectured that retrodiction is unique. In this talk\, I will present
a proof of this conjecture for the special case of classical probability
theory on finite state spaces.\n\nIn this special case\, the category in q
uestion has non-degenerate probability distributions on finite sets as its
objects and stochastic matrices as its morphisms. After preliminary defin
itions and lemmas\, the proof proceeds in three main steps.\n\nIn the firs
t step\, we focus on certain groups of automorphisms of certain objects. A
s a consequence of the axioms\, it follows that these groups are preserved
under any retrodiction functor and that the restriction of the functor to
such a group is a certain kind of group automorphism. Since this group is
isomorphic to a Lie group\, it is easy to prove that the restriction of a
retrodiction to such a group must equal Bayesian inversion if we assume c
ontinuity. If we do not make that assumption\, we need to work harder and
derive continuity "from scratch" starting from the positivity condition in
the definition of stochastic matrix.\n\nIn the second step\, we broaden o
ur attention to the full automorphism groups of objects of our category co
rresponding to uniform distributions. We show that these groups are genera
ted by the union of the subgroup consisting of permutation matrices and th
e subgroup considered in the first step. From this fact\, it follows that
the restriction of a retrodiction to this larger group must equal Bayesian
inversion.\n\nIn the third step\, we finally consider all the objects and
morphisms of our category. As a consequence of what we have shown in the
first two steps and some preliminary lemmas\, it follows that retrodiction
is given by matrix conjugation. Furthermore\, Bayesian inversion is the s
pecial case where the conjugating matrices are diagonal matrices. Because
the hom sets of our category are convex polytopes and a retrodiction funct
or is a continuous bijection of such sets\, a retodiction must map polytop
e faces to faces. By an algebraic argument\, this fact implies that the co
njugating matrices are diagonal\, answering the conjecture in the affirmat
ive.\n\nIN PERSON TALK.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emilio Minichiello (The CUNY Graduate Center)
DTSTART:20240522T230000Z
DTEND:20240523T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/68
DESCRIPTION:Title: Presenting Profunctors\nby Emilio Minichiello (The CUN
Y Graduate Center) as part of New York City Category Theory Seminar\n\n\nA
bstract\nIN PERSON TALK.\nAbstract: In categorical database theory\, profu
nctors are ubiquitous. For example\, they are used to define schemas in th
e algebraic data model. However\, they can also be used to query and migra
te data. In this talk\, we will discuss an interesting phenomenon that ari
ses when trying to model profunctors in a computer. We will introduce two
notions of profunctor presentations: the UnCurried and Curried presentatio
ns. They are modeled on thinking of profunctors as functors P: C^op x D ->
Set and as functors P: C^op -> Set^D\, respectively. Semantically of cour
se\, these are equivalent\, but their syntactic properties are quite diffe
rent. The UnCurried presentations are more intuitive and easier to work wi
th\, but they carry a fatal flaw: there does not exist a semantics-preserv
ing composition operation of UnCurried presentations that also preserves f
initeness. Therefore we introduce the Curried presentations and show that
they remedy this flaw. In the process\, we characterize which UnCurried Pr
esentations can be made Curried\, and discuss some applications. This talk
will be based off of this recent preprint which is joint work with Gabrie
l Goren Roig and Joshua Meyers.\n\nIN PERSON TALK.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samuel Mimram (École Polytechnique)
DTSTART:20240529T230000Z
DTEND:20240530T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/69
DESCRIPTION:by Samuel Mimram (École Polytechnique) as part of New York Ci
ty Category Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jake Araujo-Simon (Cornell Tech---In-Person)
DTSTART:20240918T230000Z
DTEND:20240919T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/70
DESCRIPTION:Title: Categorifying the Volterra series: towards a compositional
theory of nonlinear signal processing.\nby Jake Araujo-Simon (Cornell
Tech---In-Person) as part of New York City Category Theory Seminar\n\n\nA
bstract\nAbstract: The Volterra series is a model of nonlinear behavior th
at extends the convolutional representation of linear and time-invariant s
ystems to the nonlinear regime. Though well-known and applied in electrica
l\, mechanical\, biomedical\, and audio engineering\, its abstract and esp
ecially compositional properties have been less studied. In this talk\, we
present an approach to categorifying the Volterra series\, in which a Vol
terra series is defined as a functor on a category of signals and linear m
aps\, a morphism between Volterra series is a lens map and natural transfo
rmation\, and together\, Volterra series and their morphisms assemble into
a category\, which we call Volt. We study three monoidal structures on Vo
lt\, and outline connections of our work to the field of time-frequency an
alysis. We also include an audio demo.\n\nPaper link: https://arxiv.org/ab
s/2308.07229\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam McCrosson (Montana State University---Zoom Talk)
DTSTART:20241009T230000Z
DTEND:20241010T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/71
DESCRIPTION:Title: Exodromy.\nby Sam McCrosson (Montana State University-
--Zoom Talk) as part of New York City Category Theory Seminar\n\n\nAbstrac
t\nAbstract: A favorite result of first semester algebraic topology is the
“monodromy theorem\,” which states that for a suitable topological sp
ace X\, there is a triple equivalence between the categories of covering s
paces of X\, sets with an action from the fundamental group of X\, and loc
ally constant sheaves on X. This result has recently been upgraded by MacP
herson and others to a stratified setting\, where the underlying space may
be carved into a poset of subspaces. In this talk\, we’ll look at the m
ain ingredients of the so-called “exodromy theorem\,” reviewing strati
fied spaces and developing “constructible sheaves” and the “exit-pat
h category” along the way.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bruno Gavranović (Symbolica AI--- Zoom Talk - Special Time)
DTSTART:20241030T180000Z
DTEND:20241030T200000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/72
DESCRIPTION:Title: Categorical Deep Learning: An Algebraic Theory of Architec
tures---NOTE SPECIAL TIME.\nby Bruno Gavranović (Symbolica AI--- Zoom
Talk - Special Time) as part of New York City Category Theory Seminar\n\n
\nAbstract\nWe present our position on the elusive quest for a general-pur
pose framework for specifying and studying deep learning architectures. Ou
r opinion is that the key attempts made so far lack a coherent bridge betw
een specifying constraints which models must satisfy and specifying their
implementations. Focusing on building such a bridge\, we propose to apply
category theory— precisely\, the universal algebra of monads valued in a
2-category of parametric maps—as a single theory elegantly subsuming bo
th of these flavours of neural network design. To defend our position\, we
show how this theory recovers constraints induced by geometric deep learn
ing\, as well as implementations of many architectures drawn from the dive
rse landscape of neural networks\, such as RNNs. We also illustrate how th
e theory naturally encodes many standard constructs in computer science an
d automata theory.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Jaz Myers (Symbolica AI... 2PM TALK)
DTSTART:20241107T000000Z
DTEND:20241107T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/73
DESCRIPTION:Title: The Para and Kleisli constructions as wreath products.
\nby David Jaz Myers (Symbolica AI... 2PM TALK) as part of New York City C
ategory Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emilio Minichiello (CUNY CityTech.--- In-Person)
DTSTART:20241114T000000Z
DTEND:20241114T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/74
DESCRIPTION:Title: Decision Problems on Graphs with Sheaves. IN PERSON\nb
y Emilio Minichiello (CUNY CityTech.--- In-Person) as part of New York Cit
y Category Theory Seminar\n\n\nAbstract\nThis semester I don’t feel like
talking about my research. Instead I’ll talk about what I’ve learned
from reading the paper "Compositional Algorithms on Compositional Data: De
ciding Sheaves on Presheaves" by Althaus\, Bumpus\, Fairbanks and Rosiak.
This paper is about how we can use sheaf theory to break apart a computati
onal problem\, solve it on small pieces\, and then glue the solutions toge
ther to get a global solution to the computational problem. I’ll go thro
ugh the main ideas of this paper\, using the category of simple graphs wit
h monomorphisms as a main example to showcase their results.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noah Chrein (University of Maryland --- In-Person)
DTSTART:20240925T230000Z
DTEND:20240926T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/75
DESCRIPTION:Title: A formal category theory for oo-T-multicategories.\nby
Noah Chrein (University of Maryland --- In-Person) as part of New York Ci
ty Category Theory Seminar\n\n\nAbstract\nAbstract: We will explore a fram
ework for oo-T-multicategories. To begin\, we build a schema for multicate
gories out of the simplex schema and the monoid schema. The multicategory
schema\, D_m\, inherits the structure of a monad from the +1 monad on the
monoid schema. Simplicial T-multicategories are monad preserving functors
out of the multicategory schema\, [D_m\, T]\, into another monad T. The fr
amework is larger than just [D_m\,T]. A larger structure describes notions
of yoneda lemma and fibration. Inner fibrant\, simplicial T-multicategori
es are oo-T-multicategories. oo-T-multicategories generalize oo-categories
and oo-operads: oo-operads are fm-multicategories\, oo-categories are Id-
multicategories.\n\nWe use this framework to study oo-fc-multicategories\,
or "oo - virtual double categories". In general\, under various assumptio
ns on T (which hold for fc)\, the collection of oo-T-multicategories [D_m\
, T] has other useful structure. One such structure is a join operation. T
his join operation points towards a synthetic definition of op/cartesian c
ells\, which we hope will model oo-virtual equipments. If there is time\,
I will explain the motivation for this study as it relates to ontologies\,
meta-theories and type theories.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tim Hosgood (Topos Institute. ZOOM TALK)
DTSTART:20241127T190000Z
DTEND:20241127T200000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/76
DESCRIPTION:Title: Loose simplicial objects.\nby Tim Hosgood (Topos Insti
tute. ZOOM TALK) as part of New York City Category Theory Seminar\n\n\nAbs
tract\nThere are two stories that are historically reasonably unrelated\,
but that both lead to the same definition of a "loose simplicial object"\,
namely (i) the proof that totalisation of a Reedy fibrant cosimplicial si
mplicial set computes the homotopy limit (via the Bousfield–Kan map)\, a
nd (ii) the construction of global simplicial resolutions of coherent anal
ytic sheaves (via Toledo–Tong twisting cochains). In this talk\, we will
look at both of these stories and see what common definition they suggest
\, and then examine how this definition might be useful.\n\nThis talk is o
n (incomplete) work in progress\, joint with Cheyne Glass.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthew Cushman (CUNY -- IN PERSON TALK)
DTSTART:20241212T000000Z
DTEND:20241212T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/77
DESCRIPTION:Title: Recollements: gluing and fracture for categories.\nby
Matthew Cushman (CUNY -- IN PERSON TALK) as part of New York City Category
Theory Seminar\n\nAbstract: TBA\n\nRecollements provide a way of gluing t
wo categories together along a left-exact functor\, or conversely of obtai
ning a semi-orthogonal decomposition of a category by two full subcategori
es. Every recollement comes with a fracture square\, which in some circums
tances can be extended to a hexagon-shaped diagram of fiber sequences. In
this talk we will discuss concrete examples from topological spaces and gr
aphs before moving to smooth manifolds and the recollement that gives rise
to differential cohomology theories.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Charlotte Aten (University of Colorado Boulder--- ZOOM TALK)
DTSTART:20241205T000000Z
DTEND:20241205T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/78
DESCRIPTION:Title: Invariants of structures.\nby Charlotte Aten (Universi
ty of Colorado Boulder--- ZOOM TALK) as part of New York City Category The
ory Seminar\n\n\nAbstract\nI will discuss one part of my PhD thesis\, in w
hich I provide a categorification of the notion of a mathematical structur
e originally given by Bourbaki in their set theory textbook. The main resu
lt is that any isomorphism-invariant property of a finite structure can be
checked by computing the number of isomorphic copies of small substructur
es it contains. A special case of this theorem is the classical result of
Hilbert about elementary symmetric polynomials generating the algebra of a
ll symmetric polynomials. I will also discuss how the logical complexity o
f a positive formula controls the size of the small substructures one must
count.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arnon Avron (Tel Aviv U -- In Person talk)
DTSTART:20241121T000000Z
DTEND:20241121T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/79
DESCRIPTION:Title: What is the Structure of the Natural numbers?\nby Arno
n Avron (Tel Aviv U -- In Person talk) as part of New York City Category T
heory Seminar\n\n\nAbstract\nWe present some theorems that show that the n
otion of a structure\,\nwhich is central for both Structuralism and catego
ry theory\, has the very serious\ndefect of having no satisfactory notion
of identity which can be associated with it.\nWe use those theorems to sho
w that in particular\, there are at least two completely \ndifferent struc
tures that are entitled to be taken as `the structure of the natural \nnum
bers'\, and any choice between them would arbitrarily favor one of them ov
er\nthe equally legitimate other. This fact refutes (so we believe) the st
ructuralist thesis\nthat the natural numbers are just positions (or places
) in "the structure of the natural numbers". Finally\, we argue for the hi
gh plausibility of the identification of the natural numbers with the fini
te von Neumann ordinals.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raymond Puzio (IN PERSON TALK)
DTSTART:20250206T000000Z
DTEND:20250206T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/80
DESCRIPTION:Title: Gentle Introduction to Synthetic Differential Geometry Par
t 1\nby Raymond Puzio (IN PERSON TALK) as part of New York City Catego
ry Theory Seminar\n\n\nAbstract\nAbstract: Calculations and constructions
with infinitesimals make for a handy\, intuitive way of doing calculus and
differential geometry. They went out of favor in the nineteenth century w
hen the real number system was defined precisely but were rehabilitated a
century later when various people such as Robinson\, Lawvere\, and Kock re
alized that it is nonetheless possible to produce logically rigorous justi
fications for manipulations involving infinitesimals.\n\nOne such justific
ation emerged from scheme theory and category theory and goes by the name
"synthetic differential geometry". This talk will be an elementary pedagog
ical introduction to the subject. We will begin by showing how one can re-
interpret computing with square zero infinitesimals in terms of homomorphi
sms from an algebra of smooth functions to the algebra of dual numbers. Us
ing concepts from scheme theory\, we will correctly interpret the of corre
sponding picture of infinitesimally near points. Moving on\, we will intro
duce the axiomatic approach to synthetic differential geometry and describ
e how passing to the presheaf topos allows one to treat such infinite-dime
nsional entities as the totality of mappings between two given manifolds a
s well-defined spaces. We round off this introduction with a few words abo
ut Lie groups\, making precise the idea that the Lie algebra with its comm
utation relations forms a group presentation in terms of infinitesimal gen
erators and their relations.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jacob Szelko (Northeastern University-IN PERSON TALK)
DTSTART:20250220T000000Z
DTEND:20250220T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/81
DESCRIPTION:Title: An Introduction to Compositional Public Health.\nby Ja
cob Szelko (Northeastern University-IN PERSON TALK) as part of New York Ci
ty Category Theory Seminar\n\n\nAbstract\nCompositional public health is a
n emerging research field that exists to address the complexity in public
health responses. The field lies at the intersection of category theory\,
epidemiology\, and engineering and utilizes tools from applied category th
eory for public health applications. This talk will present the motivation
of this field\, an overview of the mathematics involved in its approaches
\, current state of the art\, live demonstrations\, and future research di
rections within this developing field.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thiago Alexandre (IN PERSON TALK)
DTSTART:20250227T000000Z
DTEND:20250227T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/82
DESCRIPTION:Title: Topological Derivators\nby Thiago Alexandre (IN PERSON
TALK) as part of New York City Category Theory Seminar\n\n\nAbstract\nAbs
tract: The theory of derivators was originally developed by Grothendieck w
ith high inspiration in topos cohomology. In a letter sent to Thomason\, w
here he explains the main ideas and motivations guiding the formal reasoni
ng of derivators\, Grothendieck also remarks that those are Morita-invaria
nt. This means that\, if two small categories A and B have equivalent topo
i of presheaves\, then the categories D(A) and D(B ) are also equivalent f
or any derivator D. This observation suggests that it may be possible to e
xtend any derivator D to the entire 2-category of topoi and geometric morp
hisms between them. Grothendieck conjectures that such an extension is alw
ays possible and essentially unique. In this case\, every derivator D defi
ned over small categories would be coming from a derivator D′ defined ov
er topoi via natural equivalences of categories of the form D(A) = D′(A^
)\, where A varies through small categories and A^ denotes the category of
presheaves over A. However\, despite these considerations\, a theory of d
erivators over topoi has not yet been developed. To address this gap\, I a
m currently developing a theory of topological derivators. With this theor
y\, I aim to provide answers to Grothendieck’s conjecture. Beyond applic
ations in geometry\, the theory of topological derivators has strong conne
ctions to first-order categorical logic. In fact\, it lies in the intersec
tion between the later and homotopical algebra. In my talk\, I would like
to present the theory of topological derivators and some of its main resul
ts.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Grigorios Giotopoulos (NYU Abu Dhabi-ZOOM TALK 10AM-NYCity Time)
DTSTART:20250306T000000Z
DTEND:20250306T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/83
DESCRIPTION:Title: Thickened smooth sets as a natural setting for Lagrangian
field theory.\nby Grigorios Giotopoulos (NYU Abu Dhabi-ZOOM TALK 10AM-
NYCity Time) as part of New York City Category Theory Seminar\n\n\nAbstrac
t\nAbstract: I will describe how a particularly convenient model for synth
etic differential geometry -- the sheaf topos of infinitesimally thickened
smooth sets -- serves as a powerful context to host classical Lagrangian
field theory. As motivation\, I will recall the textbook description of va
riational Lagrangian field theory\, and list desiderata for an ambient cat
egory in which this can rigorously be formalized. I will then explain how
sheaves over infinitesimally thickened Cartesian spaces naturally satisfy
all the desiderata\, and furthermore allow to rigorously formalize several
more field theoretic concepts. Time permitting\, I will indicate how the
setting naturally generalizes to include the description of fermionic fiel
ds\, and (gauge) fields with internal symmetries. This is based on joint w
ork with Hisham Sati and Urs Schreiber.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathon Funk (Queensborough\, CUNY- IN PERSON TALK)
DTSTART:20250312T230000Z
DTEND:20250313T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/84
DESCRIPTION:Title: Toposes and Rings\nby Jonathon Funk (Queensborough\, C
UNY- IN PERSON TALK) as part of New York City Category Theory Seminar\n\n\
nAbstract\nI shall attempt to explain a part of a broader program of how t
opos theory and operator algebra theory match. Following the example of wh
at I call a supported C*-algebra [1]\, such as a von Neumann algebra\, we
extend to an arbitrary ring the notions and constructions introduced there
. (Familiarity with [1] is not necessary for the purposes of this talk.) I
have included an explanation of the Zariski spectrum of a commutative rin
g in terms of the constructions I explain. Ultimately\, our goal is to ret
urn to C*-algebras in order to generalize [1] to all C*-algebras\, not jus
t the supported ones.\n\nThis is joint work with Simon Henry.\n\n[1] J. Fu
nk\, Toposes and C*-algebras\, preprint\, March 2024.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emilio Minichiello (CUNY CityTech - IN PERSON TALK)
DTSTART:20250409T230000Z
DTEND:20250410T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/85
DESCRIPTION:Title: Structured Decomposition Categories.\nby Emilio Minich
iello (CUNY CityTech - IN PERSON TALK) as part of New York City Category T
heory Seminar\n\n\nAbstract\nAbstract: In this talk I’ll report on some
new work\, joint with Ben Bumpus\, Zoltan Kocsis and Jade Master. The idea
here is to come up with a categorical framework to talk about decompositi
ons. In graph theory\, there are all kinds of ways of decomposing graphs\,
the most important being tree decompositions. This is a way to decompose
a graph into pieces in such a way that if you squint at it\, it looks like
a tree. By looking at the biggest piece and minimizing over all tree deco
mpositions\, one obtains treewidth\, the most important graph invariant in
algorithmics. In this paper\, we abstract this notion\, coming up with th
e definition of structured decomposition categories. To each such category
\, we can assign to each of its objects a width number. We prove that this
number is monotone under monomorphisms\, and come up with an appropriate
definition of structured decomposition functor such that we get a relation
ship between widths. We construct several examples of structured decomposi
tion categories\, whose widths coincide with several important examples fr
om the literature.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrei Rodin (University of Lorraine - IN PERSON TALK)
DTSTART:20250423T230000Z
DTEND:20250424T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/86
DESCRIPTION:Title: The concept of mathematical structure according to Voevods
ky.\nby Andrei Rodin (University of Lorraine - IN PERSON TALK) as part
of New York City Category Theory Seminar\n\n\nAbstract\nAbstract: In our
email exchange dating back to 2016 Vladimir Voevodsky suggested an origina
l conception of mathematical structure\, which was motivated\, on the one
hand\, by his work in the Homotopy Type theory and\, on the other hand\, b
y his reading of Proclus’ commentary on Euclid’s definition of plane a
ngle (Def. 1.8. of the Elements). In my talk I present Vladimir’s concep
tion of mathematical structure\, compare it with standard conceptions\, an
d discuss some questions asked by Vladimir during the same exchange. The t
alk is based on this paper: arXiv:2409.02935\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sophie d'Espalungue. (ZOOM TALK Note Special Time)
DTSTART:20250319T230000Z
DTEND:20250320T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/88
DESCRIPTION:Title: Building All of Mathematics Without Axioms: An n-Categoric
al Manifesto.\nby Sophie d'Espalungue. (ZOOM TALK Note Special Time) a
s part of New York City Category Theory Seminar\n\n\nAbstract\nAbstract: T
he formalization of mathematical language traditionally relies on undefine
d terms - such as Set\, Type\, universes - whose properties are specified
by axioms and inference rules. In this talk\, I present an alternative app
roach in which mathematical language is entirely built from definitions. A
t its core are n-category constructors - an internal alternative to typing
judgments - denoted as (X : Cat_n) for a variable X\, which are inductive
ly assigned a truth value - a meaning. Defining an n-category here consist
s of constructing an element (a proof) of the corresponding truth value. T
o give meaning to these constructors\, (n-1)-categories and (n-1)-functors
are inductively organised as an n-category\, resulting in a graded struct
ure of nested n-categories (Cat_{n-1} : Cat_n). By treating each mathemati
cal object as an element of another object\, this framework offers a natur
al and expressive language for higher category theory\, set theory\, and l
ogic\, all with vast generalisation potential. I will discuss key conseque
nces of this approach\, including its implications for fundamental notions
such as sameness\, size\, and ∞-categories\, as well as its connexions
to homotopy type theory.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hannah Aizenman (The Graduate Center\, CUNY)
DTSTART:20250326T230000Z
DTEND:20250327T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/89
DESCRIPTION:Title: Topologically Equivalent Artist Model.\nby Hannah Aize
nman (The Graduate Center\, CUNY) as part of New York City Category Theory
Seminar\n\n\nAbstract\nThe contract data visualization tools make with th
eir users is that a chart is a faithful and accurate visual representation
of the numbers it is made from. Motivated by wanting to make better tools
\, we propose a methodology for fully specifying arbitrary data to visuali
zation mappings in a manner that easily translates to code. We propose tha
t fiber bundles provide a uniform interface for describing a variety of un
derlying data - tables\, images\, networks\, etc. - in a manner that indep
endently encodes the mathematical structure of the topology and the fields
of the dataset. Modeling the data structures that store the datasets as s
heaves provides a method for specifying visualization methods that are des
igned to work regardless of how the dataset is stored - whether the data i
s on disk\, distributed\, or on demand. Specifying the visualization libra
ry components as natural transforms of sheaves means that the constraints
that the component must satisfy to be structure preserving can be specifie
d as the set of morphisms on the data and graphic sheaves\, including the
structure on the topology and fields of the data. Using category theory to
formally express how visual elements are constructed means we can transla
te those expectations into code\, which can then be used to enforce the ex
pectation that a visualization tool is faithfully translating between numb
ers and charts.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raymond Puzio. (IN PERSON TALK)
DTSTART:20250514T230000Z
DTEND:20250515T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/90
DESCRIPTION:Title: Gentle Introduction to Synthetic Differential Geometry - P
art two.\nby Raymond Puzio. (IN PERSON TALK) as part of New York City
Category Theory Seminar\n\n\nAbstract\nAbstract: This is part II of "Gentl
e introduction to synthetic differential geometry". This talk will be self
contained and not assume familiarity with part one. Moreover\, the approa
ch and topics covered this time will be sufficiently different that it wil
l be of interest to people who attended part one.\n\nIn part one\, we intr
oduce the topic in a "bottom-up" manner starting with the simplest instanc
e and building up in complexity. In part two\, we will introduce the subje
ct in a "top-down" manner where we begin by postulating a category with ce
rtain properties and proceeding from these postulates.\n\nAfter introducin
g the topic\, we will turn to Lie groups as an illustrative application. I
ntuitively\, to make a presentation of a Lie group by generators and relat
ions\, we would want to pick infinitessimal transformations for generators
. This is not possible in classical differential geometry so one must inst
ead employ various work-arounds. However\, in synthetic differential geome
try\, infinitessimal generators are well defined and we can build up Lie t
heory in a way which accords with naive intuition. In this talk\, we shall
go through the first few steps of this development. Then we shall note ho
w the synthetic approach is not only more intuitive but more powerful beca
use it allows us to extend the notion of Lie group beyond finite-dimension
al manifolds to which the classical approach is limited. We will also say
a few words about how the some of these infinite-dimensional generalizatio
ns are of use in in practical applications.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/90/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thiago Alexandre. (ZOOM TALK)
DTSTART:20250528T200000Z
DTEND:20250528T213000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/91
DESCRIPTION:Title: Topological Derivators --- Part two.\nby Thiago Alexan
dre. (ZOOM TALK) as part of New York City Category Theory Seminar\n\n\nAbs
tract\nAbstract: In this second part\, I begin by recalling the axioms of
topological derivators and presenting some elementary consequences of thes
e axioms. Following this\, I explain how topological derivators can be con
structed by sheafifying homotopy theories. I conclude with the deepest the
orem I have obtained in the theory of topological derivators\, which provi
des strong evidence for Grothendieck’s conjecture: if a derivator can be
extended to a topological derivator\, then this extension is essentially
unique.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergei Artemov (IN PERSON TALK)
DTSTART:20250507T230000Z
DTEND:20250508T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/92
DESCRIPTION:Title: Consistency of PA is a serial property\, and it is provabl
e in PA.\nby Sergei Artemov (IN PERSON TALK) as part of New York City
Category Theory Seminar\n\n\nAbstract\nAbstract: We show that PA consisten
cy is mathematically equivalent to the serial property\, which we call the
consistency scheme ConS(PA):\n\n"n is not a proof of 0=1"\, for n=0\,1\,2
\,... .\nThe proof of this equivalence is formalizable in PA. Since the st
andard consistency formula Con(PA)\n\n"for all x\, x is not a code of a pr
oof of 0=1"\nis strictly stronger than the scheme ConS(PA) in PA\, Goedel'
s Second Incompleteness theorem\, stating that PA |-\\- Con(PA) does not y
ield the unprovability of PA consistency. Hence\, the widespread belief th
at a consistent theory cannot establish its consistency has never been jus
tified.\n\nMoreover\, we show that this belief is false. The question of p
roving PA consistency in PA reduces to proving the scheme ConS(PA) in PA.
We build on Hilbert's ideas and prove ConS(PA) in PA.\n\nThis talk is a "d
ress rehearsal" for the speaker's plenary talk at the ASL meeting on May 1
3\, 2025.\n\nReference:\nS.Artemov "Serial Properties\, Selector Proofs\,
and the Provability of Consistency\," Journal of Logic and Computation\, V
olume 35\, Issue 3\, April 2025.\nhttps://doi.org/10.1093/logcom/exae034\n
Published: 26 July 2024.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam McCrosson (Montana State University...Zoom Talk)
DTSTART:20250917T230000Z
DTEND:20250918T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/93
DESCRIPTION:Title: TALK CANCELED.\nby Sam McCrosson (Montana State Univer
sity...Zoom Talk) as part of New York City Category Theory Seminar\n\n\nAb
stract\nMicrolocal sheaf theory has been gaining popularity recently for i
ts applications to symplectic geometry. In this talk\, we’ll explore a m
ore topological application of this subject: how the notion of the microsu
pport of a sheaf can be used to tell if a sheaf is “constructible\,” i
.e. locally constant on strata\, and if so\, what the coarsest stratificat
ion is with this property.\n\nVersions of this result can be found as far
back as Kashiwara and Schapira’s 1990 book “Sheaves on Manifolds” (w
hich pioneered the subject of microlocal sheaf theory). Today\, all sorts
of generalizations are possible using schemes\, \\infty-categories\, and o
ther fancy machinery. This talk will focus on a particularly simple case:
using 1-category theory and sheaves of sets on topological spaces to illus
trate the key ideas with concrete examples.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amartya Shekhar Dubey (National Institute of Science Education and
Research... Zoom Talk 2PM NYC time)
DTSTART:20251022T180000Z
DTEND:20251022T190000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/94
DESCRIPTION:Title: Unital k-restricted Infinity Operads.\nby Amartya Shek
har Dubey (National Institute of Science Education and Research... Zoom Ta
lk 2PM NYC time) as part of New York City Category Theory Seminar\n\n\nAbs
tract\nThe goal is to understand unital \\infty-operads by their arity res
trictions. Given k \\geq 1\, we develop a model for unital k-restricted \\
infinty-operads\, which are variants of \\infinity-operads with (\\leq k)-
arity morphisms\, as complete Segal presheaves on closed k-dendroidal tree
s built from corollas with valence \\leq k. Furthermore\, we prove that th
e restriction functors from unital \\infty-operads to unital k-restricted
\\infty-operads admit fully faithful left and right adjoints by showing th
at the left and right Kan extensions preserve complete Segal objects. Vary
ing k\, the left and right adjoints give a filtration and a co-filtration
for any unital \\infty-operads by k-restricted \\infty-operads. This is jo
int work with Yu Leon Liu.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathon Funk (Queensborough\, CUNY.. In Person Talk)
DTSTART:20251029T230000Z
DTEND:20251030T003000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/95
DESCRIPTION:Title: More Toposes and C*-algebras.\nby Jonathon Funk (Queen
sborough\, CUNY.. In Person Talk) as part of New York City Category Theory
Seminar\n\n\nAbstract\nLet \\( R \\) be a PID and let \\( \\vec{s} = (n_1
\, \\dots\, n_k) \\) be a "shape" vector with \\( k\, n_i \\ge 2 \\).\nWri
te \\( X_\\bullet(\\vec{s}) \\) for the simplicial \\( R \\)-module genera
ted by tensors of shape \\( \\vec{s} \\)\,\nwhere the simplicial structure
is induced by diagonal application of the usual coface and codegeneracy o
perators along all axes\;\nset \\( n = \\min(\\vec{s}) - 1 \\).\n
\nFor
a nondegenerate \\( n \\)-simplex \\( T \\) (with nondegenerate boundary\
, i.e.\, no face \\( d_i T \\) lies in the degenerate submodule)\nand \\(
0 \\le j \\le n \\)\, let \\( \\Lambda^n_j(T) \\) be the corresponding hor
n and let \\( L_\\bullet \\) denote the simplicial submodule\ngenerated by
the faces of the horn.\n
\n\n\nUsing the standard fact that the set of
fillers of \\( \\Lambda^n_j(T) \\) is a torsor under\n
\n\n\\[\nR_{n\,
j} = \\bigcap_{i \\ne j} \\ker\\!\\left(d_i : X_n(\\vec{s}) \\to X_{n-1}(\
\vec{s})\\right)\,\n\\]\n
\nwe show that \\( \\operatorname{rank}_R R_{
n\,j} \\) admits an explicit inclusion–exclusion formula counting the
“missing indices.”\nIn particular\, non-unique fillers occur iff \\( k
\\ge n \\).\n\n
\nUnder the Dold–Kan correspondence\, the re
lative homology\n\\[\nH_n\\left(N(X_\\bullet(\\vec{s}))\, N(L_\\bullet)\\r
ight)\n\\]\nis the quotient of \\( R_{n\,j} \\) by the image of the normal
ized boundary \\( \\partial_{n+1} \\)\,\nand is nontrivial iff \\( k \\ge
n+1 \\)\; when nontrivial its rank equals the missing–index count.\n
\nI will present the combinatorial formula\, and accompanying code
that verifies the counts and the two dichotomies.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/95/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Florian Lengyel (CUNY. Zoom Talk)
DTSTART:20251106T000000Z
DTEND:20251106T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/96
DESCRIPTION:Title: Horn-filling thresholds in simplicial tensor modules.\
nby Florian Lengyel (CUNY. Zoom Talk) as part of New York City Category Th
eory Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/96/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emilio Minichiello (CUNY CityTech. In Person Talk)
DTSTART:20251113T000000Z
DTEND:20251113T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/97
DESCRIPTION:Title: Model Structures for Simplicial Complexes and Graphs.\
nby Emilio Minichiello (CUNY CityTech. In Person Talk) as part of New York
City Category Theory Seminar\n\n\nAbstract\nI’ll talk about my new paper which constructs model st
ructures on the category of simplicial complexes and on reflexive graphs w
hich are reminiscent of the Thomason model structure on categories. I’ll
give some background and motivation for studying this and the surrounding
questions of graph homotopy theory.\n
\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/97/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Evan Misshula (CUNY. In-Person Talk)
DTSTART:20251204T000000Z
DTEND:20251204T013000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/98
DESCRIPTION:Title: From Outer Measures to Adjunctions: A Category-Theoretic R
ecasting of Caratheodory’s Extension Theorem.\nby Evan Misshula (CUN
Y. In-Person Talk) as part of New York City Category Theory Seminar\n\n\nA
bstract\nAbstract: The Caratheodory Extension Theorem underpins modern mea
sure theory by extending a pre-measure on an algebra to a complete measure
on the generated -algebra. Traditionally\, the proof proceeds through ou
ter measures\, Caratheodory measurability\, and a series of delicate techn
ical verifications – a process that reveals the "monsters" lurking in th
e power set but often obscures the structural simplicity of the result. In
this talk\, I will first outline the classical construction to highlight
these subtleties\, and then present a category-theoretic reformulation: th
e extension theorem emerges naturally from an adjunction. The categorical
perspective streamlines the argument\, but seeing both sides illuminates w
hy rigor is indispensable and how category theory captures the essence of
the construction.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/98/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergei Artemov (The CUNY Graduate Center.)
DTSTART:20260204T190000Z
DTEND:20260204T203000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/99
DESCRIPTION:Title: Non-compact proofs\nby Sergei Artemov (The CUNY Gradua
te Center.) as part of New York City Category Theory Seminar\n\nLecture he
ld in Room 4214.03 in The Graduate Center\, CUNY.\n\nAbstract\nNon-compact
proofs are used in mathematics but overlooked in the analysis of (un)prov
ability of consistency. We focus on arithmetical proofs of universal state
ments (*) "for any natural number n\, F(n)." A proof of (*) is compact if
all proofs of F(n)'s for n=0\,1\,2\,... fit into some finitely axiomatized
fragment of Peano Arithmetic PA. An example of non-compact reasoning is t
he standard proof of Mostowski's 1952 reflexivity theorem: PA proves the c
onsistency of its finite fragments.\n\nIt turns out that Gödel's Second I
ncompleteness Theorem\, G2\, prohibits compact proofs but does not rule ou
t non-compact proofs of PA-consistency formalizable in PA. This explains w
hy and how the recent proofs of PA-consistency in PA work: they essentiall
y formalize in PA the explicit version of Mostowski's non-compact proof an
d use Gödelian provable explicit reflection to rid redundant provability
operators. \n\nThese findings yield a new foundational reading of G2: "the
consistency of PA is not provable within a finite fragment of PA\," compl
emented with the positive message: "the consistency of PA is provable with
in the whole PA." This perspective suggests that Gödel's theorem does not
represent a failure of the system to "know" its own consistency\, but rat
her a structural limit on how that knowledge can be packaged into a single
finite string.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/99/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Ellerman (University of Ljubljana)
DTSTART:20260211T190000Z
DTEND:20260211T203000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/100
DESCRIPTION:Title: A Fundamental Duality in the Exact Sciences: An Introduct
ion to Mathematical Metaphysics.\nby David Ellerman (University of Lju
bljana) as part of New York City Category Theory Seminar\n\nLecture held i
n Room 4214.03 in The Graduate Center\, CUNY.\n\nAbstract\nThere is a fund
amental duality that runs through the exact sciences. At the logical level
\, it is the duality between (Boolean) logic of subsets and the logic of p
artitions. The quantitative versions of the dual logics are logical probab
ility theory and logical information theory. The duality accounts for the
duality in the category of Sets and its opposite Sets^{op}. The partial or
der in the two dual logics gives the two fundamental canonical functions a
nd the claim is that all canonical morphisms in Sets arise from those two
morphisms. In physics\, there is the notion of "definiteness all the way d
own" which arises in classical physics (Boolean logic of subsets) and dual
ly there is the notion of definiteness only down to a certain level and th
en objective indefiniteness that arises in quantum physics (logic of parti
tions).\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/100/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ellis Cooper
DTSTART:20260218T190000Z
DTEND:20260218T203000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/101
DESCRIPTION:Title: Algebraic String Diagrams and a Manifest Covariance Theor
em\nby Ellis Cooper as part of New York City Category Theory Seminar\n
\nLecture held in Room 4214.03 in The Graduate Center\, CUNY.\n\nAbstract\
nBook titles such as "Covariant Physics" (Moataz H. Emam) and "Covariant L
oop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spi
nfoam Theory" (Carlo Rovelli and Francesca Vidotto) shout the important of
covariance in modern mathematical physics. In categorical terms\, covaria
nce is a family of natural isomorphisms of pairs of functors defined on th
e groupoid of diffeomorphisms in a category of ``domains." "Manifest covar
iance" is a syntactic concept arising from preservation of covariance of b
asic covariant tensor calculations combined by composition and product map
s. Differential geometry and general relativity theory calculations are ex
pressed by algebraic string diagrams\, including the Einstein Curvature Te
nsor. Physical nature is categorically natural.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/101/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Austin Myer (CUNY)
DTSTART:20260304T190000Z
DTEND:20260304T203000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/102
DESCRIPTION:Title: The Bloch Material: a Simplicial Set Whose Homology is th
e Higher Chow Groups of Spencer Bloch.\nby James Austin Myer (CUNY) as
part of New York City Category Theory Seminar\n\nLecture held in Room 421
4.03 in The Graduate Center\, CUNY.\n\nAbstract\nTo this day\, it is still
unknown whether every variety enjoys a resolution of its singularities. D
ennis Sullivan suggests in a 2004 memorial article for René Thom that the
obstructions constructed to attack Steenrod’s problem could be adapted
to handle the outstanding scenario in positive characteristic. We will dis
cuss a construction en route to the realization of this dream: to each var
iety\, we prescribe a simplicial set whose homology is the higher Chow gro
ups of (Spencer) Bloch. In particular\, this simplicial set recovers the t
opology of the analytic space associated to a variety over the complex num
bers\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/102/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kristaps John Balodis (University of Calgary)
DTSTART:20260311T180000Z
DTEND:20260311T193000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/103
DESCRIPTION:Title: A geometric introduction to the local Langlands correspon
dence.\nby Kristaps John Balodis (University of Calgary) as part of Ne
w York City Category Theory Seminar\n\nLecture held in Room 4214.03 in The
Graduate Center\, CUNY.\n\nAbstract\nIn this talk I will provide a non-tr
aditional introduction to the local Langlands program for p-adic groups by
framing the so-called "Galois side" geometrically. For simplicity\, the p
rimary focus will be on the "unramified" version for GL(n). The ultimate g
oal will be to articulate the p-adic Kazhdan-Lusztig hypothesis with accom
panying examples. Along the way\, I will stop to discuss some categorical
aspects of the theory.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/103/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Morgan Rogers (University of Sorbonne)
DTSTART:20260325T180000Z
DTEND:20260325T193000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/104
DESCRIPTION:Title: Ultrarings. A categorical approach to unifying boolean an
d algebraic descriptive complexity\nby Morgan Rogers (University of So
rbonne) as part of New York City Category Theory Seminar\n\nLecture held i
n Room 4214.03 in The Graduate Center\, CUNY.\n\nAbstract\nThe presentatio
n of a commutative ring by generators and relations is (at least superfici
ally) similar to the presentation of a first order theory in terms of sort
s\, function/relation symbols and axioms. More concretely\, we can associa
te categories to rings and to theories:\n\n In commutative algebra we c
an associate to a ring its category of finitely generated projective modul
es\, which is a monoidal category with coproducts (over which the monoidal
product distributes) having several further special properties.\n Clas
sical first-order theories are classified by Boolean lextensive categories
. That is\, if we take a first order theory 𝕋\, we can associate to it
a category ℬ𝕋 (its "syntactic category") with finite limits and finit
e (pullback-stable\, disjoint) coproducts in which every subobject has a c
omplement\, such that models of 𝕋 in the category of sets correspond to
functors ℬ𝕋 → Set preserving all that structure.\n\nOf course\, th
ere are many other categories we could have chosen on each side\, but thes
e particular constructions admit a mutual generalization which we call ult
rarings. In this talk we explain what ultrarings are\, how their presentat
ions generalize those of commutative rings and first-order theories\, and
how they connect to the logical approach to complexity theory (descriptive
complexity). We will also sketch how we hope to exploit this connection i
n the future to transport tools from algebraic geometry.\n\nThis talk is b
ased on work with Baptiste Chanus and Damiano Mazza\, https://doi.org/10.4
230/LIPIcs.FSCD.2025.13\, with slightly updated definitions.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/104/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Lesnick (University at Albany -- SUNY)
DTSTART:20260415T180000Z
DTEND:20260415T193000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/105
DESCRIPTION:Title: Limit Computation Over Posets via Minimal Initial Functor
s\nby Michael Lesnick (University at Albany -- SUNY) as part of New Yo
rk City Category Theory Seminar\n\nLecture held in Room 4214.03 in The Gra
duate Center\, CUNY.\n\nAbstract\nJoint work with Tamal Dey\, Department o
f CS\, Purdue University. \n\nIt is well known that limits can be computed
by restricting along an initial functor\, and that this often simplifies
limit computation. We systematically study the algorithmic implications of
this idea for diagrams indexed by a finite poset. We say an initial funct
or $F\\colon C\\to D$ with $C$ small is \\emph{minimal} if the sets of obj
ects and morphisms of $C$ each have minimum cardinality\, among the source
s of all initial functors with target $D$. For $Q$ a finite poset or $Q\\s
ubseteq \\mathbb{N}^d$ an interval (i.e.\, a convex\, connected subposet)\
, we describe all minimal initial functors $F\\colon P\\to Q$ and in parti
cular\, show that $F$ is always a subposet inclusion. We give efficient al
gorithms to compute a choice of minimal initial functor. In the case that
$Q\\subseteq \\mathbb{N}^d$ is an interval\, we give asymptotically optima
l bounds on $|P|$\, the number of relations in $P$ (including identities)\
, in terms of the number $n$ of minima of $Q$: We show that $|P|=\\Theta(n
)$ for $d\\leq 3$\, and $|P|=\\Theta(n^2)$ for $d>3$. We apply these resul
ts to give new bounds on the cost of computing $\\lim G$ for a functor $G
\\colon Q\\to \\mathbf{Vec}$ valued in vector spaces. For $Q$ connected\,
we also give new bounds on the cost of computing the \\emph{generalized ra
nk} of $G$ (i.e.\, the rank of the induced map $\\lim G\\to \\operatorname
{colim} G$)\, which is of interest in topological data analysis.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/105/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gabriel Goren Roig
DTSTART:20260422T180000Z
DTEND:20260422T193000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/106
DESCRIPTION:by Gabriel Goren Roig as part of New York City Category Theory
Seminar\n\nLecture held in Room 4214.03 in The Graduate Center\, CUNY.\nA
bstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/106/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emilio Minichiello (CUNY CityTech)
DTSTART:20260513T180000Z
DTEND:20260513T193000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/107
DESCRIPTION:Title: Introduction to locally presentable categories\nby Em
ilio Minichiello (CUNY CityTech) as part of New York City Category Theory
Seminar\n\nLecture held in Room 4214.03 in The Graduate Center\, CUNY.\n\n
Abstract\nThis will be an expository talk about locally presentable catego
ries and surrounding ideas\, corresponding to Appendix C of my notes https
://arxiv.org/pdf/2503.20664.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/107/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cheyne Glass (The Graduate Center CUNY)
DTSTART:20260506T180000Z
DTEND:20260506T193000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/108
DESCRIPTION:Title: Homotopy theoretic least squares models.\nby Cheyne G
lass (The Graduate Center CUNY) as part of New York City Category Theory S
eminar\n\nLecture held in Room 4214.03 in The Graduate Center\, CUNY.\n\nA
bstract\nThis talk will explore a potential application of "higher sheaf t
heory" in data analysis. Given a fixed ambient space\, a presheaf of compl
exes will be constructed on the poset of finite subsets\, which encodes in
formation about least squares solutions (LS) for a choice particular model
(ie "y=mx+b"). Given a choice of cover of a data set\, the presheaf of co
mplexes evaluated on the Čech nerve naturally assembles into a Čech-LS t
otal complex where 0-cocycles represent choices of parameters on each cove
ring subset\, and homotopies between the discrepancies on overlaps. For th
e purposes of moving toward a predictive model\, additional evaluation and
localization procedures (and data) are described which assemble instead i
nto a twisted complex. With time permitting we will try to work through a
toy example with 5 data points.\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/108/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph Helfer CANCELED (Simons Center in Stony Brook)
DTSTART:20260225T190000Z
DTEND:20260225T203000Z
DTSTAMP:20260404T095717Z
UID:Category_Theory/109
DESCRIPTION:Title: Set-theoretic universes and paradoxes in elementary 2-top
oi.\nby Joseph Helfer CANCELED (Simons Center in Stony Brook) as part
of New York City Category Theory Seminar\n\nLecture held in Room 4214.03 i
n The Graduate Center\, CUNY.\n\nAbstract\nAbstract: In 1966\, Lawvere pro
posed an axiomatization of the category of categories as a foundational th
eory of mathematics. Nowadays\, we recognize that the totality of categori
es should rather be considered a *2-category*. Making this modification to
Lawvere's idea results in the notion of elementary 2-topos\, introduced b
y M. Weber: an axiomatization of the 2-category of categories. This is par
t of the more general\, ongoing program of "higher-categorical foundations
" which includes M. Makkai's theory of First-Order Logic with Dependent So
rts\, and V. Voevodsky's Homotopy Type Theory.\n\nFollowing ideas of M. Ma
kkai and B. Boshuk\, I have been continuing to develop Weber's theory\, in
particular adding crucial axioms which were missing from his definition.
After giving some general explanations of these notions\, I will explain t
wo related results about elementary 2-topoi which serve as "tests" of its
adequacy as a foundational theory: that is subsumes usual ZF set theory\,
and that it reproduces a version of the classical set-theoretic paradoxes\
, provided one is not careful to impose some "size restrictions".\n
LOCATION:https://stable.researchseminars.org/talk/Category_Theory/109/
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