This tutorial will be an introduction to the noti on of connection introduced in [1] in the setting of tangent categories an d its equivalent characterizations in [2]. Building on two formulations o f connections on vector bundles that are due to Ehresmann and to Patterson [3]\, respectively\, Cockett and Cruttwell [1] defined a notion of connec tion in the abstract setting of tangent categories. Equivalent definition s of connections in tangent categories were developed in [2] using biprodu cts in the additive category of differential bundles over an object of a t angent category\, leading also to an economical definition of connections in tangent categories as vertical connections with the property that a cer tain cone is a limit cone. In this tutorial\, we shall survey these equiv alent definitions of connection and some aspects of their equivalence.\n\n
[1] J. R. B. Cockett and G. S. H. Cruttwell\, Connections in tangent ca tegories\, Theory Appl. Categ. 32 (2017)\, 835-888.\n\n
[2] R. B. B. Luc yshyn-Wright\, On the geometric notion of connection and its expression in tangent categories\, Theory Appl. Categ. 33 (2018)\, 832-866.\n\n
[3] L .-N. Patterson\, Connexions and prolongations\, Canad. J. Math. 27 (1975)\ , 766-791.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Richard Garner (Macquarie University) DTSTART:20210615T220000Z DTEND:20210615T224500Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/12 DESCRIPTION:Title: The free tangent category on an affine connection\nby Ric hard Garner (Macquarie University) as part of BIRS workshop : Tangent Cate gories and their Applications\n\n\nAbstract\n
The purpose of this talk is to sketch a construction of the free tangent category containing an obj ect M with a connection on its tangent bundle. It turns out that the maps of this tangent category are completely determined by the calculus of mult ilinear maps on the tangent\nbundle\; and that this calculus can be encode d by a certain kind of operad\, which comes endowed with an operation of c ovariant derivative O(n)->O(n+1) and constants T in O(2) (torsion) and R in O(3) (curvature)\, with as axioms the chain rule\, the two Bianchi iden tities\nand the Ricci identity. Any such operad generates a tangent categ ory\; the free such operad generates the free tangent category on an affin e connection.\n\n
This is work-in-progress with Geoff Cruttwell.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Ben MacAdam (University of Calgary) DTSTART:20210615T230000Z DTEND:20210615T234500Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/13 DESCRIPTION:Title: New tangent structures for Lie algebroids and Lie groupoids\nby Ben MacAdam (University of Calgary) as part of BIRS workshop : Tang ent Categories and their Applications\n\n\nAbstract\nThe tangent bundle on a smooth manifold is\, in a sense\, sufficient structure for Lagrangian m echanics. In a famous note from 1901\, Poincare reformulated Lagrangian me chanics by replacing the tangent bundle with a Lie algebra acting on a smo oth manifold [1\, 2]. Poincare's formalism leads to the Euler-Poincare equ ations\, which capture the usual Euler-Lagrange equations as a specific ex ample. In 1996\, Weinstein sketched out a general program building on Poin care's ideas to formulate mechanics on Lie groupoids using Lie algebroids [3]\, which motivates the work of Martinez et al. [4\,5]\, Libermann [6]\, and the recent thesis by Fusca [7].\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Tom Goodwillie (Brown University) DTSTART:20210616T150000Z DTEND:20210616T154500Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/14 DESCRIPTION:Title: Functor calculus\nby Tom Goodwillie (Brown University) as part of BIRS workshop : Tangent Categories and their Applications\n\n\nAb stract\nFunctor calculus is a way of organizing the interplay between homo topy theory and stable homotopy theory. Its name reflects an analogy with differential calculus. There are derivatives\, $n$th derivatives\, and Tay lor polynomials in functor calculus. \n\nFunctors between homotopical cate gories (categories which\, like the category of topological spaces\, have a suitable structure for “doing homotopy theory”) can be thought of as resembling smooth maps between manifolds. Homotopical categories that are stable correspond to manifolds that are vector spaces. I will sketch the high points of functor calculus with this geometric analogy in mind. \n\nU ntil recently the relation with smooth geometry has existed mostly as a su ggestive analogy. It is being pursued in detail now by Bauer\, Burke\, Chi ng\, and others using the framework of tangent structures on categories.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Brenda Johnson (Union College) DTSTART:20210616T160000Z DTEND:20210616T164500Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/15 DESCRIPTION:Title: An example of a cartesian differential category from functor calculus\nby Brenda Johnson (Union College) as part of BIRS workshop : Tangent Categories and their Applications\n\n\nAbstract\nIn this talk\, I will provide an introduction to abelian functor calculus\, a version of f unctor calculus inspired by classical constructions of Dold and Puppe\, an d of Eilenberg and Mac Lane. I will then explain how the analog of a dire ctional derivative in abelian functor calculus gives rise to the structure of a cartesian differential category for a particular category of functor s of abelian categories.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/15/ END:VEVENT BEGIN:VEVENT SUMMARY:Eric Finster (University of Cambridge) DTSTART:20210616T170000Z DTEND:20210616T174500Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/16 DESCRIPTION:Title: The Nilpotence Tower\nby Eric Finster (University of Camb ridge) as part of BIRS workshop : Tangent Categories and their Application s\n\n\nAbstract\nMuch like the theory of affine schemes and commutative ri ngs\, the\ntheory of (higher) topoi leads a dual life: one algebraic and o ne\ngeometric. In the geometric picture\, a topos is a kind of\ngeneraliz ed space whose points carry the structure of a category.\nDually\, in the algebraic point of view\, a topos may be thought of\nas the "ring of conti nuous functions on a generalized space with\nvalues in homotopy types".\n\ nIn this talk\, I will explain the connection between Goodwillie's\ncalcul us of functors and this algebro-geometric picture of the\ntheory of higher topoi. Specifically\, I will describe how one can\nview the topos of n-e xcisive functors as an analog of the\ncommutative k-algebra k[x]/xⁿ⁺¹ \, freely generated by a nilpotent\nelement of order n+1.\n\nMore generall y\, I will show how every left exact localization E → F\nof topoi may be extended to a tower of such localizations\n\nE ⋯ → Fₙ → Fₙ₋ ₁ → ⋯ F₀ = F\n\nwhich we refer to as the Nilpotence Tower\, and wh ose values at an\nobject of E may be seen as a generalized version of the Goodwillie\ntower of a functor with values in spaces. Under the analogy w ith\nscheme theory described above\, this construction corresponds to the\ ncompletion of a commutative ring along an ideal\, or\, geometrically\,\nt o the filtration of the formal neighborhood of a subscheme by it's\nn-th o rder sub-neighborhood. I will also explain how\, in addition\nto the homo topy calculus\, the orthogonal calculus of Michael Weiss\ncan be seen as a n instance of this same construction.\n\nThis is joint work with M. Anel\, G. Biedermann and A. Joyal.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Kristine Bauer (University of Calgary) DTSTART:20210616T210000Z DTEND:20210616T214500Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/17 DESCRIPTION:Title: Tangent Infinity Categories\nby Kristine Bauer (Universit y of Calgary) as part of BIRS workshop : Tangent Categories and their Appl ications\n\n\nAbstract\nThis is joint work with M. Burke and M. Ching. In this talk\, I will present the definition of a tangent infinity category as a generalization of Leung's presentation of tangent categories as Weil- modules. A key example of a tangent structure on the infinity category of infinity categories is an extension of Lurie’s tangent bundle functor. We call this the Goodwillie tangent structure\, since it encodes the theo ry of Goodwillie calculus. The differential objects in this tangent in finity category are precisely the stable infinity categories. Following C ockett-Cruttwell these form a cartesian differential category. I will exp lain that the derivative in this CDC is the same as the BJORT derivative f or abelian functor calculus\, showing that the Goodwillie tangent structur e is an extension of BJORT.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael Ching (Amherst College) DTSTART:20210616T220000Z DTEND:20210616T224500Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/18 DESCRIPTION:Title: Dual tangent structures for infinity-toposes\nby Michael Ching (Amherst College) as part of BIRS workshop : Tangent Categories and their Applications\n\n\nAbstract\nI will describe two tangent infinity-cat egories whose objects are the infinity-toposes: one algebraic and one geom etric. The algebraic version is a restriction to infinity-toposes of the G oodwillie tangent structure defined by Bauer\, Burke and myself\, in which the tangent bundle consists of the stabilizations of slice infinity-topos es. The geometric structure is dual to the algebraic with tangent bundle f unctor given by an adjoint to that of the Goodwillie structure. There is a useful analogy to tangent structures on the category of commutative rings and its opposite (the category of affine schemes).\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/18/ END:VEVENT BEGIN:VEVENT SUMMARY:André Joyal (Université du Québec à Montréal) DTSTART:20210616T230000Z DTEND:20210616T233000Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/19 DESCRIPTION:Title: The (higher) topos classifying ∞-connected objects\nby André Joyal (Université du Québec à Montréal) as part of BIRS worksho p : Tangent Categories and their Applications\n\n\nAbstract\nJoint work wi th Mathieu Anel\, Georg Biedermann and Eric Finster.\n\nI will present an application of Goodwillie’s calculus to higher topos theory. The (higher ) topos which classifies $\\infty$-connected objects is formally the "dual " of the (higher) logos $S[U_\\infty]$ freely generated by an $\\infty$-co nnected object $U_\\infty$. The logos $S[U_\\infty]$ is a left exact topol ogical localisation of the logos $S[U] = Fun[Fin\, S]$ freely generated by an object $U$. We show that a functor\n$ Fin \\to S$ belongs to $S[U_\\i nfty]$ if and only if it is $\\infty$-excisive if and only if it is the ri ght Kan extension of its restriction to the subcategory of finite n-connec ted spaces $C_n \\subset Fin$ for every $n \\geq 0$. There is a morphism o f logoi from $S[U_\\infty]$ to the category of Goodwillie towers of functo rs $Fin \\to S$\, but we do not know if it is an equivalence of categorie s. We also consider the logos $S[U_\\infty′ ]$ freely generated by a poi nted $\\infty$-connected object $U'_\\infty$ .\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Jonathan Gallagher (Dalhousie University) DTSTART:20210617T150000Z DTEND:20210617T154500Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/20 DESCRIPTION:Title: Differential programming\nby Jonathan Gallagher (Dalhousi e University) as part of BIRS workshop : Tangent Categories and their Appl ications\n\n\nAbstract\nDifferential and tangent categories have been appl ied to providing the\nsemantics of differential programming languages. As interest in\ndifferential programming langauges continues to grow due to\ napplications in machine learning\, many differential programming\nlanguag es are being extended with features for probabilistic\nprogramming and in some cases quantum programming. In this talk\, we\nwill investigate struc tures on top of differential and tangent\ncategories that allow modelling probabilistically extended programming\nlanguages. To do this\, we will develop some of the basics of\nfunctional analysis and distribution theory in the context of\ndifferential categories. We will also develop differe nt approaches to\nencoding probabilistic computations in a differential la nguage.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Bruno Gavranovic (University of Strathclyde) DTSTART:20210617T160000Z DTEND:20210617T164500Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/21 DESCRIPTION:Title: Categorical Foundations of Gradient-Based Learning\nby Br uno Gavranovic (University of Strathclyde) as part of BIRS workshop : Tang ent Categories and their Applications\n\n\nAbstract\nWe propose a categori cal foundation of gradient-based machine learning algorithms in\nterms of lenses\, parametrised maps\, and reverse derivative categories.\n\nThis fo undation provides a powerful explanatory and unifying framework: it encomp asses a variety of gradient\ndescent algorithms such as ADAM\, AdaGrad\, a nd Nesterov momentum\,\nas well as a variety of loss functions such as as MSE and Softmax cross-entropy\, shedding new light on their similarities a nd differences.\nOur approach also generalises beyond neural networks (mod elled in categories of smooth maps)\,\naccounting for other structures rel evant to gradient-based learning such as boolean circuits.\n\nFinally\, we also develop a novel implementation of gradient-based learning in\nPython \, informed by the principles introduced by our framework.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/21/ END:VEVENT BEGIN:VEVENT SUMMARY:Mario Alvarez-Picallo (Huawei Research) DTSTART:20210617T170000Z DTEND:20210617T174500Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/22 DESCRIPTION:Title: Soundness for automatic differentiation via string diagrams\nby Mario Alvarez-Picallo (Huawei Research) as part of BIRS workshop : Tangent Categories and their Applications\n\n\nAbstract\nReverse-mode auto matic differentiation\, especially in the presence of complex language\nfe atures\, is notoriously hard to implement correctly\, and most implementat ions focus on\ndifferentiating straight-line imperative first-order code. Generalisations exist\, however\,\nthat can tackle more advanced features\ ; for example\, the algorithm described by Pearlmutter\nand Siskind in the ir 2008 paper can differentiate (pure) code containing closures.\n\nWe sho w that AD algorithms can benefit enormously from being translated into the language\nof string diagrams in two steps: first\, we rephrase Pearlmutte r and Siskind's algorithm as\na set of rules for transforming hierarchical graphs\; rules which can -and indeed have been-\nbe implemented correctly and efficiently in a non-trivial language. Then\, we sketch a proof\nof s oundness for it by reducing its transformations to the axioms of Cartesian reverse\ndifferential categories\, expressed as string diagrams.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/22/ END:VEVENT BEGIN:VEVENT SUMMARY:Dorette Pronk (Dalhousie University) DTSTART:20210617T210000Z DTEND:20210617T212000Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/23 DESCRIPTION:Title: Exponentials and Enrichment for Orbispaces\nby Dorette Pr onk (Dalhousie University) as part of BIRS workshop : Tangent Categories a nd their Applications\n\n\nAbstract\nOrbifolds are defined like manifolds\ , by local charts. Where manifold charts are open subsets of Euclidean spa ce\, orbifold charts consist of an open subset of Euclidean space with an action by a finite group (thus allowing for local singularities). However\ , a more useful way to represent them is in terms of proper étale groupoi ds (which we will call orbispaces) and the maps between them are obtained as a bicategory of fractions of the 2-category of proper étale groupoids with respect to the class of essential equivalences. In recent work with B ustillo and Szyld we have shown that in any bicategory of fractions the ho m-categories form a pseudo colimit of the hom categories of the original b icategory.\n\n \n\nWe will show that this result can be extended to our to pological context: for topological groupoids the hom-groupoids can again b e topologized and under suitable conditions on the spaces these groupoids form both exponentials and enrichment. We will show that taking the approp riate pseudo colimit of these hom-groupoids within the 2-category of topol ogical groupoids gives us a notion of hom-groupoids for the bicategory of orbispaces. When the domain orbispace is orbit compact\, we see show that this groupoid is proper and satisfies the conditions to be an exponential. When we further cut back our morphisms between orbispaces to so-called ad missible maps\, we obtain a proper étale groupoid that is essentially equ ivalent to the pseudo colimit and hence is also the exponential. Furthermo re\, we show that the bicategory of orbit-compact orbispaces is enriched o ver orbispaces: the composition is given by a map of orbispaces rather tha n a continuous functor.\n\nThis work rephrases the result from [Chen] in t erms of groupoid representations for orbifolds and strengthens his result on enrichment: he expressed this in terms of a map between the quotient sp aces of the mapping orbispaces\, where we are able to give this in terms o f a map between the orbispaces.\n\nI will end the talk with several exampl es of mapping spaces. This is joint work with Laura Scull and started out as a project of the first Women in Topology workshop.\n\n[Chen] Weimin Che n\, On a notion of maps between orbifolds I: function spaces\, Communicati ons in Contemporary Mathematics 8 (2006)\, pp. 569-620.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/23/ END:VEVENT BEGIN:VEVENT SUMMARY:Susan Niefield (Union College) DTSTART:20210617T213000Z DTEND:20210617T215000Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/24 DESCRIPTION:Title: Linear Bicategories: Quantales and Quantaloids\nby Susan Niefield (Union College) as part of BIRS workshop : Tangent Categories and their Applications\n\n\nAbstract\nLinear bicategories were introduced by Cockett\, Koslowski and Seely as the\nbicategorical version of linearly di stributive categories. Such a bicategory B\nhas two forms of composition r elated by a linear distribution. In this talk\, we\nconsider locally order ed linear bicategories of the form Q-Rel\, i.e.\, relations\nvalued in a q uantale Q\; as well as those B which are Girard bicategories.\nThe latter provide examples which are not locally ordered\; and they have\nthe same r elation to linear bicategories as ∗-autonomous categories have to\nlinea rly distributive categories. Examples include the bicategories Quant and\n Qtld\, whose 1-cell are bimodules and objects are quantales and quantaloid s\,\nrespectively.\n\nThis is joint work with Rick Blute.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Priyaa Srinivasan (University of Calgary) DTSTART:20210617T220000Z DTEND:20210617T222000Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/25 DESCRIPTION:by Priyaa Srinivasan (University of Calgary) as part of BIRS w orkshop : Tangent Categories and their Applications\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/25/ END:VEVENT BEGIN:VEVENT SUMMARY:Bryce Clarke (Macquarie University) DTSTART:20210617T223000Z DTEND:20210617T225000Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/26 DESCRIPTION:Title: Lenses as algebras for a monad\nby Bryce Clarke (Macquari e University) as part of BIRS workshop : Tangent Categories and their Appl ications\n\n\nAbstract\nLenses are a family of mathematical structures use d in computer science to specify bidirectional transformations between sys tems. In many instances\, lenses can be understood as morphisms equipped w ith additional algebraic structure\, and admit a characterisation as algeb ras for a monad on a slice category. For example\, very well-behaved lense s between sets were shown to be algebras for a monad on Set / B\, while c- lenses between categories (better known as split opfibrations) are algebra s for a monad on Cat / B. Delta lenses are another kind of lens between ca tegories which generalise both of these previous examples\, however they h ave only been previously characterised as certain algebras for a semi-mona d. In this talk\, I will improve this result to show that delta lenses als o arise as algebras for a monad\, and discuss several interesting conseque nces of this characterisation.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Anders Kock (Aarhus\, Denmark) DTSTART:20210618T150000Z DTEND:20210618T152000Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/27 DESCRIPTION:Title: Barycentric calculus\, and the log-exp bijection\nby Ande rs Kock (Aarhus\, Denmark) as part of BIRS workshop : Tangent Categories a nd their Applications\n\n\nAbstract\nIn terms of synthetic differential ge ometry\, it makes sense to compare the infinitesimal structure of a space and of its tangent bundle. This hinges of the possibility to form certain affine combinations (barycentic calculus) of the algebra maps from A to B\ , where A and B are arbitrary commutative rings.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Kadri Ilker Berktav (Middle East Technical University\, Turkey) DTSTART:20210618T153000Z DTEND:20210618T155000Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/28 DESCRIPTION:Title: Higher structures in physics\nby Kadri Ilker Berktav (Mid dle East Technical University\, Turkey) as part of BIRS workshop : Tangent Categories and their Applications\n\n\nAbstract\nThis is a talk on higher structures in geometry and physics. We\, indeed\, intend to overview the basics of derived algebraic geometry and its essential role in encoding th e formal geometric aspects of certain moduli problems in physics. Througho ut the talk\, we always study objects with higher structures in a functori al perspective\, and we shall focus on algebraic local models for those st ructures. With this spirit\, we will investigate higher spaces and structu res in a variety of scenarios. In that respect\, we shall also mention som e of our works in this research direction.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/28/ END:VEVENT BEGIN:VEVENT SUMMARY:Rowan Poklewski-Koziell (University of Cape Town) DTSTART:20210618T160000Z DTEND:20210618T162000Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/29 DESCRIPTION:Title: Frobenius-Eilenberg-Moore objects in dagger 2-categories\ nby Rowan Poklewski-Koziell (University of Cape Town) as part of BIRS work shop : Tangent Categories and their Applications\n\n\nAbstract\nA Frobeniu s monad on a category is a monad-comonad pair whose multiplication and com ultiplication are related via the Frobenius law. Street has given several equivalent definitions of Frobenius monads. In particular\, they are those monads induced from ambidextrous adjunctions. On a dagger category\, much of this comes for free: every monad on a dagger category is equivalently a comonad\, and all adjunctions are ambidextrous. Heunen and Karvonen call a monad on a dagger category which satisfies the Frobenius law a dagger F robenius monad. They also define the appropriate notion of an algebra for such a monad\, and show that it captures quantum measurements and aspects of reversible computing. In this talk\, we will show that these definition s are exactly what is needed for a formal theory of dagger Frobenius monad s\, with the usual elements of Eilenberg-Moore object and completion of a 2-category under such objects having dagger counterparts. This may pave th e way for characterisations of categories of Frobenius objects in dagger m onoidal categories and generalisations of distributive laws of monads on d agger categories.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/29/ END:VEVENT BEGIN:VEVENT SUMMARY:Tarmo Uustalu (Reykjavik University) DTSTART:20210618T163000Z DTEND:20210618T165000Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/30 DESCRIPTION:Title: Monad-comonad interaction laws (co)algebraically\nby Tarm o Uustalu (Reykjavik University) as part of BIRS workshop : Tangent Catego ries and their Applications\n\n\nAbstract\nI will introduce monad-comonad interaction laws as mathematical objects to describe how an effectful comp utation (in the sense of functional programming) can run in an environment serving its requests. Such an interaction law is a natural transformation typed\n\n$T X \\times DY \\to R (X \\times Y)$\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/30/ END:VEVENT BEGIN:VEVENT SUMMARY:Nicolas Blanco (University of Birmingham) DTSTART:20210618T170000Z DTEND:20210618T172000Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/31 DESCRIPTION:Title: Bifibrations of polycategories and MLL\nby Nicolas Blanco (University of Birmingham) as part of BIRS workshop : Tangent Categories and their Applications\n\n\nAbstract\nPolycategories are structures genera lising categories and multicategories by letting both the domain and codom ain of the morphisms to be lists of objects. This provides an interesting framework to study models of classical multiplicative linear logic. In par ticular the interpretation of the connectives ise given by objects defined by universal properties in contrast to their interpretation in a *-autono mous category.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/31/ END:VEVENT BEGIN:VEVENT SUMMARY:Simona Paoli (Leicester University) DTSTART:20210618T173000Z DTEND:20210618T175000Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/32 DESCRIPTION:Title: Weakly globular double categories and weak units\nby Simo na Paoli (Leicester University) as part of BIRS workshop : Tangent Categor ies and their Applications\n\n\nAbstract\nWeakly globular double categorie s are a model of weak 2-categories based on the notion of weak globularity \, and they are known to be suitably equivalent to Tamsamani 2-categories. Fair 2-categories\, introduced by J. Kock\, model weak 2-categories with strictly associative compositions and weak unit laws. In this talk I will illustrate how to establish a direct comparison between weakly globular do uble categories and fair 2-categories and prove they are equivalent after localisation with respect to the 2-equivalences. This comparison sheds new light on weakly globular double categories as encoding a strictly associa tive\, though not strictly unital\, composition\, as well as the category of weak units via the weak globularity condition. \n\nReference: S. Paoli\ , Weakly globular double categories and weak units\, arXiv:2008.11180v1\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/32/ END:VEVENT BEGIN:VEVENT SUMMARY:Chad Nester (Union College) DTSTART:20210618T190000Z DTEND:20210618T192000Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/33 DESCRIPTION:Title: Concurrent Material Histories\nby Chad Nester (Union Coll ege) as part of BIRS workshop : Tangent Categories and their Applications\ n\n\nAbstract\nThe resource-theoretic interpretation of symmetric monoidal categories allows us to express pieces of material history as morphisms. In this talk we will see how to extend this to capture concurrent interact ion. \nSpecifically\, we will see that the resource-theoretic interpretati on extends to single object double categories with companion and conjoint structure\, and that in this setting material history may be decomposed in to interacting concurrent components. \nAs an example\, we will show how t ransition systems with boundary (spans of reflexive graphs) can be equippe d to generate material history in a compositional way as transitions unfol d. Some directions for future work will also be proposed.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/33/ END:VEVENT BEGIN:VEVENT SUMMARY:Cole Comfort (University of Oxford) DTSTART:20210618T193000Z DTEND:20210618T195000Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/34 DESCRIPTION:Title: A graphical calculus for Lagrangian relations\nby Cole Co mfort (University of Oxford) as part of BIRS workshop : Tangent Categories and their Applications\n\n\nAbstract\n
Symplectic vector spaces are the phase space of linear mechanical systems. The symplectic form describes\, for example\, the relation between position and momentum as well as curre nt and voltage. The category of linear Lagrangian relations between symple ctic vector spaces is a symmetric monoidal subcategory of relations which gives a semantics for the evolution -- and more generally linear constrain ts on the evolution -- of various physical systems.\n\n
We give a new pr esentation of the category of Lagrangian relations over an arbitrary field as a `doubled' category of linear relations. More precisely\, we show tha t it arises as a variation of Selinger's CPM construction applied to linea r relations\, where the covariant orthogonal complement functor plays of t he role of conjugation. Furthermore\, for linear relations over prime fiel ds\, this corresponds exactly to the CPM construction for a suitable choic e of dagger. We can furthermore extend this construction by a single affin e shift operator to obtain a category of affine Lagrangian relations. Usin g this new presentation\, we prove the equivalence of the prop of affine L agrangian relations with the prop of qudit stabilizer theory in odd prime dimensions. We hence obtain a unified graphical language for several dispa rate process theories\, including electrical circuits\, Spekkens' toy theo ry\, and odd-prime-dimensional stabilizer quantum circuits.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/34/ END:VEVENT BEGIN:VEVENT SUMMARY:Nuiok Dicaire (University of Edinburgh) DTSTART:20210618T200000Z DTEND:20210618T202000Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/35 DESCRIPTION:Title: Localization of monads via subunits\nby Nuiok Dicaire (Un iversity of Edinburgh) as part of BIRS workshop : Tangent Categories and t heir Applications\n\n\nAbstract\nGiven a “global” monad\, one wishes t o obtain “local” monads such that these locally behave like the global monad. In this talk\, I will provide an overview of how subunits can be u sed to provide a notion of localisation on monads. I will start by introdu cing subunits\, a special kind of subobject of the unit in a monoidal cate gory. Afterwards\, I will provide two equivalent ways of understanding the localisation of monads. The first involves a strength on subunits\, while the second relies on the formal theory of graded monads. I will also expl ain how to construct one from the other.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/35/ END:VEVENT BEGIN:VEVENT SUMMARY:Sacha Ikonicoff (University of Calgary) DTSTART:20210618T203000Z DTEND:20210618T205000Z DTSTAMP:20260404T041448Z UID:BIRS_21w5251/36 DESCRIPTION:Title: Divided power algebras with derivation\nby Sacha Ikonicof f (University of Calgary) as part of BIRS workshop : Tangent Categories an d their Applications\n\n\nAbstract\nClassical divided power algebras are c ommutative associative algebras endowed with `divided power' monomial oper ations. They were introduced by Cartan in the 1950's in the study of the h omology of Eilenberg-MacLane spaces\, and appear in several branches of ma thematics\, such as crystalline cohomology and deformation theory.\n \nIn this talk\, we will investigate divided power algebras with derivation\, a nd identify the most natural compatibility relation between a derivation a nd the divided power operations. The work of Keigher and Pritchard on form al divided power series (also called Hurwitz series) suggests a certain `p ower rule'. We will prove\, using the framework of operads\, that this pow er rule gives a reasonable definition for a divided power algebra with der ivation. We will extend this result to a more general notion of divided po wer algebras\, such as restricted Lie algebras\, with derivation.\n LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5251/36/ END:VEVENT END:VCALENDAR