BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:T. Fritz (University of Innsbruck) DTSTART:20230427T130000Z DTEND:20230427T133000Z DTSTAMP:20260404T111324Z UID:ItaCa-Fest-2023/1 DESCRIPTION:Title: What is probability theory?\nby T. Fritz (University of Innsbruck) as part of ItaCa Fest 2023\n\n\nAbstract\nWhat is probability theory\, and what should it be? I will argue that these are important ques tions\, and that probability theory here is special in that these question s are not as meaningful when asked about other areas of mathematics. The g oal of the talk is then to discuss these questions as well as a proposed p artial answer with the audience. This partial answer is based on Markov ca tegories and axioms for probability formulated in terms of Markov categori es.\n LOCATION:https://stable.researchseminars.org/talk/ItaCa-Fest-2023/1/ END:VEVENT BEGIN:VEVENT SUMMARY:E. Di Lavore (Tallinn University of Technology) DTSTART:20230427T134000Z DTEND:20230427T141000Z DTSTAMP:20260404T111324Z UID:ItaCa-Fest-2023/2 DESCRIPTION:Title: Evidential decision theory via partial Markov categories\nby E. Di Lavore (Tallinn University of Technology) as part of ItaCa Fes t 2023\n\n\nAbstract\nI will present partial Markov categories. In the sam e way that Markov categories encode stochastic processes\, partial Markov categories encode stochastic processes with constraints\, observations and updates. In particular\, we prove a synthetic Bayes theorem\, and we use it to define a syntactic partial theory of observations on any Markov cate gory whose normalisations can be computed in the original Markov category. Finally\, we formalise Evidential Decision Theory in terms of partial Mar kov categories. This is recent joint work with Mario Román.\n LOCATION:https://stable.researchseminars.org/talk/ItaCa-Fest-2023/2/ END:VEVENT BEGIN:VEVENT SUMMARY:D. Trotta (Università di Pisa) DTSTART:20230427T142000Z DTEND:20230427T145000Z DTSTAMP:20260404T111324Z UID:ItaCa-Fest-2023/3 DESCRIPTION:Title: Gödel doctrines and Dialectica logical principles\nby D. Trotta (Università di Pisa) as part of ItaCa Fest 2023\n\n\nAbstract\n In this talk\, I will introduce the notion of Gödel doctrine\, which is a doctrine categorically embodying both the logical principles of tradition al Skolemization and the existence of a prenex normal form presentation fo r every formula\, and I will explain how this notion is related to the Dia lectica construction. In particular\, building up from Hofstra’s earlier fibrational characterization of de Paiva’s categorical Dialectica const ruction\, I will show that a doctrine is an instance of the Dialectica con struction if and only if it is a Gödel doctrine. This result establishes an intrinsic presentation of the Dialectica doctrine\, contributing to the understanding of the Dialectica construction itself and its properties fr om a logical perspective. Finally\, I will show how this notion allows us to provide a simple presentation and an explanation in terms of universal properties of the two crucial logical principles involved in the Dialectic a interpretation\, namely Markov's principle and the principle of independ ence of premise.\n LOCATION:https://stable.researchseminars.org/talk/ItaCa-Fest-2023/3/ END:VEVENT BEGIN:VEVENT SUMMARY:F. Guffanti (Università degli Studi di Milano) DTSTART:20230524T130000Z DTEND:20230524T133000Z DTSTAMP:20260404T111324Z UID:ItaCa-Fest-2023/4 DESCRIPTION:Title: A doctrinal view of logic\nby F. Guffanti (Università degli Studi di Milano) as part of ItaCa Fest 2023\n\n\nAbstract\nThe aim o f this talk is to offer an interpretation via doctrines of a classical res ult in first-order logic\, i.e. Henkin’s Theorem (“Every consistent th eory has a model”). The theorem is generalized in the language of implic ational existential doctrines\, focusing on the translation of some key st eps in the original proof\, such as adding constants to a language and axi oms to a theory.\n LOCATION:https://stable.researchseminars.org/talk/ItaCa-Fest-2023/4/ END:VEVENT BEGIN:VEVENT SUMMARY:M. Menni DTSTART:20230524T134000Z DTEND:20230524T141000Z DTSTAMP:20260404T111324Z UID:ItaCa-Fest-2023/5 DESCRIPTION:Title: Decidable objects and molecular toposes\nby M. Menni as part of ItaCa Fest 2023\n\n\nAbstract\nConsider an extensive category wit h finite products and its full subcategory of decidable objects. Assuming that this inclusion has a finite-product preserving left adjoint then the adjunction has stable units. It follows as a corollary that every pre-cohe sive topos over a Boolean base is molecular. Is every pre-cohesive topos m olecular?\n LOCATION:https://stable.researchseminars.org/talk/ItaCa-Fest-2023/5/ END:VEVENT BEGIN:VEVENT SUMMARY:P. Freni (University of Leeds) DTSTART:20230524T142000Z DTEND:20230524T145000Z DTSTAMP:20260404T111324Z UID:ItaCa-Fest-2023/6 DESCRIPTION:Title: What should Strong Vector Spaces be?\nby P. Freni (Univ ersity of Leeds) as part of ItaCa Fest 2023\n\n\nAbstract\nSpaces of gener alized power series have been important objects in asymptotic analysis and in the algebra and model theory of valued structures ever since the intro duction of the first instances of them by Levi-Civita and Hahn. A key feat ure in this sort of structures is a notion of formal summability and often "natural" linear maps built in this context (such as derivations) are req uired to preserve this stronger form of linearity\, whence they are called strongly linear. In the talk we will propose a framework for strong linea rity: we will argue about a notion of reasonable category of strong vector spaces (r.c.s.v.) generalizing the usual setting for strong linearity and show that up to equivalence there is a universal locally small r.c.s.v. ∑Vect and it can be construed as a torsion free part of Ind(Vect^op) wit h respect to an appropriate torsion theory. We will then give a brief desc ription of a monoidal closed structure for ∑Vect and the relation ∑Vec t has with another orthogonal subcategory of Ind(Vect^op) equivalent to th e category of linearly topologized vector spaces that are colimits of line arly compact spaces. Finally\, we will present some open questions in this setting.\n LOCATION:https://stable.researchseminars.org/talk/ItaCa-Fest-2023/6/ END:VEVENT BEGIN:VEVENT SUMMARY:E. Vitale (Université Catholique de Louvain) DTSTART:20230616T130000Z DTEND:20230616T133000Z DTSTAMP:20260404T111324Z UID:ItaCa-Fest-2023/7 DESCRIPTION:Title: The completion under strong homotopy cokernels\nby E. V itale (Université Catholique de Louvain) as part of ItaCa Fest 2023\n\n\n Abstract\nFor A a category with finite colimits\, we show that the embeddi ng of A into the category of arrows Arr(A) determined by the initial objec t is the completion of A under strong homotopy cokernels. The nullhomotopy structure of Arr(A) is the usual one induced by the canonical string of a djunctions between A and Arr(A).\n LOCATION:https://stable.researchseminars.org/talk/ItaCa-Fest-2023/7/ END:VEVENT BEGIN:VEVENT SUMMARY:A. Cappelletti (Università degli Studi di Milano) DTSTART:20230616T142000Z DTEND:20230616T145000Z DTSTAMP:20260404T111324Z UID:ItaCa-Fest-2023/8 DESCRIPTION:Title: Protoadditive Functors and Pretorsion Theories in Multipoin ted Context\nby A. Cappelletti (Università degli Studi di Milano) as part of ItaCa Fest 2023\n\n\nAbstract\nWe explore a multipointed version o f the definition of a protoadditive functor\, in relation to pretorsion th eories. We show that a pretorsion theory\, whose reflector into the torsio n-free part is protoadditive\, gives rise to an admissible Galois structur e which admits a simple characterization of central extensions. Moreover\, multipointed pretorsion theories satisfying mild additional assumptions c orrespond to stable factorization systems. Interesting examples of such pr etorsion theories can be found in MV-algebras\, in Heyting algebras\, and in the dual of two-valued elementary toposes. Joint work with Andrea Monto li.\n LOCATION:https://stable.researchseminars.org/talk/ItaCa-Fest-2023/8/ END:VEVENT BEGIN:VEVENT SUMMARY:S. Awodey DTSTART:20230928T130000Z DTEND:20230928T133000Z DTSTAMP:20260404T111324Z UID:ItaCa-Fest-2023/9 DESCRIPTION:Title: Algebraic Type Theory\nby S. Awodey as part of ItaCa Fe st 2023\n\n\nAbstract\nA type theoretic universe E —> U bears a certain algebraic structure resulting from the type-forming operations of unit typ e\, identity type\, dependent sum\, and dependent product (as in [1]) whic h may be generalized to form the concept of a “Martin-Löf algebra”. A free ML-algebra is then a model of type theory\, perhaps with special pro perties. The general theory of such ML-algebras is then a proof-relevant v ersion of the theory of Zermelo-Fraenkel algebras from the algebraic set t heory of Joyal & Moerdijk [2]. \n\n[1] S. Awodey. Natural models of homoto py type theory. Math.Stru.Comp.Sci.\, 28(2)\, 2008.\n\n[2] A. Joyal and I. Moerdijk\, Algebraic Set Theory\, Cambridge University Press\, 1995.\n LOCATION:https://stable.researchseminars.org/talk/ItaCa-Fest-2023/9/ END:VEVENT BEGIN:VEVENT SUMMARY:J. Wrigley DTSTART:20230928T134000Z DTEND:20230928T141000Z DTSTAMP:20260404T111324Z UID:ItaCa-Fest-2023/11 DESCRIPTION:Title: Topological groupoids for classifying toposes\nby J. W rigley as part of ItaCa Fest 2023\n\n\nAbstract\nGrothendieck toposes\, an d by extension\, logical theories\, can be represented by topological stru ctures. In [1]\, Butz and Moerdijk showed that every topos with enough poi nts is equivalent to a topos of sheaves on an open topological groupoid. T he next obvious question is: which topological groupoids represent a parti cular topos? This talk presents a model-theoretic terms characterisation o f which open topological groupoids represent the classifying topos of a th eory. Intuitively\, this characterises which groupoids of models contain e nough information to reconstruct the theory.\n\n[1] C. Butz & I. Moerdijk\ , Representing topoi by topological groupoids. J. Pure Appl. Algebra 130 ( 1998)\, no. 3\, 223–235.\n\n[2] J. Wrigley\, On topological groupoids th at represent theories\, arXiv:2306.16331 (2023).\n LOCATION:https://stable.researchseminars.org/talk/ItaCa-Fest-2023/11/ END:VEVENT BEGIN:VEVENT SUMMARY:G. Lobbia DTSTART:20231025T130000Z DTEND:20231025T133000Z DTSTAMP:20260404T111324Z UID:ItaCa-Fest-2023/12 DESCRIPTION:Title: A skew approach to enrichment for Gray-categories\nby G. Lobbia as part of ItaCa Fest 2023\n\n\nAbstract\nThe category of Gray-c ategories does not admit a monoidal biclosed structure that models weak hi gher-dimensional transformations. In this talk\, I will outline these prob lems and show how skew structures can provide a solution. In particular\, I will describe closed skew monoidal structures on the category of Gray-ca tegories capturing higher lax transformations and higher pseudo-transforma tions.\n\nIf time will allows it\, we will give some intuition on the inte raction between these skew monoidal structures and the model structure on Gray-Cat\, and what categories enriched in these skew structures — the r esulting semi-strict 4-categories — look like.\n\nThis is joint work wit h John Bourke and results are based on our paper of the same name.\n LOCATION:https://stable.researchseminars.org/talk/ItaCa-Fest-2023/12/ END:VEVENT BEGIN:VEVENT SUMMARY:S. R. Koudenburg DTSTART:20231025T142000Z DTEND:20231025T145000Z DTSTAMP:20260404T111324Z UID:ItaCa-Fest-2023/13 DESCRIPTION:Title: Formal Day convolution and low-dimensional monoidal fibrat ions\nby S. R. Koudenburg as part of ItaCa Fest 2023\n\n\nAbstract\nLe t T be a monad on an augmented virtual double category K. The main result of this talk describes conditions ensuring that a formal Yoneda embedding y: A → P in K can be lifted along the forgetful functor U: Lax-T-Alg → K\, where Lax-T-Alg is the augmented virtual double category of lax T-alg ebras.\n\nTaking K = Prof the augmented virtual double category of profunc tors and T the ``free strict monoidal category''-monad the main result rec overs the Day convolution monoidal structure on the category of presheaves P = Set$^{A^{op}}$ on a monoidal category A. Taking the same monad on the augmented virtual double category K = dFib of two-sided discrete fibratio ns instead\, the main result implies the ``monoidal Grothendieck equivalen ce'' of lax monoidal functors A → Set and monoidal discrete opfibrations with base A (a variation on a result of Moeller and Vasilakopoulou).\n\nM oving up a dimension\, given a 2-monoidal 2-category A the main result lik ewise implies the equivalence of lax 2-monoidal 2-functors A → Cat and 2 -monoidal locally discrete split 2-opfibrations with base A. The main ingr edient here is that (somewhat surprisingly) there exists an augmented virt ual double category ldSp2Fib that accommodates the lax natural transformat ions required to define the formal Yoneda embedding induced by the Grothen dieck equivalence for locally discrete split 2-opfibrations.\n\nTime permi tting I will report on work in progress on a similar equivalence for monoi dal double split opfibrations (double fibrations in the sense of Cruttwell \, Lambert\, Pronk and Szyld).\n LOCATION:https://stable.researchseminars.org/talk/ItaCa-Fest-2023/13/ END:VEVENT BEGIN:VEVENT SUMMARY:M. Volpe DTSTART:20231123T140000Z DTEND:20231123T143000Z DTSTAMP:20260404T111324Z UID:ItaCa-Fest-2023/14 DESCRIPTION:Title: Traces of dualizable categories and functoriality of the B ecker-Gottlieb transfers\nby M. Volpe as part of ItaCa Fest 2023\n\n\n Abstract\nFor any fiber bundle with compact smooth manifold fiber X ⟶ Y\ , Becker and Gottlieb have defined in [1] a "wrong way" map S[Y] ⟶ S[X] at the level of homology with coefficients in the sphere spectrum. Later o n\, these wrong way maps have been defined more generally for continuous f unctions whose homotopy fibers are finitely dominated\, and have been sinc e referred to as the Becker-Gottlieb transfers. It has been a long standin g open question whether these transfers behave well under composition\, i. e. if they can be used to equip homology with a contravariant functorialit y. Previous attempts to prove such functoriality contained unfixable mista kes (see [2]\, [3]).\n\nIn this talk\, we will approach the transfers from the perspective of sheaf theory. We will recall the notion of a locally c ontractible geometric morphism\, and then define a Becker-Gottlieb transfe r associated to any proper\, locally contractible map between locally cont ractible and locally compact Hausdorff spaces. We will then use techniques coming from recent work of Efimov on localizing invariants and dualizable stable infinity-categories to construct fully functorial "categorified tr ansfers". Functoriality of the Becker-Gottlieb transfers is then obtained by applying topological Hochschild homology to the categorified transfers. \nThis is a joint work with Maxime Ramzi and Sebastian Wolf.\n\n[1] James Becker\, Daniel Gottlieb\, The transfer map and fiber bundles\, Topology \ , 14 (1975) (pdf\, doi:10.1016/0040-9383(75)90029-4)\n\n[2] Rune Haugseng\ , The Becker-Gottlieb Transfer Is Functorial (arXiv:1310.6321)\n\n[3] John Klein\, Cary Malkiewich\, The transfer is functorial\n LOCATION:https://stable.researchseminars.org/talk/ItaCa-Fest-2023/14/ END:VEVENT BEGIN:VEVENT SUMMARY:S. Henry DTSTART:20231123T144000Z DTEND:20231123T151000Z DTSTAMP:20260404T111324Z UID:ItaCa-Fest-2023/15 DESCRIPTION:Title: Pro-completion of the category of sets and prodiscrete spa ces\nby S. Henry as part of ItaCa Fest 2023\n\n\nAbstract\nIt is a wel l-known result that the pro-completion of the category of finite sets is e quivalent to the category of profinite spaces. But there seems to be no si milar description of the pro-completion of the category of all sets - it d oes not corresponds to pro-discrete spaces. A first obstruction for this i s that many pro-sets give rise to "pointfree" spaces (locales)\, so at the very least we need to look at this question in terms of locales. But this is far from being enough\, and the category of prodiscrete locale is stil l not the pro-completion of the category of sets.\n\nAfter reviewing brief ly the claims above\, I will clarify this gap by showing that the category of prodiscrete (or strongly zero-dimensional) locales is the "extensive p rocompletion" of the category of sets. That is\, it is the minimal complet ion of the category of sets as an (infinitarilly) extensive category. To p ut it another way\, the category of strongly zero-dimensional locales is t he initial complete and infinitarilly extensive category. This involves a characterization of prodiscrete locales as "special" pro-sets that satisfy a local version of extensivity expressed which can be expressed as a limi t preservation condition.\n LOCATION:https://stable.researchseminars.org/talk/ItaCa-Fest-2023/15/ END:VEVENT BEGIN:VEVENT SUMMARY:R.Stenzel DTSTART:20231123T152000Z DTEND:20231123T155000Z DTSTAMP:20260404T111324Z UID:ItaCa-Fest-2023/16 DESCRIPTION:Title: Homotopy-coherent Category Theory à la Benabou\nby R. Stenzel as part of ItaCa Fest 2023\n\n\nAbstract\nIn [1] Benabou argued th at the theory of fibered categories ought to be understood as a natural fr amework for the development of formal categorical logic. At the center of his paper are various instances of categorical comprehension schemes\, eac h of which expresses that some relative categorical structure associated t o a given fibered B-category E can be internalized in B. Which particular instances hold\, and how one instance relates to another\, depends not onl y on the base B (together with some given B-category E)\, but also on the meta-theory we consider the base B to be enriched over in the first place. This becomes evident when one defines the notion of comprehension (over a n ordinary or more generally over an ∞-categorical base B) with respect to the larger ambient ∞-category of spaces rather than that of sets\, se e [2]. For instance\, ∞-category theory is univalent (in the sense of Vo evodsky)\, while ordinary category theory is not. This has non-trivial imp lications regarding both the structural theory of comprehension schemes (o ver a fixed ∞-categorical base B) as well as the validity of particular instances of such. In this talk\, I want to discuss a few of the arising d ifferences between the 1-categorical and ∞-categorical theories.\n\n[1] J. Benabou - Fibered Categories and the foundations of naive category theo ry\, The Journal of Symbolic Logic\, Volume 50\, Number 1\, 1985.\n\n[2] R . Stenzel - (∞\,1)-Categorical comprehension schemes\, arxiv: 2010.09663 \, 2020.\n LOCATION:https://stable.researchseminars.org/talk/ItaCa-Fest-2023/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Julia Ramos Gonzales (Université catholique de Louvain) DTSTART:20230616T134000Z DTEND:20230616T141000Z DTSTAMP:20260404T111324Z UID:ItaCa-Fest-2023/17 DESCRIPTION:Title: Bicategorical presentations of étendues\nby Julia Ram os Gonzales (Université catholique de Louvain) as part of ItaCa Fest 2023 \n\n\nAbstract\nÉtendues are the Grothendieck topoi that “locally look like a locale”. They are known to admit presentations in terms of étale groupoids\, left cancellative Grothendieck sites or Ehresmann sites. On o ne hand\, the presentations in terms of étale groupoids are well-behaved bicategorically: Pronk showed in [2] that the 2-category of étendues is b iequivalent to a bicategory of fractions of the 2-category of étale group oids. On the other hand\, left cancellative Grothendieck sites and Ehresma nn sites\, while enough to provide presentations at the level of the objec ts\, turn out to be too restrictive in order to allow for a bicategory of fractions presentation of the whole 2-category of étendues.\nIn this talk we introduce a family of Grothendieck sites\, the torsion-free generated Grothendieck sites\, which contains the left cancellative ones and does al low to recover the 2-category of étendues as a bicategory of fractions. I n addition\, we introduce the family of generalized Ehresmann sites\, a ne w family of presentations of étendues enlarging that of Ehresmann sites. In parallel with the bicategorical comparison between left cancellative Gr othendieck sites and Ehresmann sites carried out by DeWolf and Pronk in [1 ]\, we study the connection between the torsion-free generated Grothendiec k sites and the generalized Ehresmann sites with the final goal of proving that generalized Ehresmann sites also allow to recover étendues as a sui table bicategory of fractions.\nThis is joint work in progress with Darien DeWolf and Dorette Pronk.\n[1] D. DeWolf and D. Pronk. A double categoric al view on representations of etendues. Cahiers de Topologie et Géométri e Différentielle Catégoriques\, LXI:3–56\, 2020.\n[2] D. A. Pronk. Ete ndues and stacks as bicategories of fractions. Compositio Math.\, 102(3):2 43–303\, 1996.\n LOCATION:https://stable.researchseminars.org/talk/ItaCa-Fest-2023/17/ END:VEVENT BEGIN:VEVENT SUMMARY:N. Gambino DTSTART:20231025T134000Z DTEND:20231025T141000Z DTSTAMP:20260404T111324Z UID:ItaCa-Fest-2023/18 DESCRIPTION:Title: Monoidal bicategories\, differential linear logic\, and an alytic functors\nby N. Gambino as part of ItaCa Fest 2023\n\n\nAbstrac t\nI will explain how the bicategory of analytic functors\, introduced in [FGHW]\, can be seen as a bicategorical model of differential linear logic . This is joint work in progress with Marcelo Fiore and Martin Hyland.\n\n [FGHW] M. Fiore\, M. Hyland\, N. Gambino and G. Winskel\, The cartesian cl osed bicategory of generalised species of structures\, Journal of the Lond on Mathematical Society 77 (2) 2008\, pp. 203-220.\n LOCATION:https://stable.researchseminars.org/talk/ItaCa-Fest-2023/18/ END:VEVENT END:VCALENDAR