Itaca Fest 2023

ItaCa Fest is an online webinar aimed to gather the community of ItaCa.

The seminar will be live on Zoom at this link. The time is: 3pm Italian time.

Here the list of seminars

• April 27, 2023
• May 24, 2023
• June 16, 2023
• September 28, 2023
• October 25, 2023
• November 23, 2023

April 27, 2023

Tobias Fritz

What is probability theory?

What is probability theory, and what should it be? I will argue that these are important questions, and that probability theory here is special in that these questions are not as meaningful when asked about other areas of mathematics. The goal of the talk is then to discuss these questions as well as a proposed partial answer with the audience. This partial answer is based on Markov categories and axioms for probability formulated in terms of Markov categories.

Elena Di Lavore

Evidential decision theory via partial Markov categories

I will present partial Markov categories. In the same 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 category whose normalisations can be computed in the original Markov category. Finally, we formalise Evidential Decision Theory in terms of partial Markov categories. This is recent joint work with Mario Román.

Davide Trotta

Gödel doctrines and Dialectica logical principles

In this talk, I will introduce the notion of Gödel doctrine, which is a doctrine categorically embodying both the logical principles of traditional Skolemization and the existence of a prenex normal form presentation for every formula, and I will explain how this notion is related to the Dialectica construction. In particular, building up from Hofstra’s earlier fibrational characterization of de Paiva’s categorical Dialectica construction, I will show that a doctrine is an instance of the Dialectica construction 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 from 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 Dialectica interpretation, namely Markov's principle and the principle of independence of premise.

May 24, 2023

Francesca Guffanti

A doctrinal view of logic

The aim of this talk is to offer an interpretation via doctrines of a classical result in first-order logic, i.e. Henkin’s Theorem (“Every consistent theory has a model”). The theorem is generalized in the language of implicational existential doctrines, focusing on the translation of some key steps in the original proof, such as adding constants to a language and axioms to a theory.

Matias Menni

Decidable objects and molecular toposes

Consider an extensive category with 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-cohesive topos over a Boolean base is molecular. Is every pre-cohesive topos molecular?

Pietro Freni

What should Strong Vector Spaces be?

Spaces of generalized power series have been important objects in asymptotic analysis and in the algebra and model theory of valued structures ever since the introduction of the first instances of them by Levi-Civita and Hahn. A key feature in this sort of structures is a notion of formal summability and often "natural" linear maps built in this context (such as derivations) are required to preserve this stronger form of linearity, whence they are called strongly linear. In the talk we will propose a framework for strong linearity: 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) with respect to an appropriate torsion theory. We will then give a brief description of a monoidal closed structure for ∑Vect and the relation ∑Vect has with another orthogonal subcategory of Ind(Vect^op) equivalent to the category of linearly topologized vector spaces that are colimits of linearly compact spaces. Finally, we will present some open questions in this setting.

June 16, 2023

Enrico Vitale

The completion under strong homotopy cokernels

For A a category with finite colimits, we show that the embedding of A into the category of arrows Arr(A) determined by the initial object 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 adjunctions between A and Arr(A).

Julia Ramos Gonzales

Bicategorical presentations of étendues

Étendues 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 one hand, the presentations in terms of étale groupoids are well-behaved bicategorically: Pronk showed in [2] that the 2-category of étendues is biequivalent to a bicategory of fractions of the 2-category of étale groupoids. On the other hand, left cancellative Grothendieck sites and Ehresmann sites, while enough to provide presentations at the level of the objects, turn out to be too restrictive in order to allow for a bicategory of fractions presentation of the whole 2-category of étendues.
In this talk we introduce a family of Grothendieck sites, the torsion-free generated Grothendieck sites, which contains the left cancellative ones and does allow to recover the 2-category of étendues as a bicategory of fractions. In addition, we introduce the family of generalized Ehresmann sites, a new family of presentations of étendues enlarging that of Ehresmann sites. In parallel with the bicategorical comparison between left cancellative Grothendieck sites and Ehresmann sites carried out by DeWolf and Pronk in [1], we study the connection between the torsion-free generated Grothendieck sites and the generalized Ehresmann sites with the final goal of proving that generalized Ehresmann sites also allow to recover étendues as a suitable bicategory of fractions.
This is joint work in progress with Darien DeWolf and Dorette Pronk.
[1] D. DeWolf and D. Pronk. A double categorical view on representations of etendues. Cahiers de Topologie et Géométrie Différentielle Catégoriques, LXI:3–56, 2020.
[2] D. A. Pronk. Etendues and stacks as bicategories of fractions. Compositio Math., 102(3):243–303, 1996.

Andrea Cappelletti

Protoadditive Functors and Pretorsion Theories in Multipointed Context

We explore a multipointed version of the definition of a protoadditive functor, in relation to pretorsion theories. We show that a pretorsion theory, whose reflector into the torsion-free part is protoadditive, gives rise to an admissible Galois structure which admits a simple characterization of central extensions. Moreover, multipointed pretorsion theories satisfying mild additional assumptions correspond to stable factorization systems. Interesting examples of such pretorsion theories can be found in MV-algebras, in Heyting algebras, and in the dual of two-valued elementary toposes. Joint work with Andrea Montoli.

September 28, 2023

Steve Awodey

Algebraic Type Theory

A type theoretic universe E —> U bears a certain algebraic structure resulting from the type-forming operations of unit type, identity type, dependent sum, and dependent product (as in [1]) which 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 properties. The general theory of such ML-algebras is then a proof-relevant version of the theory of Zermelo-Fraenkel algebras from the algebraic set theory of Joyal & Moerdijk [2]
[1] S. Awodey. Natural models of homotopy type theory. Math.Stru.Comp.Sci., 28(2), 2008.
[2] A. Joyal and I. Moerdijk, Algebraic Set Theory, Cambridge University Press, 1995.

Joshua Wrigley

Topological groupoids for classifying toposes

Grothendieck toposes, and by extension, logical theories, can be represented by topological structures. In [1], Butz and Moerdijk showed that every topos with enough points is equivalent to a topos of sheaves on an open topological groupoid. The next obvious question is: which topological groupoids represent a particular topos? This talk presents a model-theoretic terms characterisation of which open topological groupoids represent the classifying topos of a theory. Intuitively, this characterises which groupoids of models contain enough information to reconstruct the theory.
[1] C. Butz & I. Moerdijk, Representing topoi by topological groupoids. J. Pure Appl. Algebra 130 (1998), no. 3, 223–235.
[2] J. Wrigley, On topological groupoids that represent theories, arXiv:2306.16331 (2023).

October 25, 2023

Gabriele Lobbia

A skew approach to enrichment for Gray-categories

The category of Gray-categories does not admit a monoidal biclosed structure that models weak higher-dimensional transformations. In this talk, I will outline these problems and show how skew structures can provide a solution. In particular, I will describe closed skew monoidal structures on the category of Gray-categories capturing higher lax transformations and higher pseudo-transformations.
If time will allows it, we will give some intuition on the interaction between these skew monoidal structures and the model structure on Gray-Cat, and what categories enriched in these skew structures — the resulting semi-strict 4-categories — look like.
This is joint work with John Bourke and results are based on our paper of the same name.

Nicola Gambino

Monoidal bicategories, differential linear logic, and analytic functors

I 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.
[FGHW] M. Fiore, M. Hyland, N. Gambino and G. Winskel, The cartesian closed bicategory of generalised species of structures, Journal of the London Mathematical Society 77 (2) 2008, pp. 203-220.

Seerp R. Koudenburg

Formal Day convolution and low-dimensional monoidal fibrations

Let 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-algebras.
Taking K = Prof the augmented virtual double category of profunctors and T the ``free strict monoidal category''-monad the main result recovers 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 fibrations instead, the main result implies the ``monoidal Grothendieck equivalence'' of lax monoidal functors A → Set and monoidal discrete opfibrations with base A (a variation on a result of Moeller and Vasilakopoulou).
Moving up a dimension, given a 2-monoidal 2-category A the main result likewise implies the equivalence of lax 2-monoidal 2-functors A → Cat and 2-monoidal locally discrete split 2-opfibrations with base A. The main ingredient here is that (somewhat surprisingly) there exists an augmented virtual double category ldSp2Fib that accommodates the lax natural transformations required to define the formal Yoneda embedding induced by the Grothendieck equivalence for locally discrete split 2-opfibrations.
Time permitting I will report on work in progress on a similar equivalence for monoidal double split opfibrations (double fibrations in the sense of Cruttwell, Lambert, Pronk and Szyld).

November 23, 2023

Marco Volpe

Traces of dualizable categories and functoriality of the Becker-Gottlieb transfers

For 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 on, these wrong way maps have been defined more generally for continuous functions whose homotopy fibers are finitely dominated, and have been since referred to as the Becker-Gottlieb transfers. It has been a long standing open question whether these transfers behave well under composition, i.e. if they can be used to equip homology with a contravariant functoriality. Previous attempts to prove such functoriality contained unfixable mistakes (see [2], [3]).
In this talk, we will approach the transfers from the perspective of sheaf theory. We will recall the notion of a locally contractible geometric morphism, and then define a Becker-Gottlieb transfer associated to any proper, locally contractible map between locally contractible 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 transfers". Functoriality of the Becker-Gottlieb transfers is then obtained by applying topological Hochschild homology to the categorified transfers.
This is a joint work with Maxime Ramzi and Sebastian Wolf.

[1] James Becker, Daniel Gottlieb, The transfer map and fiber bundles, Topology , 14 (1975) (pdf, doi:10.1016/0040-9383(75)90029-4)
[2] Rune Haugseng, The Becker-Gottlieb Transfer Is Functorial (arXiv:1310.6321)
[3] John Klein, Cary Malkiewich, The transfer is functorial

Simon Henry

Pro-completion of the category of sets and prodiscrete spaces

It is a well-known result that the pro-completion of the category of finite sets is equivalent to the category of profinite spaces. But there seems to be no similar description of the pro-completion of the category of all sets - it does not corresponds to pro-discrete spaces. A first obstruction for this is 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 still not the pro-completion of the category of sets.

After reviewing briefly the claims above, I will clarify this gap by showing that the category of prodiscrete (or strongly zero-dimensional) locales is the "extensive procompletion" of the category of sets. That is, it is the minimal completion of the category of sets as an (infinitarilly) extensive category. To put it another way, the category of strongly zero-dimensional locales is the 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 limit preservation condition.

Raffael Stenzel

The Comparison Lemma in higher topos theory

In ordinary topos theory, the Comparison Lemma states that every suitably dense embedding of a small category C into a potentially large site (D,K) induces the structure of a site (C,J) such that the given embedding induces an equivalence of associated sheaf theories (see e.g. [1, Theorem B.2.2.3]). The Lemma is useful in various situations; it for instance can be used to prove that the canonical topology on an infinitary pretopos with a small generating set has a small presentation ([1, Theorem B.2.2.7]). In this talk, I show that the Comparison Lemma fails in the usual context of Grothendieck topologies on infinity-categories. This in particular means that Lurie's canonical Grothendieck topologies ([2, Section 6.2.4]) generally fail to induce an equivalence of sheaf theories even in best case scenarios. I state a version of the Comparison Lemma which does hold instead, and discuss some of the arising questions and consequences.

[1] P.T. Johnstone - Sketches of an Elephant Volume 1, Clarendon Press, Oxford , 2002.
[2] J. Lurie - Higher Topos Theory, Princeton University Press, Princeton, 2009.