Itaca Fest 2024

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 10, 2024
• May 7, 2024
• June 5, 2024
• September 25, 2024
• October 22, 2024
• November 20, 2024

April 10, 2024

Pierre-Alain Jacqmin

Surjection-like classes of morphisms

Many classes of epimorphisms are considered in the literature with the aim of generalizing surjective functions from the category Set of sets to an arbitrary category. However, some of them fail to have specific desirable properties. In this talk, we are interested in classes of morphisms which interact with finite limits as surjections do in Set. More precisely, we study classes of morphisms in finitely complete categories which admit a "good" embedding in a presheaf category. By good embedding, we mean a functor which preserves and reflects finite limits and the classes of morphisms involved. We will examine both the conservative faithful and the fully faithful cases. Our main result is a complete characterization of those classes of morphisms via simple and well-known properties.

Manuel Mancini

On the representability of actions of non-associative algebras

It is well known that in the semi-abelian category Grp of groups, internal actions are represented by automorphisms. This means that the category Grp is action representable and the actor of a group X is the group Aut(X). The notion of action representable category has proven to be quite restrictive: for instance, if a non-abelian variety of non-associative algebras, over an infinite field of characteristic different from two, is action representable, then it is the category of Lie algebras. More recently G. Janelidze introduced the notion of weakly action representable category, which includes a wider class of categories.
In this talk we show that for an algebraically coherent variety of algebras and an object X of it, it is always possible to construct a partial algebra E(X), called external weak actor of X, which allows us to describe internal actions on X. Moreover, we show that the existence of a weak representation is connected to the amalgamation property and we give an application of the construction of the external weak actor in the context of varieties of unitary algebras.

Alan Cigoli

From Yoneda's additive regular spans to fibred cartesian monoidal opfibrations

It is well known that group cohomology can be interpreted in terms of equivalence classes of crossed extensions, the abelian group structure being given by the so-called Baer sums. By analogy, an intrinsic definition of cohomology in strongly semi-abelian categories, or more generally in exact Mal'tsev categories (Bourn-Rodelo), is given. In this talk, I will explain how Baer sums can be formally derived from the fibred/cofibred nature of the category of all crossed extensions of a given length. This point of view turns out to be very close to Yoneda's theory of Ext groups. We will see how his notion of additive regular span is actually an instance of fibred cartesian monoidal opfibration. Time permitting, I will give some hint on how this formal point of view can be carried on to a 2-dimensional level, thus giving a notion of cohomology 2-group.
(based on joint works with S. Mantovani, G. Metere and E.M. Vitale

May 7, 2024

Sebastian Wolf

Higher Category Theory Internal to an Infinity Topos

The goal of this talk will be to give a brief introduction to the theory of higher categories internal to an infinity-topos, developed in joint work with Louis Martini. I will also indicate why such a theory is useful to get a better understanding of the geometry of infinity topoi. If time permits, I will conclude by explaining how one can use this language to give a characterization of proper morphism of infinity topoi in the sense of Lurie.

Franco Rota

Exceptional collections and pseudolattices in mirror symmetry

In the 1990s, theoretical physicists correctly predicted curve counts in an algebraic variety (a quintic threefold X) inferring them from a “mirror variety” Y. This was the start of mirror symmetry - a field of algebraic geometry that investigates how to construct mirrors and how to make the duality precise. A modern incarnation of the theory is the homological mirror symmery (HMS) conjecture by Kontsevich, which states that the duality first observed geometrically reflects an equivalence between a category built from X and one obtained from Y.

Describing and motivating some of the structures carried by these categories, I’ll briefly mention how to interpret the HMS equivalence as a quasi-isomorphism of Ainfty-algebras, and then elaborate on necessary conditions for the equivalence, which rephrase the question into multilinear algebra.

Viktoriya Ozornova

Equivalences in higher categories

There are different notions of ‘sameness’ arising in mathematics. The first one we usually encounter is equality of elements in a set. In our ‘daily life’, we are used to identify isomorphic objects, and we are secretly doing so in our favorite category. For categories themselves, we look for equivalences between those. But when should we consider two 2-categories to be ‘the same’? And how does the pattern continue? This talk is based upon joint work with Amar Hadzihasanovich, Félix Loubaton and Martina Rovelli.

June 5, 2024

Luigi Santocanale

Complete congruences of completely distributive lattices

All the binomial lattices embed into the quantale Q(I) of sup-preserving endomaps of the unit interval. Elements of these lattices can be seen as monotone paths from (0,0) to (1,1), discrete paths for the binomial lattices, continuous paths for Q(I). We aim at extending a natural geometric interpretation of lattice congruences of binomial lattices to congruences of Q(I). This is, in particular, a completely distributive lattice. Relying on Lawson-Hoffmann duality, we characterise those maps between continuous domains that give rise to complete maps between completely distributive lattices. This allows to describe the complete congruences of an arbitrary completely distributive lattice by means of an interior operator on the collection of the closed sets of an associated topological space. In particular, we show that these congruences form a frame. We study this frame for the unit interval lattice, arguing that this frame is not a Boolean algebra, nor it is a (co)spatial. For the quantale Q(I), we give a geometrical interpretation of these congruences by means of directed homotopies.

Jonathan Weinberger

The dependent Gödel fibration

Gödel‘s Dialectica proof interpretation from the 1950s was used as a tool for consistency proofs. In the late 80s, de Paiva introduced several categorified version of it, leading to notions of Dialectica categories. These, in turn, have later been generalized to the level of fibered categories. We present a characterization of Dialectica fibrations via the notion of Gödel fibration, generalizing earlier work by Spadetto—Trotta—de Paiva. This is joint work with Davide Trotta and Valeria de Paiva.

Luca Spada

2-Weil 2-rigs

Among commutative unital semirings (rigs, for short), let us call 2-Weil the ones that have a unique homomorphism into the distributive lattice 2. As 2 is the initial algebra in the category of additively idempotent rigs (2-rigs, for short), 2-Weil 2-rigs can be thought of as coordinate algebras of spaces with a single point. I will show how to characterize 2-Weil rigs as those that have unique saturated prime ideal and will provide an axiomatization thereof in geometric logic. Further we will see that the category of 2-Weil 2-rigs is a co-reflective full subcategory of the category of 2-rigs.

September 25, 2024

Dario Stein

Random Variables and Categories of Abstract Sample Spaces

Two high-level "pictures" of probability theory have emerged: one that takes as central the notion of random variable, and one that focuses on channels and distributions (Markov kernels).

While the channel-based picture has been captured and widely generalized using the notion of Markov category, the categorical analogue of the random variable picture is less clear. I will discuss the conceptual interplay between the two pictures: A crucial step is to understand the category of sample spaces associated to a given Markov category. This construction gives rise to a host of well-known examples. Building on the work of Simpson, we can describe random variables in the sheaf topos over those sample spaces.

Danel Ahman

Comodule Representations of Second-Order Functionals

In information-theoretic terms, a map is continuous when a finite amount of information about the input suffices for computing a finite amount of information about the output. Already Brouwer observed that this allows one to represent a continuous functional from sequences to numbers with a certain well-founded question-answer tree.

In type theory, a second-order functional is a (dependently typed) map

F : (∏(a : A) . P a) → (∏(b : B) . Q b).

Its continuity is once again witnessed by (B-many) well-founded trees whose nodes are “questions” a : A, the branches are indexed by “answers” p : P a, and the leaves are “results” Q b. In this work, we observe that such tree representations can be expressed in purely category-theoretic terms, using the notion of right T-comodules for the monad T of well-founded trees on the category of containers. A tree representation for F is then just a Kleisli map for the monad T.

Doing so exposes a rich underlying structure, and immediately suggests generalisations: any right T-comodule for any monad T on containers gives rise to a representation theorem for second-order functionals. We give several examples of these, ranging from finitely supported functionals, to functionals that can query their input just once (or sometimes not at all), to functionals that can additionally interact with their environment, to partial functionals, to observing that any functional can be trivially represented by itself.

This is joint work with Andrej Bauer from the University of Ljubljana.

Matthew Di Meglio

Abstraction of contraction

The theory of contractions on a Hilbert space plays an important role in modern functional analysis. It is built upon Sz.-Nagy's unitary dilation theorem, which says that every contraction on a Hilbert space admits a minimal unitary dilation (a unitary dilation of a contraction T: X → X is a unitary U: Y → Y on a Hilbert space Y containing X via an isometry M: X → Y such that T = M*UM). This talk is about an abstraction of the notion of contraction to suitably nice *-categories, and will build to a category-theoretic proof of a variant of Sz.-Nagy's theorem.

October 22, 2024

John Bourke

Bicategorical enrichment in algebra

In category theory, sometimes one does not wish to work with categories per se but instead categories over a fixed base. Such concrete categories can be viewed as categories enriched in a quantoloid, a certain bicategory. Garner showed this perspective is illuminating, using it to characterise topological categories as bicategory-enriched categories which are total.

In this talk, I will explain how the same enrichment is useful in algebra, where we also sometimes work over a fixed base. We will use the bicategorically-enriched perspective to show that Eilenberg-Moore categories of monads are free cocompletions of their Kleisli categories, which is false from the traditional point of view, and use this to give a nice proof of Beck's monadicity theorem. This is a report on ongoing work with Soichiro Fujii.

Giacomo Tendas

Regular theories from the enriched point of view

In logic, regular theories are those whose axioms are built from atomic formulas using conjunctions and existential quantifiers. The categories of models of such theories have been widely studied and characterised in purely category theoretical terms through the notion of injectivity class and through certain closure properties, that I will recall during the talk.

When moving to the context of enriched category theory, a corresponding notion of "enriched injectivity class" has been studied by several authors, but no enriched notion of regular logic was considered in the literature before. The aim of this talk, which is based on joint work with Rosicky, is to fill this gap by introducing a version of "enriched regular logic" that interacts well with the category theoretical counterparts mentioned above. I will also explain how this is related to the internal logic of a topos, and that internal to Banach and metric spaces.

Luca Mesiti

Towards elementary 2-toposes

In this talk we will discuss which axioms we should require for a good notion of 2-categorical elementary topos. 2-dimensional elementary topos theory has originated with the work of Weber, who proposed to upgrade subobject classifiers to discrete opfibration classifiers. In the archetypal case of Cat, the discrete opfibration classifier is exhibited by the Grothendieck construction, suggesting that we can think of 2-dimensional classifiers as internal Grothendieck constructions in a 2-category. The theory of elementary 2-toposes has then been further developed in my PhD thesis, where I proposed a stronger better-behaved notion of discrete opfibration classifier called good 2-classifier. We will see that a powerful theorem of reduction of the study of 2-dimensional classifiers to dense generators provides a good 2-classifier in the 2-category of stacks over a site. Exactly as sheaves give Grothendieck toposes, stacks give 2-dimensional Grothendieck toposes and they should thus be a preeminent example of elementary 2-topos. We can then study this preeminent example to try and understand which further axioms we should require to reach a notion of elementary 2-topos.

November 20, 2024

Nicola Carissimi

Enriched bicategories for enrichies bi(co)ends

Two main generalizations of category theory are bicategories and enriched categories. The first one allows morphisms one level up, the other one allows morphisms to be objects in any monoidal category. This talk will be about what happens if we do the two at the same time. We will see the notion of monoidal bicategory and the main available results and tools (such as strictification and string diagrammatic language) with which enriched bicategories and their rich algebraic structures can be tamed. Thus, the assumption of a suitable notion of braiding on the base monoidal bicategory will allow to generalize the fundamental constructions of forming the opposite and the tensor product of enriched bicategories, and possibly more.

Giuseppe Leoncini

Enriched Homotopy Cocompletions

Starting from a 1-categorical base V which is not assumed endowed with a choice of model structure (or any kind of homotopical structure), we propose a definition of homotopy colimits enriched in V in such a way that: (i) for V = Set, we retrieve the classical theory of homotopy colimits, and (ii) restricting to isomorphisms as weak equivalences, we retrieve ordinary and enriched 1-colimits. We construct the free homotopy V-cocompletion of a small V-category in such a way that it satisfies the expected universal property. Over the base V = Set, we retrieve Dugger’s construction of the universal model category on a small category C. We interpret the homotopy V-enriched cocompletion of a point as the analogue of homotopy theory of spaces in the enriched context. We compare our approach with some previous definitions of enriched homotopy colimits, such as those given by Shulman, Lack & Rosicky, and Vokrinek, and we show that, when the latter are defined and well behaved, they can be retrieved within our framework, up to Quillen homotopy.

Ivan Di Liberti

Taking ItaCa seriously

This talk is a short presentation of the ItaCa's historical progression, its most important milestones and its possible future perspectives.