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

TBA

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Giacomo Tendas

TBA

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Luca Mesiti

TBA

TBA

November 20, 2024

Nicola Carissimi

TBA

TBA

TBA

TBA

TBA

TBA

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