ItaCa Fest 2022

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

The seminar will be live on Zoom at this link

April 20, 2022

Antonio Lorenzin

Formality and strongly unique enhancements

Inspired by the intrinsic formality of graded algebras, we give a characterization of strongly unique DG-enhancements for a large class of algebraic triangulated categories, linear over a commutative ring. We will discuss applications to bounded derived categories and bounded homotopy categories of complexes. For the sake of an example, the bounded derived category of finitely generated abelian groups has a strongly unique enhancement.

Martti Karvonen

Inner automorphisms as 2-cells

Thinking of groups as one-object categories makes the category of groups naturally into a 2-category. We observe that a similar construction works for any category: a 2-cell f → g is given by an inner automorphism of the codomain that takes f to g, where inner automomorphisms are defined in general using isotropy groups. We will explore the behavior of limits and colimits in the resulting 2-category: when the underlying category is cocomplete, the resulting 2-category has coequalizers iff the isotropy functor is representable - in the case of groups, this amounts to deducing the existence of HNN-extensions from the representability of id : Grp → Grp. Under reasonable conditions, limits and connected colimits in the underlying category are 2-categorical limits/colimits in the resulting 2-category. However, many other 2-dimensional limits and colimits fail to exist, unless the underlying category has only trivial inner automorphisms.

May 19, 2022

Greta Coraglia

Comonads for dependent types

In exploring the relation between a classical model of dependent types (comprehension categories) and a new one (judgemental dtts) we pin-point the comonadic behaviour of weakening and contraction. We describe three different 2-categories and show that they are 2-equivalent, then proceed to analyze the benefits of each of the three. The fact that one can precisely relate such different perspectives allows, for example, for a swift and cleaner treatment of type constructors: we show how certain categorical models for dependent types come inherently equipped with some due to the choices one makes in introducing tools to interpret context extension.

Joachim Kock

Decomposition spaces, right fibrations, and edgewise subdivision

Decomposition spaces are simplicial infinity-groupoids subject to an exactness condition weaker than the Segal condition. Where the Segal condition expresses composition, the weak condition expresses decomposition. The motivation for studying decomposition spaces is that they have incidence coalgebras and Möbius inversion. The most important class of simplicial maps for decomposition spaces are the CULF maps (standing for ‘conservative’ and ‘unique-lifting-of-factorisation’), first studied by Lawvere; they induce coalgebra homomorphisms. The theorem I want to arrive at in the talk says that the infinity-category of (Rezk-complete) decomposition spaces and CULF maps is locally an infinity-topos. More precisely for each (Rezk-complete) decomposition space D, the slice infinity-category Decomp/D is equivalent to PrSh(Sd(D)), the infinity-topos of presheaves on the edgewise subdivision of D. Most of the talk will be spent on explaining preliminaries, though.

This is joint work with Philip Hackney.

June 28, 2022

Filippo Bonchi

Deconstructing Tarski’s calculus of relations with Tape diagrams

The calculus of (binary) relations has been introduced by Tarski as a variable-free alternative to first order logic. In this talk we introduce tape diagrams, a graphical language for expressing arrows of arbitrary finite biproduct rig categories, and we show how the calculus of relation can be encoded within tape diagrams.

Ingo Blechschmidt

Reifying dynamical algebra

Traveling the mathematical multiverse to apply tools for the countable also to the uncountable

Commutative algebra abounds with proofs which are quite elegant and at the same time quite abstract. Even for concrete statements, proofs often appeal to transfinite methods like the axiom of choice or the law of excluded middle. Following Hilbert’s call, we should work to elucidate how these abstract proofs can be recast in more concrete, computational terms, regarding abstract proofs as intriguing guiding templates for formulating concrete proofs and regarding objects concocted by Zorn’s lemma such as maximal ideals as convenient fictions. One such technique for making computational sense of abstract proofs is dynamical algebra, going back to the work of Dominique Duval and her coauthors in the 1980’s. The talk will first present the basic story of dynamical algebra with an illustrative example. Then we will report on joint work with Peter Schuster how to reify dynamical algebra using formal metatheorems of categorical logic, supplying a firm foundation to dynamical algebra, complementing previous approaches. A particular feature of our approach is that we apply a construction devised by Berardi and Valentini for the special case of countable rings, which indeed fundamentally requires the countability assumption, by a logical sleight of hand by Joyal and Tierney to arbitrary rings. This trick is applicable quite generally which is why we believe that it is of interest to a larger group of people. It is unlocked by categorical logic running on a certain fractal without points, the pointfree space of enumerations of a given set.

September 20, 2022

Giuseppe Metere*

*Due to an unexpected event, Cigoli couldn’t give the talk; nevertheless, Metere talked about the same topic.

Groupal Pseudofunctors

Let B be an additive category and let Set denote the category of sets. A finite product preserving functor F from B to Set necessarily factors through the category Ab of abelian groups. This simple and important observation has no straightforward generalization when F and Set are replaced by a pseudo-functor and the 2-category Cat of categories, respectively. The latter situation occurs precisely when B is the base category of an opfibration. In this talk, we will focus on pseudo-functors corresponding to cartesian monoidal opfibrations of codomain B. Among such, we will eventually characterize, in terms of oplax and lax monoidal structure, those factorizing through the bicategory of symmetric categorical groups. This is the case, for example, when the starting opfibration has groupoidal fibres. This is joint work with S. Mantovani and G. Metere.

Luca Reggio

Arboreal categories and homomorphism preservation theorems

Game comonads, introduced by Abramsky, Dawar et al. in 2017, provide a categorical approach to (finite) model theory. In this framework one can capture, in a purely syntax-free way, various resource-sensitive logic fragments and corresponding combinatorial parameters. After an introduction to game comonads, I shall present an axiomatic framework which captures the essential common features of these constructions. This is based on the notion of arboreal category, in which every object is generated by its `paths’. I will then show how (resource-sensitive) homomorphism preservation theorems in logic can be recast and proved at this axiomatic level. This is joint work with Samson Abramsky.

October 18, 2022

Matteo Capucci

Triple categories of open cybernetic systems

Categorical system theory (in the sense of Myers) is a double categorical yoga for describing the compositional structure of open dynamical systems. It unifies and improves on previous work on operadic notions of system theory, and provides a strong conceptual scaffolding for behavioral system theory. However, some of the most interesting systems out there escape the simple model of dynamical systems. They are instead cybernetic systems, or in other words, controllable dynamical systems. Notable and motivating examples are strategic games and machine learning models. In this talk I’m going to outline an upgrade of categorical system theory to deal with such systems by resorting to triple categories.

Martín Escardó

The talk has been canceled due to an unforeseen personal event. Click.

November 22, 2022¹

[¹] The timetable for November 2022 event has been changed to accomodate the first speaker from Macquarie University

Nicola di Vittorio

A gentle introduction to 2-derivators

Derivators originated in the 1980s from independent efforts by Grothendieck and Heller aimed at formalising homotopy theory. They realised that the collection of homotopy categories of diagram categories retains enough information to capture homotopy limits and colimits using just old-fashioned category theory. Going one dimension up we could ask how much of $(\infty,1)$-category theory can be developed in this way. Progress in this direction has been done by Riehl and Verity in their work on $\infty$-cosmoi by showing that similar ideas allow even for internalisation of adjunctions from 2-categorical data. In this talk I will explain to which extent the theory of derivators can be enhanced to a theory of $2$-derivators having $\infty$-cosmology as a model.

Georgios Raptis

What is a stable n-category?

Triangulated categories provide a convenient framework for the study of derived functors in algebra and geometry. In most cases of interest, triangulated structures can be enhanced to more highly structured objects with better properties. The search for appropriate enhancements of triangulated categories has led to various foundational approaches in stable homotopy theory. In the context of \infty-categories (or quasi-categories), this involves the notion of stable \infty-category. Indeed, the homotopy 1-category of a stable \infty-category is canonically triangulated. But what about n-categories for 1 < n < \infty? Is there an appropriate notion of stable (or triangulated) category in the context of n-categories that interpolates between stable \infty-categories and triangulated categories? The main examples should again be the homotopy n-categories of stable \infty-categories. In this talk, I will discuss the relevant properties of higher homotopy categories leading to a notion of stable n-category. If time permits, I will also mention some uses of this notion of stable n-category for (higher) Brown representability and algebraic K-theory.