# A Categorical Day in Turin

## February 25th, 2019

### Department of Mathematics "G. Peano", University of Turin.

**Aims: **This Mini-Workshop consists of introductory lectures at the level of students of the Master program in Mathematics. The main aim is to present some techniques in Category Theory which are of interest in different areas of Mathematics. This edition is related to Algebra, Algebraic Geometry, Computer Science and Logic.

**Registration**: Although there is no registration fee, we ask the participants to fill out the following form where one can also subscribe to the social dinner to be held on the day of the conference.

**Math-Lab**: Gli studenti della Laurea Magistrale del Dipartimento di Matematica "G. Peano" sono invitati e, partecipando ad almeno due delle quattro lezioni, potranno avere una firma per Math-Lab. E' richiesta l'iscrizione, vedere link qui sotto.

## Invited Speakers:

- Filippo Bonchi (Univ. Pisa) <filippo.bonchi@unipi.it>

Title: **Interacting Hopf Algebras: the theory of linear systems. ( SLIDES)**

__Abstract.__ Signal Flow Graphs (SFGs) were introduced in the 1940s by Shannon as a formal circuit model of a class of simple analog computing machines. They are a common abstraction in control theory and signal processing, used for modelling physical systems and their controllers.

*While Signal Flow Graphs are usually presented as combinatorial structures, in this talk we take a purely algebraic outlook: we provide a string diagrammatic syntax for SFGs and a sound and complete axiomatisation of their Laurent-Series semantics.*

*With tiny variations to the syntax and to the axioms, we are able to characterise other structures which appears to be fundamental in different fields, like Petri nets, passive and non-passive electrical circuits, the ZX calculus (a diagrammatic language for Quantum Mechanics) and Graphical Linear Algebras (https://graphicallinearalgebra.net). *

- Marino Gran (Univ. catholique de Louvain ) <marino.gran@uclouvain.be>

Title: **Semi-abelian categories and non-additive torsion theories. ( SLIDES)**

__Abstract.__ In his celebrated article [1] S. Mac Lane first mentioned the problem of finding an axiomatic context capturing some typical properties of the category of groups. The introduction of the notion of semi-abelian category [2] made it possible to treat many fundamental properties the categories of groups, Lie algebras, crossed modules and compact groups have in common, in a similar way to the one the notion of abelian category is useful to study module categories and their categories of sheaves. The theory of semi-abelian categories provides a suitable categorical setting to investigate some fundamental aspects of non-abelian homological algebra, and to develop a categorical approach to torsion theories and commutators.

*This introductory talk will focus on some basic properties and examples of semi-abelian categories. It will be shown that the notion of non-additive torsion theory is related to the notion of semi-localization [3,4], and then an abstract characterization of semi-localizations of semi-abelian categories will be given [5]. *

*References*

*[1] S. Mac Lane, Duality of groups, Bull. Am. Math. Soc. 56 (6), 486-516 (1950)*

*[2] G. Janelidze, L. Marki and W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra 168, 367-386 (2002)*

*[3] D. Bourn and M. Gran, Torsion theories in homological categories, J. Algebra 305, 18-47 (2006)*

*[4] S. Mantovani, Semilocalizations of exact and lextensive categories, Cah. Topol. Géom. Différ. Catég. 39, 27-44 (1998)*

*[5] M. Gran and S. Lack, Semi-localizations of semi-abelian categories, J. Algebra 454, 206-232 (2016) *

*Pietro Polesello**(Univ. Padova) <pietro@math.unipd.it>*

Title: **Variations on the fundamental groupoid **

__Abstract.__

A basic result in algebraic topology is the Seifert-Van Kampen theorem, which allows to compute the fundamental groupoid $\Pi_1(X)$ of a space, by means of the $\Pi_1(U_i)$'s for an open cover $\{U_i\}_i$ of $X$. This can be expressed by saying that $\Pi_1(X)$ is equivalent to the 2-colimit of the $\Pi_1(U_i)$'s. If we take this result as a defining property, we get the notion of a costack (which is dual to that of a stack): the data of a category $C(U)$ for any open subset $U$ and compatible "extension functors" $C(V)\toC(U)$ for any open inclusion $V\subsetU$, for which $C(U)$ is equivalent to the 2-colimit of the $C(U_i)$'s for any open covers $\{U_i\}_i$ of $U$.

In this talk, I will show that, for a locally 1-connected space $X$, we may characterise the fundamental costack $\Pi_1$ among costaks of groupoids on $X$ by a "local triviality" property. Then I will consider two generalizations of $\Pi_1(X)$: first, the topological fundamental groupoid $\Pi^\tau_1(X)$ (defined by Brazas), which allows to work on a wider class of spaces (including pathological ones); second, the fundamental category $\Pi_1(X,\Sigma)$ (defined by MacPherson), which allows to consider stratified spaces. * *

## PROGRAM (Aula Spallanzani)

- 9:00 - 10:00
**Bonchi**.

10:00-11:00** Coffee Break**

- 11:00 - 12:00
**Gran**.

12:00-13:45** Lunch break**

- 13:45 - 14:45
**Polesello**.

**Organizers:** Alessandro Ardizzoni, Cristiana Bertolin, Felice Cardone, Matteo Viale

**Poster:** Pdf