ItaCa Fest 2021
ItaCa Fest is an online webinar aimed to gather the community of ItaCa. You can find us on researchseminars.
ItaCa Fest will come back in September! Stay tuned. Here you can find our past events and speakers.
September 28, 2021
|14:30||Porta||IRMA, Université de Strasbourg||Pro and Ind-categories in Algebra and Geometry||▤ ▶|
|15:30||Van der Linden||Université catholique de Louvain||Algebras with representable representations||▤ ▶|
Pro and Ind-categories in Algebra and Geometry
In this talk we are going to discuss some natural instances of pro and ind categories in algebraic and geometric contexts, highlighting the importance of working with objects in Ind(Cat∞) and Pro(Cat∞) instead of their Cat∞ realizations. Towards the end we will raise some questions, with the intent of determining what is the “correct” object to consider in these contexts, so as to optimize the generalization/applicability trade-off.
Tim Van Der Linden
Algebras with representable representations
(Joint work with Xabier García-Martínez, Matsvei Tsishyn and Corentin Vienne)
Just like group actions are represented by group automorphisms, Lie algebra actions are represented by derivations: up to isomorphism, a split extension of a Lie algebra B by a Lie algebra X corresponds to a Lie algebra morphism B → Der(X) from B to the Lie algebra Der(X) of derivations on X. The aim of this talk is to elaborate on the question, whether the concept of a derivation can be extended to other types of non-associative algebras over a field K, in such a way that these generalised derivations characterise the K-algebra actions. We prove that the answer is no, as soon as the field K is infinite. In fact, we prove a stronger result: already the representability of all abelian actions – which are usually called representations or Beck modules – suffices for this to be true. Thus we characterise the variety of Lie algebras over an infinite field of characteristic different from 2 as the only variety of non-associative algebras which is a non-abelian category with representable representations. This emphasises the unique role played by the Lie algebra of linear endomorphisms gl(V) as a representing object for the representations on a vector space V.
October 21, 2021
|14:30||Trimble||Western Connecticut State University||TBA||▤ ▶|
|15:30||Zanfa||Università degli studi dell'Insubria||TBA||▤ ▶|
November 18, 2021
|14:30||Fiorenza||Università degli Studi di Roma "La Sapienza"||TBA||▤ ▶|
|15:30||Olimpieri||LIPN, Université Sorbonne Paris Nord||TBA||▤ ▶|
December 2021 - ItaCa Fest XMas edition
|14:30||Rogers||Università degli Studi dell'Insubria||TBA||▤ ▶|
|15:30||Gran||Université catholique de Louvain||TBA||▤ ▶|
June 15, 2021
|14:30||Caramello||Università dell'Insubria||Relative topos theory via stacks||▤ ▶|
|15:30||Saracco||Université Libre de Bruxelles||Globalization for geometric partial comodules||▤ ▶|
May 20, 2021
14:30 UTC+2 on zoom
|14:30||Tholen||York University||Revisiting Burroni's T-categories||▤ ▶|
|15:30||Hadzihasanovic||TalTech||The smash product of monoidal theories||▤ ▶|
April 19, 2021
|14:30||Dell'Ambrogio||Université de Lille||Mackey 2-functors||▤ ▶|
|15:30||Sobociński||TalTech||Rewriting Modulo Symmetric Monoidal Structure||▤ ▶|
Abstracts of the past events
Mathematicians of different stripes like to have groups act on different sorts of objects: vector spaces, topological spaces, C*-algebras, spectra, and so on. At the heart of all flavours of “equivariant mathematics” are operations such as restrictions and inductions (and conjugations, inflations, etc). The latter have been successfully axiomatized more than half a century ago (at least for finite groups) by the algebraic notion of Mackey functors. But Mackey functors take values in abelian groups, and the operations are modeled by homomorphisms between them; however, what gives rise to most Mackey functors found in Nature is a collection of categories of equivariant objects together with restriction and induction functors between them. These functors enjoy properties such as being adjoint, which are invisible to the classical axioms. In this talk I will introduce the recent theory of Mackey 2-functors, algebraic gadgets similar to additive derivators whose purpose is precisely to capture this higher-categorical layer of information. In order to motivate our 2-categorical flavour of axiomatic representation theory, I will evoke exemples from throughout mathematics and I will outline our first notable applications. For instance, we can export results from the usual theory of linear representations to more geometric and topological settings. This is joint work with Paul Balmer.
Rewriting Modulo Symmetric Monoidal Structure
String diagrams are an elegant, convenient and powerful syntax for arrows of symmetric monoidal categories. In recent years, they have been used as compositional descriptions of computational systems from various fields, including quantum foundations, linear algebra, control theory, automata theory, concurrency theory, and even linguistics. All of these applications rely on diagrammatic reasoning, which is to string diagrams as equational reasoning is to ordinary terms.
If we are to take string diagrams out of research papers and into more practical applications, we need to ask ourselves about how to implement diagrammatic reasoning. This is the focus of my talk.
It turns out that there is a tight correspondence between symmetric monoidal categories where every object has a coherent special Frobenius algebra structure and categories of cospans of hypergraphs. The correspondence, therefore, takes us from a topological understanding of string diagrams to a combinatorial data-structure-like description. Moreover, diagrammatic reasoning translates via this correspondence exactly to DPO rewriting with interfaces.
Given the above, a natural question is how much of this correspondence survives if we drop the assumption about Frobenius structure: i.e. can we use this correspondence to implement diagrammatic reasoning on vanilla symmetric monoidal categories. The answer is yes, but we need to restrict the kinds of cospans we consider: the underlying hypergraph has to be acyclic and satisfy an additional technical condition called monogamy. Moreover, we must restrict the DPO rewriting mechanism to a variant that we call convex DPO rewriting. The good news is that none of these modifications come with a significant algorithmic cost.
The material in this talk is with Filippo Bonchi, Fabio Gadducci, Aleks Kissinger and Fabio Zanasi, and has been published in a series of papers:
- “Rewriting modulo symmetric monoidal structure”, Proceedings of LiCS 2016
- “Confluence of Graph Rewriting with Interfaces”, Proceedings of ESOP 2017
- “Rewriting with Frobenius”, Proceedings of LiCS 2018
Revisiting Burroni’s T-categories and the Street-Walters comprehensive factorization system
Following the appearance of Lambek’s multicategories and Barr’s presentation of topological spaces as relational T-algebras, Albert Burroni introduced T-categories and T-functors in 1971. They provide an overarching environment for the general study of algebras and spaces, which encompasses elements of monad theory, internal category theory, and categorical topology.
In this talk we have a fresh look at Burroni’s paper and point to the Street-Walters comprehensive factorization system for functors and the (antiperfect, perfect) factorization system for continuous maps of Tychonoff spaces to demonstrate that, despite its generality, Burroni’s setting allows for the establishment of non-trivial results and the discovery of unexpected connections between seemingly unrelated theorems. (Joint work with Leila Yeganeh)
The smash product of monoidal theories
The smash product of pointed spaces is a classical construction of topology. The tensor product of props, which extends both the Boardman-Vogt product of symmetric operads and the tensor product of Lawvere theories, seems firmly like a piece of universal algebra.
In this talk, we will see that the two are facets of the same construction: a “smash product of pointed directed spaces”. Here, “directed spaces” are modelled by combinatorial structures called diagrammatic sets, developed as a homotopically sound foundation for diagrammatic rewriting in higher dimensions, while the cartesian product of spaces is replaced by a form of Gray product.
Most interestingly, the smash product applies to presentations of higher-dimensional theories and systematically produces oriented equations and higher-dimensional coherence data (oriented syzygies). This introduces a synthetic, compositional method in rewriting on higher structures.
This talk is based on my preprint arXiv:2101.10361 with the same title.
Relative topos theory via stacks
In this talk, based on joint work with Riccardo Zanfa, we shall introduce new foundations for relative topos theory based on stacks. One of the central results in our theory is an adjunction between the category of (relatively small) toposes over the topos of sheaves on a given site (C, J) and that of C-indexed categories. This represents a wide generalization of the classical adjunction between presheaves on a topological space and bundles over it, and allows one to interpret several constructions on sheaves and stacks in a geometrical way; in particular, it leads to fibrational descriptions of direct and inverse images of sheaves and stacks, as well as to a geometric understanding of the sheafification process. It also naturally allows one to regard any Grothendieck topos as a ‘petit’ topos associated with a ‘gros’ topos, thereby providing an answer to a problem posed by Grothendieck in the seventies.
Globalization for geometric partial comodules
The study of partial symmetries (e.g. partial dynamical systems, (co)actions, (co)representations, comodule algebras) is a relatively recent research area in continuous expansion, whose origins can be traced back to the study of C*-algebras generated by partial isometries. One of the central questions in the field is the existence and uniqueness of a so-called globalization or enveloping (co)action.
In the framework of partial actions of groups, any global action of a group on a set induces a partial action of the group on any subset by restriction. The idea behind the concept of globalization of a given partial action is to find a (universal) global action such that the initial partial action can be realized as the restriction of this global one. The importance of this procedure is testified by the numerous globalization results already existing in the literature which, however, are based on some ad hoc constructions, depending on the nature of the objects carrying the partial action.
We propose here a unified approach to globalization in a categorical setting, explaining several of the existing results from the literature and, at the same time, providing a procedure to construct globalizations in concrete contexts of interest. Our approach relies on the notion of geometric partial comodules (recently introduced by Hu and Vercruysse in [HV]) which –unlike classical partial actions, that exist only for (topological) groups and Hopf algebras– can be defined over any coalgebra in an arbitrary monoidal category with pushouts.
[HV] J. Hu, J. Vercruysse, Geometrically partial actions. Trans. Amer. Math. Soc. 373 (2020), no. 6, 4085–4143.
[PJ] P. Saracco, J. Vercruysse, Globalization for geometric partial comodules. Preprint (2020).
This seminars series is supported by the “National Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA – INdAM)