# ItaCa Fest 2021

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

## Events

## November 18, 2021

Time | Speaker | Affiliation | Talk | Material |
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14:30 | Fiorenza |
Università degli Studi di Roma "La Sapienza" | Categorical shadows lurking behind integral formulas for genera |
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15:15 | Question time | |||

15:30 | Olimpieri |
University of Leeds | Categorifying Intersection Types |
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16:15 | Question time | |||

16:30 | Free chat |

## October 21, 2021

Time | Speaker | Affiliation | Talk | Material |
---|---|---|---|---|

14:30 | Zanfa |
Università degli studi dell'Insubria | Generalized presheaf-bundle adjunctions |
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15:15 | Question time | |||

15:30 | Trimble |
Western Connecticut State University | Grothendieck groups of 2-rigs as lambda rings |
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16:15 | Question time | |||

16:30 | Free chat |

## September 28, 2021

Time | Speaker | Affiliation | Talk | Material |
---|---|---|---|---|

14:30 | Porta |
IRMA, Université de Strasbourg | Pro and Ind-categories in Algebra and Geometry |
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15:15 | Question time | |||

15:30 | Van der Linden |
Université catholique de Louvain | Algebras with representable representations |
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16:15 | Question time | |||

16:30 | Free chat |

## June 15, 2021

Time | Speaker | Affiliation | Talk | Material |
---|---|---|---|---|

14:30 | Caramello |
Università dell'Insubria | Relative topos theory via stacks |
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15:15 | Question time | |||

15:30 | Saracco |
Université Libre de Bruxelles | Globalization for geometric partial comodules |
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16:15 | Question time | |||

16:30 | Free chat |

## May 20, 2021

Time | Speaker | Affiliation | Talk | Material |
---|---|---|---|---|

14:30 | Tholen |
York University | Revisiting Burroni's T-categories |
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15:15 | Question time | |||

15:30 | Hadzihasanovic |
TalTech | The smash product of monoidal theories |
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16:15 | Question time | |||

16:30 | Free chat |

## April 19, 2021

Time | Speaker | Affiliation | Talk | Material |
---|---|---|---|---|

14:30 | Dell'Ambrogio |
Université de Lille | Mackey 2-functors |
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15:15 | Question time | |||

15:30 | Sobociński |
TalTech | Rewriting Modulo Symmetric Monoidal Structure |
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16:15 | Question time | |||

16:30 | Free chat |

## Abstracts of the past events

### Ivo Dell’Ambrogio

#### Mackey 2-functors

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.

### Paweł Sobociński

#### 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

### Walter Tholen

#### 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)

### Amar Hadzihasanovic

#### 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.

### Olivia Caramello

#### 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.

### Paolo Saracco

#### 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).

### Mauro Porta

#### 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*.

### Riccardo Zanfa

#### Generalized presheaf-bundle adjunctions

We present two new results generalizing the well-known presheaf-bundle adjunction for topological spaces, which relate indexed categories (and presheaves) over a site (C,J) with toposes over the topos Sh(C,J). The content of this seminar can be found in a joint work with Olivia Caramello titled Relative topos theory via stacks, and currently available on arXiv.

### Todd Trimble

#### Grothendieck groups of 2-rigs as lambda rings

This talk will report on recent joint work with John Baez and Joe Moeller. We introduce a notion of 2-rig as a way of categorifying the usual notion of rig (ring without negatives); examples include categories of group representations, categories of vector bundles over spaces, categories of coherent sheaves over projective varieties, and many others. We describe Schur functors on general 2-rigs, and indicate how the free 2-rig on one generator encodes Schur functors on 2-rigs. Finally, we indicate how the Grothendieck group of a 2-rig yields a lambda ring, where the theory of lambda-rings is a “plethory” obtained by decategorifying a conceptually simple 2-plethory that is associated with the free 2-rig.

### Domenico Fiorenza

#### Categorical shadows lurking behind integral formulas for genera

*Professor Friedrich*: And so, if you have a complex genus taking rational values, i.e., a ring homomorphism from the complex cobordism ring to the field **Q** of rational numbers, you have an integral formula expressing it.

*The Categorist*: If a formula is true, it must be expressed by a commutative diagram.

*Professor Friedrich*: But my formula is true!

*The Categorist*: Then it must be expressed by a commutative diagram.

*Professor Friedrich*: Show me.

*The Categorist*: Let us consider the category of spectra…

*Professor Friedrich*: And so?

*The Categorist*: Well… I don’t see a commutative diagram here.

*Professor Friedrich*: And so?

*The Categorist*: And so your formula must be false.

*Professor Friedrich*: But my formula is true!

*The Categorist*: Impossible.

*Sir Michael*: There is something I think I know on the Spanier-Whitehead dual of a smooth manifold that may happen to be of some relevance here.

(Emil Ionesco, Triceratops)

### Federico Olimpieri

#### Categorifying Intersection Types

We study a family of distributors-induced bicategorical models of lambda-calculus, proving that they can be syntactically presented via intersection type systems. We first introduce a class of 2-monads whose algebras are monoidal categories modelling resource management. We lift these monads to distributors and define a parametric Kleisli bicategory, giving a sufficient condition for its cartesian closure. In this framework we define a proof-relevant semantics: the interpretation of a term associates to it the set of its typing derivations in appropriate systems. We prove that our model characterize solvability, adapting reducibility techniques to our setting. We conclude by describing wo examples of our construction.

## Programme Committee

This seminars series is supported by the “National Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA – INdAM)