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**Title:** Tangent categories, Cartesian differential categories and how they are related

**Abstract:** Tangent categories are a categorical generalization of the category of manifolds by having maps like the projection and the zero-section of the tangent bundle that fulfill certain relations. With a similar strategy, Cartesian differential categories generalize the category of finite-dimensional R-vector spaces and smooth maps.Unsurprisingly the construction generalizing manifolds and the construction generalizing vector-spaces are related. More precisely there is an adjunction between the category of Cartesian tangent categories and the category of Cartesian differential categories. Explaining this adjunction is the main goal of this talk.

**Title:** Categories of Kirchoff Relations

**Abstract:** It is known that the category of affine Lagrangian relations over a field F, of integers modulo a prime p (with p > 2) is isomorphic to the category of stabilizer quantum circuits for p-dits. Furthermore, it is known that electrical circuits (generalized for the field F) occur as a natural subcategory of affine Lagrangian relations. The purpose of this paper is to provide a characterization of the relations in this subcategory — and in important subcategories thereof — in terms of parity-check and generator matrices as used in error detection.

In particular, we introduce the subcategory consisting of Kirchhoff relations to be (affinely) those Lagrangian relations that conserve total momentum or equivalently satisfy Kirchhoff’s current law. We characterize these Kirchhoff relations in terms of parity-check matrices and, study two important subcategories: the deterministic Kirchhoff relations and the lossless relations.

**Title:** Cartesian Differential Monads

**Abstract:** Cartesian Differential Categories are defined to introduce and study the notion of differential from calculus in a category theory point of view. In a Cartesian Differential Category, morphisms between objects can be « derived », and this differentiation operation must satisfy a list of properties, including a version of the chain rule. The most predominant source of Cartesian Differential Categories is obtained by studying the free (co)algebras of a (co)monad equipped with a heavy structure – a differential storage structure – via the concept of (co)Kleisli category.

In this talk, we will introduce the notion of a Cartesian Differential (co)Monad on a Category with finite biproducts, which gives the lightest apparatus on a (co)monad which allows us to define a Cartesian Differential Category structure on its (co)Kleisli category. We will then list quantity of examples of such monads, most of which could not be given a differential storage structure, thus motivating our new construction.

This joint work with J-S Pacaud Lemay is available on the ArXiv: https://arxiv.org/abs/2108.04304

Pre-requisite: students wishing to attend the talk are welcome to do so! They should make sure they know the definition of a monad and of an algebra over a monad, a simple search on a favourite browser is probably enough.

**Title:** Operadic Tangent Categories

**Abstract: **One of the main questions I posed to my supervisor Geoffrey Cruttwell when I applied for the PhD program, was whether non-commutative geometry could be described using the language of tangent categories. My background in theoretical physics makes me care about non-commutative geometry because it could be a valid mathematical framework to describe general relativity in a way that is compatible with quantum mechanics.

Before Christmas 2021, he showed me his research on tangent category theory, applied to algebraic geometry. This new work on algebraic geometry, initially introduced by Geoff and Robin Cockett and recently further developed by Geoff and J.S. Lemay, allowed me to reformulate my question in the following terms: can we extend this tangent category construction, defined for commutative algebras, to general associative algebras?

In this talk, I present an answer to this question showing how the construction presented by Geoff can be extended to non-commutative geometry and more generally to algebras of (algebraic symmetric) operads.

The talk will be structured as follows: I will start by giving the main motivation for the talk, and then I will briefly recall the key definitions and constructions of tangent category theory. I will then spend some time describing the construction for commutative algebras. I will then give the main definitions and results of operad theory. Following that, I will show how to construct a canonical tangent structure on the category of algebras over an operad. Thereafter, I will discuss the corresponding tangent structure over the opposite category, showing its geometrical meaning. Finally, I will give some of the results that I found so far that extend the constructions of the commutative case.

This work is in collaboration with my supervisors Geoffrey Cruttwell and Dorette Pronk. I also would like to thank J.S. Lemay for the great discussions and ideas he shared with me about his work and mine.

**Title:** Decomposition of topological Azumaya algebras in the stable range

**Abstract:** Topological Azumaya algebras are topological shadows of more complicated algebraic Azumaya algebras defined over, for example, schemes. Tensor product is a well-defined operation on topological Azumaya algebras. Hence given a topological Azumaya algebra A of degree mn, where m and n are positive integers, it is a natural question to ask whether A can be decomposed according to this factorization of mn. In this talk, I explain the definition of a topological Azumaya algebra over a topological space X, and present a result about what conditions should m, n, and X satisfy so that A can be decomposed.

**Title:** Koszul duality of algebras for operads

**Abstract:** Ginzburg-Kapranov and Getzler-Jones exhibited a duality between algebras for an operad O and coalgebras (with divided powers) for a “Koszul dual” cooperad BO, taking the form of an adjoint pair of functors between these categories. Instances of this duality include that between Lie algebras and cocommutative coalgebras, as in Quillen’s work on rational homotopy theory, and bar-cobar duality for associative (co)algebras, as in the work of Moore. I will review this formalism and discuss the following basic question: on what subcategories of O-algebras and BO-coalgebras does this duality adjunction restrict to an equivalence? I will discuss an answer to this question and explain the relation to a conjecture of Francis and Gaitsgory.

Recording (Passcode: u.2c?5w?)

**Title:** Regular logic in a `double category of relations’

**Abstract:** In this talk, we will present a definition of a `double category of relations’, inspired by that of a bicategory of relations’ due to Carboni and Walters. Roughly, a `double category of relations’ is a cartesian equipment whose horizontal bicategory satisfies a certain discreteness condition. We will then sketch how any such structure yields a sound interpretation of regular logic.

**Title:** The Eckmann-Hilton argument in duoidal categories**Abstract:** We will go over the basics of duoidal categories, illustrating with a number of examples. As monoidal categories provide a context for monoids, duoidal categories provide one for duoids and bimonoids. Our main goal is to discuss a number of versions of the classical Eckmann-Hilton argument which may be formulated in this setting. As an application we will obtain the commutativity of the cup product on the cohomology of a bimonoid with coefficients in a duoid, an extension of a familiar result for group and bialgebra cohomology.

The talk borrows on earlier work in collaboration with Swapneel Mahajan on the foundations of duoidal categories (2010). The main results are from ongoing work with Javier Coppola. We also rely on work of Richard Garner and Ignacio López-Franco (2016).

**Title:** Iterated Traces

**Abstract:** The trace of a matrix does not seem like an operation that should be iterated, but if we step back and think of trace as an operation on endomorphisms (or almost endomorphisms) that is invariant under cyclic permutation this becomes more plausible. I’ll make sense of iterated traces in monoidal bicategories, describe independence of order for iterated traces, and connect this result to (disguised) examples that have appeared in the literature.

**Title :** Graded (co-)Differential Categories

**Abstract :** Co-differential categories are additive symmetric monoidal categories where you can differentiate certain kind of morphisms with the help of a combination of a monad, monoids and a deriving transformation.

Recently, J.-S. Pacaud Lemay proved that the category of finite-dimensional vector spaces cannot be endowed with a nontrivial structure of co-differential category. And this is very unlikely that it will work better with some familiar categories such as the category of finite sets and relations or the category of all Hilbert spaces.

In this talk, I will introduce a generalisation of co-differential categories: graded co-differential categories, which basically consist in putting the word “graded” everywhere in the axioms of a co-differential category. I will explain how the preceding categories can be endowed with the structure of a graded co-differential category as example of a general categorical construction which use symmetric powers.

This is joint work in progress with Jean-Simon Pacaud-Lemay.