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**Title**: Tangent infinity categories revisited (Part 1)

**Abstract**: In this talk I will discuss recent work with Michael Ching and Florian Schwarz on Tangent Infinity Categories. Over the summer, an error was discovered in our previous work. In this talk I will explain the error and how we propose to fix it.

**Title**: Morita equivalence and bicategories of fractions: a simple geometric idea gets out of hand.

**Abstract**: We start by considering the geometry of spaces with singularities: in these spaces, the local geometry is not homogeneous as in a manifold, but varies from point to point. Lie groupoids provide a way of encoding this information. However, this encoding is not unique: different Lie groupoids may represent the same singular space. This leads to the idea of ‘Morita equivalence’ of Lie groupoids: two groupoids are Morita equivalent if they encode the same underlying geometric structure. To formalize this equivalence we consider the ‘localization’ of Lie groupoids, which creates a ‘bicategory of fractions’ in which the Morita equivalences become isomorphisms.

The goal of this talk is to describe this bicategory of fractions construction in all its diagrammatic glory, and convince you that it is geometrically motivated and meaningful. I will start with the geometric intuition of how Lie groupoids are used to encode singular spaces, and how Morita equivalence arises as a natural consequence, with examples and pictures. I will then explain the localization construction, starting from the familiar idea of localizing a prime ideal in a commutative ring and extending to more complicated contexts, culminating in the bicategory of fractions we are interested in. At the end I will mention applications and current work in this area. No background with Lie groupoids or bicategories is assumed and all scary diagrams will be thoroughly explained and motivated.

*This event will be in a hybrid format. Besides the option to attend in person the presentation will be streamed on zoom. For the zoom link and password, please contact florian.schwarz@ucalgary.ca*

**Title**: Turing Categories

**Abstract**:

This talk is based on the following papers/notes:

(1) “Introduction to Turing categories” with Pieter Hofstra

(2) “Timed set, functional complexity, and computability” with Boils, Gallagher, Hrubes

(3) “Total maps of Turing categories” with Pieter Hofstra and Pavel Hrubes

(4) “Estonia notes” on my website

Turing categories are the theory of “abstract computability”. Their development followed my meeting Pieter Hofstra. He was in Ottawa at the time and he subsequently joined me as a postdoc. The core theory was developed in Calgary before he returned to Ottawa as a faculty member. Tragically he died earlier this year when there was still so much to do and, indeed, that he had done, but had not published.

Turing categories are important because they characterize computability in a minimal traditional context. These ideas are not original to Pieter and I: De Paola, Heller, Longo, Moggi, and others had all travelled in this terrain before we did. Pieter and I simply took the ideas polished them a bit and moved them a step further on a road which still stretches ahead.

So, the purpose of the talk is to try and explain what all this was about … and what we were striving to accomplish. To do this I have to introduce restriction categories and Turing categories in that context. Then I will describe a family of models which are fundamental to computer science. Finally, I will take a quick look along the road at some open issues.

**Title**: Kan extensions as a framework to extend resource monotones

**Abstract**: A monotone for a resource theory assigns a real number to each resource in the theory signifying the utility or the value of the resource. Gour and Tomamichel studied the problem of extending monotones using set-theoretical framework when a resource theory embeds fully and faithfully into the larger theory. One can generalize the problem of computing monotone extensions to scenarios when there exists a functorial transformation of one resource theory to another instead of just a full and faithful inclusion. In this article, we show that (point-wise) Kan extensions provide a precise categorical framework to describe and compute such extensions of monotones. To set up monontone extensions using Kan extensions, we introduce partitioned categories (pCat) as a framework for resource theories and pCat functors to formalize relationship between resource theories. We describe monotones as pCat functors into ([0,∞],≤), and describe extending monotones along any pCat functor using Kan extensions. We show how our framework works by applying it to extend entanglement monotones for bipartite pure states to bipartite mixed states, to extend classical divergences to the quantum setting, and to extend a non-uniformity monotone from classical probabilistic theory to quantum theory.

**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?)