Speakers

Title: Tangent structure for operadic algebras

Abstract:

In the preface of its Lectures on Noncommutative Geometry [Gin05], Ginzburg writes:

“Each of the mathematical worlds that we study is governed by an appropriate operad. Commutative geometry is governed by the operad of commutative (associative) algebras, while noncommutative geometry ‘in the large’ is governed by the operad of associative not necessarily commutative algebras. […]

There are other geometries arising from operads of Lie algebras, Poisson algebras, etc.“

In this talk we will show that the theory of tangent categories provides a suitable language to make precise this inspiring intuition of Ginzburg: operads produce (algebraic) geometrical theories. We will construct two different tangent categories for each operad, one defined over the category of algebras of the operad and the other one, over the opposite category, using the theories of tangent categories and tangent monads. We will provide concrete examples of these two tangent categories for a few operads. Finally, we will discuss some ideas for future work on this project.

This is joint work with Sacha Ikonicoff and JS Lemay.

The slides can be found at https://www.dropbox.com/s/y67pqm2w8ih0c5f/Tangent%20Categories%20for%20operadic%20algebras%20-%20Calgary%20February%202023%205.pdf?dl=0.

Bibliography

• Gin05: Ginzburg; Lectures on Noncommutative Geometry (2005) (https://arxiv.org/pdf/math/0506603.pdf)

Title: Algebraic examples of coCartesian differential monads from operads

Abstract:

Operads are a useful tool to classify types of algebras. To each operad is associated a monad, which in turn gives rise to a category of algebras. There is an operad for associative algebras, for commutative algebras, for Lie algebras, and the list goes on.

In this talk, we will give an introduction to operads and their algebras. We will study the notion of operadic derivation, which generalises the notion of algebraic derivation. We will define an analogue of the module of Kähler differentials for an algebra over an operad. Studying the case of free algebras over a fixed operad will allow us to build a natural derivation which equips the associated monad with the structure of a coCartesian differential monad.

This is joint work with Marcello Lanfranchi and JS Lemay.

The slides can be found at https://www.dropbox.com/s/hne867oza0fkp4v/Algebraic%20examples%20of%20coCartesian%20differential%20monads%20coming%20from%20operads.pdf?dl=0

The slides can be found at https://www.dropbox.com/s/hne867oza0fkp4v/Algebraic%20examples%20of%20coCartesian%20differential%20monads%20coming%20from%20operads.pdf?dl=0

Title: Tangent categories & tangent monads

Abstract:

Tangent categories, first introduced by Rosicky [Ros84] and more recently generalized by Cockett and Cruttwell [Coc14], provide a categorical axiomatization for differential geometry. The core idea of this concept is to equip a category with an endofunctor which replicates the role of the tangent bundle functor, i.e. which associates to each object another object that looks like the fibre bundle of tangent spaces of the object. In this talk we explore the basic definitions and results of this theory, providing concrete examples of tangent categories.

In the second part, we are going to explore the theory of tangent monads, first introduced by Cockett, Lemay and Lucyshyn-Wright [Coc19]. Monads are often regarded as powerful tools to produce algebraic theories. In this regard, tangent monads produce geometrical theories. We will dig into the definition of tangent monads and the main results of the theory. We will also show that coCartesian differential monads induce tangent monads. This is a striking result, which will allow us to provide tangent categories for many types of algebras.

The slides can be found at https://www.dropbox.com/s/7mz7k2t7eqttm15/Tangent%20Categories%20%26%20Tangent%20Monads%20-%20Calgary%20February%202023%202.pdf?dl=0.

Bibliography

• Coc14: Cockett, Cruttwell; Differential Structure, Tangent Structure, and SDG (2014) (https://link.springer.com/article/10.1007/s10485-013-9312-0)
• Coc19: Cockett, Lemay, Lucyshyn-Wright; Tangent Categories from the Coalgebras of Differential Categories (2019) (https://arxiv.org/abs/1910.05617)
• Ros84: Rosicky; Abstract tangent functors (1984) (https://eudml.org/doc/91746)

Title: Cartesian differential categories and coCartesian differential monads

Abstract:

Cartesian Differential Categories (CDCs) are defined to introduce and study the notion of differential from calculus in a category theory point of view. In a CDC, morphisms between objects can be « derived », and this differentiation operation must satisfy a list of properties, including a version of the chain rule.

With JS Lemay, we introduced the notion of a coCartesian Differential Monad (cCDM) on a Category with finite biproducts to give the lightest apparatus on a monad which allows us to define a CDC structure on its Kleisli category.

In this introductory talk, we will motivate the definition for CDCs using the example of multivariable calculus on Euclidian spaces. We will define cCDMs, recall the construction of the Kleisli category for a monad, and show how to build a CDC structure on the Kleisli category of a cCDM, along with motivating examples. This joint work with JS Lemay is available on the ArXiv: https://arxiv.org/abs/2108.04304

The slides can be found at https://www.dropbox.com/s/xa5h09ch6qs3vq4/Cartesian%20differential%20categories%20and%20coCartesian%20differential%20monads.pdf?dl=0

Title: A reintroduction to proofs

Abstract: In an introduction to proofs course, students learn to write proofs informally in the language of set theory and classical logic. In this talk, I’ll explore the alternate possibility of teaching students to write proofs informally in the language of dependent type theory. I’ll argue that the intuitions suggested by this formal system are closer to the intuitions mathematicians have about their praxis. Furthermore, dependent type theory is the formal system used by many computer proof assistants both “under the hood” to verify the correctness of proofs and in the vernacular language with which they interact with the user. Thus, students could practice writing proofs in this formal system by interacting with computer proof assistants such as Coq and Lean.

This talk is given as the University of Regina PIMS Distinguished Lecture and we are going to stream it in MS 325 to watch it.

Title: Introduction to the de Rham Cohomology

Abstract: In this talk I will cover the details behind the construction of the de Rham cohomology. In particular, I will focus on its boundary map – the exterior derivative – and understanding what this cohomology tells us about closed and exact differential forms.

Title: Tangent infinity categories revisited (Part 2)

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: 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.