#### Statement on Black Lives Matter

**Title:** Partial monoids

**Abstract:** A partial monoid is a set A with a multiplication and a unit, but the multiplication is only defined on a certain subset of pairs. This means the multiplication m:AxA -> A is a partial map, a map that is only defined on a certain subset of its domain. The category of finite commutative partial monoids is an important ingredient for the construction of tangent infinity categories where a bicategory of spans of partial monoids encodes a weakly commutative ring structure. As partial maps are generalized by restriction categories one can directly generalize the definition of a partial monoid in any restriction category.

I will present a certain restriction category B of bags (a.k.a. multisets) with a partial monoid in it. Any partial monoid in any join restriction category X is induced through a functor from B to X which means the partial monoid in B is the generic partial monoid. This characterization of partial monoids is analogous to the concept of a Lawvere theory of an algebraic structure.

**Title**: Path Categories in A-Homotopy Theory

**Abstract**: A-homotopy theory is a discrete homotopy theory for graphs. While A-homotopy theory has many of the nice properties of the classical homotopy theory on topological spaces, we would like to know if it also has a nice structure to work with as well. We are pursuing this structure through path categories and homotopy type theory. In this talk, I will discuss why path categories might be the right option to get the structure that we are looking for in A-homotopy theory and how we can define a path category on the category of graphs (or something very close to it). This is joint work with Dr. Laura Scull.

**Title:** Principal bundles in Join restriction categories

**Abstract:** Principal bundles arise in different areas of matematics with different definitions. However, they all have in common some kind of local triviality. Here I will present some work in progress on generalizing these in terms of join restriction categories, a notion that means to capture properties of partial maps.

Most of the time we will spend on join-restriction categories and their properties. Then, abstracting concepts from differential geometry, we will consider fiber bundles and principal bundles and see that the existence of a global right action is a consequence and does not need to be demanded.

The slides can be found at https://www.dropbox.com/s/xaj48qzrtfvbvh6/Presentation_join_restriction_bundles.pdf?dl=0

**Title:** Connections in algebraic geometry

**Abstract:**

Any tangent category has a notion of connection for its differential bundles (the analog of vector bundles). In this talk, we’ll explore what this definition gives us in the tangent category of affine schemes. The talk will be roughly divided into three parts: (1) connections in general, and how they are described in a tangent category, (2) a review of the tangent category of affine schemes, (3) some particular examples of connections in the tangent category of affine schemes, and how the abstract tangent categorical definition relates to existing definitions of connections in algebraic geometry.

This talk will be split into two parts. The first part starts at 10am, the second one at 11:30am

The speaker’s notes on algebraic geometry can be found at https://www.reluctantm.com/gcruttw/publications/alg_geo_notes.pdf

**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.