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.