The geometric tangent category of an operad (part 2)

Marcello Lanfranchi – 14 February 2024

Algebraic geometry, synthetic differential geometry, and differential geometry are all geometrical theories devoted to capturing some notion of differentiability. In the same way as a category provides a common language to study mathematical entities, a tangent category provides a minimal context to develop geometrical theories concerned with infinitesimals and local linear behaviours. So far, tangent category theory proved to be up to this challenge by providing models for all of these three theories. But what about noncommutative geometry? The noncommutative geometry program is devoted to interpreting associative algebras as geometrical spaces. Even though noncommutative geometry shares lots of commonalities with the other mentioned theories, noncommutative spaces lack a classical notion of points so, it is not a trivial question asking whether or not tangent categories could be employed in this setting. In this talk, I will present a joint work with Sacha Ikonicoff (University of Ottawa) and JS Lemay (Macquarie University) in which we provided the first nontrivial example of a tangent category for (algebraic) noncommutative geometry. Our approach is far more general: we showed that for any operad, the corresponding category of affine schemes, which is the opposite of the category of algebras of the operad, comes equipped with a tangent structure. I divided the talk into two parts: In the first part, I aim to introduce the fundamentals of tangent category theory and discuss the notion of tangent monads and operads. In the second part, I will present our construction which relates operads with tangent categories and I will discuss two important classifications: vector fields and differential objects.