A tangent category approach to operadic geometry

Abstract

In his Lectures on Noncommutative Geometry, Ginzburg proposes a theory of algebraic noncommutative (affine) geometry. One of the fundamental insights of noncommutative geometry is to regard associative, non necessarily commutative, algebras as geometric spaces. In the last section of the aforementioned lectures, Ginzburg suggests an ambitious generalization of his work: he observes that most of the constructions he characterized in the noncommutative case, carry over into the realm of operadic algebras and he proposes a theory of operadic geometry. From a philosophical viewpoint one wonders if the similarities captured by Ginzburg could hide a deeper phenomenon: a common language which captures some important features of these examples. In this thesis, tangent category theory is applied for the first time to describe the patterns and similarities observed by Ginzburg. This work largely extends Cruttwell and Lemay's attempt to employ tangent category theory to capture significant features of commutative algebraic geometry. From the perspective of operad theory, this thesis translates in the context of tangent categories some important operadic constructions, such as derivations, enveloping operads, and modules. From the perspective of tangent category theory, it provides new examples of noncommutative non-pointwise models of geometry described with tangent categories. First, We show that each operad is canonically associated with two tangent categories: the algebraic and the geometric tangent categories. Once established this functorial correspondence between operads and tangent categories, we describe two important constructions. First, we show an equivalence between slice tangent categories and enveloping operads; second, we employ this result to classify differential bundles as modules over the operadic algebras. In the last chapter, we apply the established relationship between operads and tangent categories to the theory of algebraic deformation. First, we prove that the category of operad itself and its opposite carry two tangent structures, which are closely related to deformations. Finally, we explore some ideas, inspired by tangent category theory, to classify all infinitesimal deformations of an operadic algebra.