On the tensor product of large categories
14 September 2017
UAntwerp, Campus Middelheim, A.143 - Middelheimlaan 1 - 2020 Antwerpen (route: UAntwerpen, Campus Middelheim
Julia Ramos González
Wendy Lowen & Boris Shoykhet
PhD defence Julia Ramos González - Faculty of Science, Department of Mathematics and Computer Science
The product of schemes is an essential operation in classical algebraic geometry. The goal of this thesis is to define a product of noncommutative spaces generalizing the product of schemes.
One approach to noncommutative algebraic geometry consists in considering certain types of categories as models for noncommutative spaces. In this work we will follow this approach and focus on the following two models: Grothendieck abelian categories and well-generated dg categories (i.e. pretriangulated dg categories with well-generated homotopy category). Our aim is then to define suitable tensor products of both Grothendieck abelian categories and well-generated dg categories generalizing the product of schemes in the classical commutative setting.
The main tool to do so will be Gabriel-Popescu type theorems.
In the abelian setting, the classical Gabriel-Popescu theorem states that Grothendieck categories are the categories of sheaves over linear sites. We define a tensor product of linear sites and prove it induces a well-defined tensor product of Grothendieck categories. We further compare this tensor product with other well-known tensor products in the literature, such as the tensor product of locally presentable categories and Deligne's tensor product of abelian categories. Additionally, we analyse the monoidal structure induced by this tensor product in the category of Grothendieck categories from two different perspectives, using filtered bicolimits on the one hand, and bicategories of fractions on the other hand.
In the dg setting, an enhancement of Porta's triangulated Gabriel-Popescu theorem shows that well-generated dg categories are quasi-equivalent to quotients of derived dg categories with respect to localizing subcategories generated by a set. These representations allow us to define a tensor product with a universal property in the homotopy category of well-generated dg categories and prove its existence by providing an explicit construction.