Filtered bicolimit presentations of locally presentable linear categories, Grothendieck categories and their tensor products
J. Ramos González
TL;DR
This work develops two robust filtrations for recovering Grothendieck categories as filtered bicolimits and analyzes their interaction with tensor products. It shows that any locally presentable k-linear category, in particular any Grothendieck category, is a filtered bicolimit of its α-presentable subcategories, and that their tensor product is obtained as a filtered bicolimit of the Kelly tensor products of these α-presentable parts. It further proves that Grothendieck categories admit a filtered bicolimit via linear site presentations, but this second presentation generally fails to recover tensor products, underscoring a fundamental distinction between the two representations. A key contribution is translating the functoriality, associativity, and symmetry of the Kelly tensor product into the tensor product of locally presentable (and Grothendieck) linear categories, enabling computation of tensor products of cocontinuous functors and providing a foundation for noncommutative geometric constructions grounded in linear site and Ind$_ ext{α}$ formalisms.
Abstract
We investigate two different ways of recovering a Grothendieck category as a filtered bicolimit of small categories and the compatibility of both with the tensor product of Grothendieck categories. Firstly, we show that any locally presentable linear category (and in particular any Grothendieck category) can be recovered as the filtered bicolimit of its subcategories of $α$-presentable objects, with $α$ varying in the family of small regular cardinals. We then prove that the tensor product of locally presentable linear categories (and in particular the tensor product of Grothendieck categories) can be recovered as a filtered bicolimit of the Kelly tensor product of $α$-cocomplete linear categories of the corresponding subcategories of $α$-presentable objects. Secondly, we show that one can recover any Grothendieck category as a filtered bicolimit of its linear site presentations. We then prove that the tensor product of Grothendieck categories, in contrast with the first case, cannot be recovered in general as a filtered bicolimit of the tensor product of the corresponding linear sites. Finally, as a direct application of the first presentation, we translate the functoriality, associativity and symmetry of the Kelly tensor product of $α$-cocomplete linear categories to the tensor product of locally presentable linear categories.