Grothendieck categories and their tensor product as filtered colimits

Abstract

We present two ways of recovering a Grothendieck category as a filtered colimit of small categories by means of the construction of the (2-)filtered (bi)colimit of categories from [9]. The first one, making use of the fact that Grothendieck categories are locally presentable, allows to recover a Grothendieck category as a filtered colimit of its subcategories of alpha-presentable objects, for alpha varying in the family of small regular cardinals. The second one, making use of the fact that Grothendieck categories are precisely the linear topoi, permits to recover a Grothendieck category as a filtered colimit of its linear site presentations. We then show that the tensor product of Grothendieck categories from [18] can be recovered as a filtered colimit of Kelly's alpha-cocomplete tensor product of the categories of alpha-presentable objects with alpha varying in the family of small regular cardinals. We use this construction to translate the functoriality, associativity and simmetry of Kelly's tensor product to the tensor product of Grothendieck categories