Filtered bicolimit presentations of locally presentable linear categories, Grothendieck categories and their tensor products

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 $\alpha$-presentable objects, with $\alpha$ 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 $\alpha$-cocomplete linear categories of the corresponding subcategories of $\alpha$-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 $\alpha$-cocomplete linear categories to the tensor product of locally presentable linear categories.