4 research outputs found
Grothendieck categories as a bilocalization of linear sites
We prove that the 2-category Grt of Grothendieck abelian categories with
colimit preserving functors and natural transformations is a bicategory of
fractions in the sense of Pronk of the 2-category Site of linear sites with
continuous morphisms of sites and natural transformations. This result can
potentially be used to make the tensor product of Grothendieck categories from
earlier work by Lowen, Shoikhet and the author into a bi-monoidal structure on
Grt
Grothendieck categories and their tensor product as filtered colimits
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
Exponentiable Grothendieck categories in flat Algebraic Geometry
We introduce and describe the -category of
Grothendieck categories and flat morphisms between them. First, we show that
the tensor product of locally presentable linear categories
restricts nicely to . Then, we characterize exponentiable
objects with respect to : these are continuous Grothendieck
categories. In particular, locally finitely presentable Grothendieck categories
are exponentiable. Consequently, we have that, for a quasi-compact
quasi-separated scheme , the category of quasi-coherent sheaves
is exponentiable. Finally, we provide a family of examples
and concrete computations of exponentials.Comment: Minor revision. The proofs of Sec 5 have been expanded to make the
paper self containe