4 research outputs found

    Grothendieck categories as a bilocalization of linear sites

    Full text link
    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

    Full text link
    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

    Full text link
    We introduce and describe the 22-category Grt♭\mathsf{Grt}_{\flat} of Grothendieck categories and flat morphisms between them. First, we show that the tensor product of locally presentable linear categories ⊠\boxtimes restricts nicely to Grt♭\mathsf{Grt}_{\flat}. Then, we characterize exponentiable objects with respect to ⊠\boxtimes: 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 XX, the category of quasi-coherent sheaves Qcoh(X)\mathsf{Qcoh}(X) 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
    corecore