On the tensor product of linear sites and Grothendieck categories

We define a tensor product of linear sites, and a resulting tensor product of Grothendieck categories based upon their representations as categories of linear sheaves. We show that our tensor product is a special case of the tensor product of locally presentable linear categories, and that the tensor product of locally coherent Grothendieck categories is locally coherent if and only if the Deligne tensor product of their abelian categories of finitely presented objects exists. We describe the tensor product of non-commutative projective schemes in terms of Z-algebras, and show that for projective schemes our tensor product corresponds to the usual product scheme.Comment: New sections 5.3 on the alpha-Deligne tensor product and 5.4 on future prospect

On exact categories and applications to triangulated adjoints and model structures

We show that Quillen's small object argument works for exact categories under very mild conditions. This has immediate applications to cotorsion pairs and their relation to the existence of certain triangulated adjoint functors and model structures. In particular, the interplay of different exact structures on the category of complexes of quasi-coherent sheaves leads to a streamlined and generalized version of recent results obtained by Estrada, Gillespie, Guil Asensio, Hovey, J{\o}rgensen, Neeman, Murfet, Prest, Trlifaj and possibly others.Comment: 38 pages; version 2: major revision, more explanation added at several places, reference list updated and extended, misprints correcte

A convenient category for directed homotopy

We propose a convenient category for directed homotopy consisting of preordered topological spaces generated by cubes. Its main advantage is that, like the category of topological spaces generated by simplices suggested by J. H. Smith, it is locally presentable

Homotopy locally presentable enriched categories

We develop a homotopy theory of categories enriched in a monoidal model category V. In particular, we deal with homotopy weighted limits and colimits, and homotopy local presentability. The main result, which was known for simplicially-enriched categories, links homotopy locally presentable V-categories with combinatorial model V-categories, in the case where has all objects of V are cofibrant.Comment: 48 pages. Significant changes in v2, especially in the last sectio

Locally class-presentable and class-accessible categories

We generalize the concepts of locally presentable and accessible categories. Our framework includes such categories as small presheaves over large categories and ind-categories. This generalization is intended for applications in the abstract homotopy theory