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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
Grothendieck quasitoposes
A full reflective subcategory E of a presheaf category [C*,Set] is the
category of sheaves for a topology j on C if and only if the reflection
preserves finite limits. Such an E is called a Grothendieck topos. More
generally, one can consider two topologies, j contained in k, and the category
of sheaves for j which are separated for k. The categories E of this form, for
some C, j, and k, are the Grothendieck quasitoposes of the title, previously
studied by Borceux and Pedicchio, and include many examples of categories of
spaces. They also include the category of concrete sheaves for a concrete site.
We show that a full reflective subcategory E of [C*,Set] arises in this way for
some j and k if and only if the reflection preserves monomorphisms as well as
pullbacks over elements of E.Comment: v2: 24 pages, several revisions based on suggestions of referee,
especially the new theorem 5.2; to appear in the Journal of Algebr
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