554 research outputs found
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
Effective descent morphisms of regular epimorphisms
Let be a regular category with pushouts of regular epimorphisms by
regular epimorphism and the category of regular epimorphisms in .
We prove that every regular epimorphism in is an effective descent
morphism if, and only if, is a regular category. Then, moreover, every
regular epimorphism in is an effective descent morphism. This is the case,
for instance, when is either exact Goursat, or ideal determined, or is a
category of topological Mal'tsev algebras, or is the category of -fold
regular epimorphisms in any of the three previous cases, for any
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