Totally distributive toposes
Abstract
A locally small category E is totally distributive (as defined by RosebrughWood) if there exists a string of adjoint functors t  c  y, where y : E > E^ is the Yoneda embedding. Saying that E is lex totally distributive if, moreover, the left adjoint t preserves finite limits, we show that the lex totally distributive categories with a small set of generators are exactly the injective Grothendieck toposes, studied by Johnstone and Joyal. We characterize the totally distributive categories with a small set of generators as exactly the essential subtoposes of presheaf toposes, studied by KellyLawvere and KennettRiehlRoyZaks.
 Publication:

arXiv eprints
 Pub Date:
 August 2011
 arXiv:
 arXiv:1108.4032
 Bibcode:
 2011arXiv1108.4032L
 Keywords:

 Mathematics  Category Theory;
 Computer Science  Logic in Computer Science;
 Mathematics  General Topology
 EPrint:
 Now includes extended result: The lex totally distributive categories with a small set of generators are exactly the injective Grothendieck toposes