Totally distributive toposes
Abstract
A locally small category E is totally distributive (as defined by Rosebrugh-Wood) 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 Kelly-Lawvere and Kennett-Riehl-Roy-Zaks.
Keywords
Cite
@article{arxiv.1108.4032,
title = {Totally distributive toposes},
author = {Rory B. B. Lucyshyn-Wright},
journal= {arXiv preprint arXiv:1108.4032},
year = {2012}
}
Comments
Now includes extended result: The lex totally distributive categories with a small set of generators are exactly the injective Grothendieck toposes; Made changes to abstract and intro to reflect the enhanced result; Changed formatting of diagrams