TL;DR
This paper characterizes totally distributive toposes, especially lex totally distributive categories with small generators, as injective Grothendieck toposes and essential subtoposes of presheaf toposes, linking them to existing topos theory concepts.
Contribution
It provides a new characterization of totally distributive categories and connects them to well-studied classes like injective Grothendieck toposes and subtoposes.
Findings
Lex totally distributive categories with small generators are exactly injective Grothendieck toposes.
Totally distributive categories with small generators are precisely essential subtoposes of presheaf toposes.
The paper links the concept of total distributivity to established topos-theoretic structures.
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.
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