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.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2011
- DOI:
- 10.48550/arXiv.1108.4032
- arXiv:
- arXiv:1108.4032
- Bibcode:
- 2011arXiv1108.4032L
- Keywords:
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- Mathematics - Category Theory;
- Computer Science - Logic in Computer Science;
- Mathematics - General Topology
- E-Print:
- Now includes extended result: The lex totally distributive categories with a small set of generators are exactly the injective Grothendieck toposes