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Totally distributive toposes

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.