TL;DR
This paper establishes a correspondence between subtoposes of a classifying topos and quotients of the associated geometric theory, enabling transfer of topos-theoretic and logical constructions.
Contribution
It introduces a bijection linking subtoposes and quotient theories, bridging topos theory and logic in a novel way.
Findings
Bijection between subtoposes and quotient theories
Transfer of topos-theoretic constructions to logical lattice
Enhanced understanding of the structure of geometric theories
Abstract
We show that there is a bijection between the subtoposes of the classifying topos of a geometric theory T over a signature L and the closed geometric theories over L which are `quotients' of the theory T; next, we analyze how classical topos-theoretic constructions on the lattice of subtoposes of a given topos can be transferred, via the bijection above, to logical constructions in the corresponding lattice of theories.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Homotopy and Cohomology in Algebraic Topology · Rings, Modules, and Algebras