Abstract
In this chapter the quotients of a given theory of presheaf type are investigated by means of Grothendieck topologies that can be naturally attached to them, establishing a ‘semantic’ representation for the classifying topos of such a quotient as a subtopos of the classifying topos of the given theory of presheaf type. It is also shown that the models of such a quotient can be characterized among the models of the theory of presheaf type as those which satisfy a key property of homogeneity with respect to a Grothendieck topology associated with the quotient. A number of sufficient conditions for the quotient of a theory of presheaf type to be again of presheaf type are also identified: these include a finality property of the category of models of the quotient with respect to the category of models of the theory and a rigidity property of the Grothendieck topology associated with the quotient.
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