Generalizations of topos theory.

Question

Is there a generalisation for topos theory? If so; then can you define that particular generalisation in brevity?

Answer 1

A topos is a special kind of category, so really the notion of a category generalizes the notion of a topos.

Slightly less tongue-in-cheek, a (elementary) topos is a Cartesian Closed Category, with all Finite Colimits, and a Subobject Classifier.

As such, Cartesian Closed Categories are a nice generalization of toposes, even moreso are the Cocomplete Cartesian Closed Categories.

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In general, toposes let us interpret all of (intuitionistic) First Order Logic inside them. If we restrict to any fragment of iFOL we will get a new class of categories which contains the toposes, but contains more categories as well. These categories will be able to interpret some portion of iFOL inside them. For example:

• Finite Product Categories allow us to interpret equational structures
• Regular Categories allow us to interpret "regular logic"
• Coherent Categories allow us to interpret "coherent logic"

For an overview of these ideas, see Steve Awodey's lecture notes here

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I hope this helps ^_^