### Theory of Objects

Nov. 3rd, 2016 11:42 am1. Idea

The theory of objects is the logical theory whose models in a category 𝒞 are precisely the objects of 𝒞.

2. Definition

The theory of objects 𝕆 is the theory with no axioms over the signature with a single type and no primitive symbols except equality.

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The classifying topos 𝒮[𝕆] for the theory of objects 𝕆, or the object classifier, as it is also called, is the presheaf topos

**[FinSet,Set]**on the opposite category of

**FinSet**.

What motivates the terminology, is that for any topos

**E**, geometric morphisms

**E→𝒮[𝕆]**correspond to objects of

**E**.