Observations

on one page

1. 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.


===================

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.

r6research - just friended this person.