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Yesterday I described the essence of model theory in one tweet. I'm proud of it. 
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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.


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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.

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