Contents

# Contents

## Idea

The theory of objects is the logical theory whose models in a category $\mathcal{C}$ are precisely the objects of $\mathcal{C}$.

## Definition

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

## Properties

Of course, $\mathbb{O}$ is a geometric theory, and as models for $\mathbb{O}$ in a topos $\mathcal{E}$ correspond to objects of $\mathcal{E}$, we can use its classifying topos to get a representation of the objects of $\mathcal{E}$.

###### Proposition

The classifying topos for the theory of objects is the presheaf topos $[FinSet, Set]$ over the opposite category of the category FinSet of finite sets.

## Further Remarks

• For more general base toposes $\mathcal{S}$, it is a theorem due to Andreas Blass that the theory of objects has a classifying topos precisely if $\mathcal{S}$ has a natural numbers object.

• A step up on the ladder of logical complexity is the theory of inhabited objects $\mathbb{O}_\exists$ that adds to $\mathbb{O}$ the existential axiom $\top\vdash(\exists x)\top$. Its classifying topos $Set[\mathbb{O}_\exists]$ is the functor category $[FinSet_\exists, Set]$ with $FinSet_\exists$ the category of finite nonempty sets. It has the property that every topos $\mathcal{E}$ admits a localic morphism to $Set[\mathbb{O}_\exists]$.1

• If instead of an additional axiom one adds a single constant symbol to the signature of $\mathbb{O}$ one obtains the theory of pointed objects $\mathbb{O}_\ast$ i.e. the empty theory relative to the signature with a single sort and a single constant. Its models are pointed objects and its classifying topos is $[FinSet_\ast,Set]$. (See the discussion&references at classifying topos for the theory of objects.)

• In the syntax-free approach to geometric theories of Johnstone (2002, I B4.2) the theory of objects corresponds to the forgetful functor sending an $\mathcal{S}$-topos to its underlying category. (See at geometric theory the section on the functorial definition.)

Sections B4.2, D3.2 of

1. cf. Johnstone (2002 II, p.773) and Joyal-Tierney (1984). For some further information on $FinSet_\exists$ see the references at generic interval.