Notation classifying topos

Question

The classifying topos of a geometric theory $\mathbb T$ is a topos $\mathcal E_\mathbb T$ such that for any other Grothendieck topos $\mathcal E$, the category of geometric morphisms from $\mathcal E_\mathbb T$ to $\mathcal E$ (or the other way round -- I tend to confuse the direction of geometric morphisms) is equivalent to the $\mathbb T$-models in $\mathcal E$.

I noticed that in a document of Caramello (which I think is part of her book Theories, Sites, Toposes) a different notation is used: the classifying topos of $\mathbb T$ is denoted by $\mathbf{Set}[\mathbb T]$ instead of $\mathcal E_\mathbb T$.

Question: In which sense can the classifying topos of $\mathbb T$ be viewed as adjoining $\mathbb T$ to the category $\mathbf{Set}$ of sets?

Answer 1

There is indeed a strong analogy between this situation and adjoining a polynomial variable to a ring, except that the direction of the arrows is somewhat messed up:

A ring homomorphism $\mathbb{Z}[X] \to R$ is the same thing as a ring homomorphism $\mathbb{Z} \to R$ (of which there is exactly one) together with one arbitrarily chosen element of $R$. (More precisely, we have a bijection, natural in the ring $R$.)

A geometric morphism $\mathcal{E} \to \mathrm{Set}[\mathbb{T}]$ is the same thing as a geometric morphism $\mathcal{E} \to \mathrm{Set}$ (of which there is exactly one, up to unique isomorphism) together with one arbitrarily chosen $\mathbb{T}$-model in $\mathcal{E}$. (More precisely, we have an equivalence of categories, natural in the topos $\mathcal{E}$.)

In case you wonder how to adjoin a $\mathbb{T}$-model to another topos than $\mathrm{Set}$, you can simply take the product $\mathcal{E}[\mathbb{T}] = \mathcal{E} \times \mathrm{Set}[\mathbb{T}]$, just like we have $R[X] = R \otimes \mathbb{Z}[X]$. (But be aware that the product of toposes is not given by the product of the underlying categories.) (And one could generalize to the case where $\mathbb{T}$ is not an ordinary geometric theory but instead a geometric theory internal to $\mathcal{E}$.)

In case you wonder, if you can adjoin a model of an arbitrary geometric theory to a topos, then what else can you adjoin to a ring apart from just a new ring element: you could for example "adjoin" two elements $X$ and $Y$ with the property that $X^2 = 5Y$; this would yield $R[X, Y]/(X^2 - 5Y)$. But the analogy arguably starts breaking down here.

The fact that the geometric morphisms go in the opposite direction from the ring homomorphisms (and that we use the product of toposes where we used the tensor product (coproduct) of rings) can be explained by saying that toposes are "geometric" objects and rings are "algebraic" objects. Or alternatively by the observation that while an element of a ring can be "pushed forward" along a ring homomorphism, a $\mathbb{T}$-model in a topos can be pulled back along a geometric morphism.