### 2-Toposes

#### Posted by David Corfield

As 2-toposes seem to be cropping up a bit, here and here, let’s see if we can attract some experts to teach us about them.

On p. 36 of Mark Weber’s Strict 2-toposes, a 2-topos is defined as a finitely complete cartesian closed 2-category equipped with a duality involution and a classifying discrete opfibration. Cat is a good example of a 2-topos. Are there other familiar ones?

A 1-topos is a kind of 1-category. The 1-category of sets is a paradigmatic example, in which $1 \to 2$ (the set of truth values) is the classifier.

A 2-topos is a kind of 2-category. The 2-category of categories is a paradigmatic example, in which Pointed Set $\to$ Set is the classifier.

Hmmm, so what’s a 0-topos? It ought to be a kind of 0-category or set. The set of truth values should be a paradigmatic example. What kind of set should it be?

If in Set we have $A \to 2^A$, and in Cat we have $C \to Set^{C^op}$, is there a 0-Yoneda?

If it is possible to extract an internal language from a topos, what results from a similar process applied to a 2-topos? Do we find that it supports a higher order categorified logic?

## Re: 2-toposes

David asked:

Let me make some guesses, to get the ball rolling.

Trivially,

finitecategories must be an example. Then, since sheaves on a site give an important class of 1-toposes, surely stacks (of categories) on a site give an important class of 2-toposes.I think Urs asked somewhere whether the 2-category of toposes is a 2-topos. I would have thought not, since Mark requires a duality involution. In the case of $Cat$ this must be $(-)^{op}$, but this isn’t going to work for toposes: the opposite of a topos isn’t a topos.

I don’t think I know any examples of toposes except for sheaves on a site, finite sheaves on a site, and trivial variants thereof. I

thinkit’s the case that the effective topos, and probably other realizability toposes, provide further examples. Also, one has the classifying topos for any geometric theory; I don’t know if that’s always a topos of sheaves. Maybe there are analogues of these things for 2-toposes. And maybe there are 2-toposes that are not the analogue of anything in the 1-dimensional world!My guess: a 0-topos is just a set. Agreed: the set $2 = \{true, false\}$ should be a paradigmatic example.

Well, a $(-1)$-category is a truth value in the classical sense (i.e. $true$ or $false$), and there’s meant to be just one $(-2)$-category, somehow corresponding to the truth value $true$. (I suppose I mean “corresponding” in the same way that the $(-1)$-categories $true$ and $false$ correspond to the $0$-categories $0$ and $1$, and that a $0$-category corresponds to the discrete $1$-category on it.) Also, the exponentiation of truth values is surely implication: $\beta^\alpha$ means $(\alpha \Rightarrow \beta)$. So I think the $0$-Yoneda embedding is the statement $\alpha = (\alpha \Rightarrow true),$ where $\alpha \in 2 = \{ true, false\}$. How come that’s an equality? Well, both $\alpha$ and $(\alpha \Rightarrow \true)$ are elements of $2 = (-1)Cat$, which is a $0$-category, so they’re either equal or not — there’s no chance of mapping between them.

Now I have a question of my own. Cantor’s theorem says that $not(A \cong 2^A)$, for sets $A$. I spent a little while trying to imitate this for categories, to show that $not(C \simeq Set^{C^{op}})$ for categories $C$ (small if you like). But I couldn’t, the sticking point being that ‘op’. Can anyone (dis)prove this?