2-natural operations on toposes

Question

Any pseudonatural endomorphism $\Phi$ of the forgetful 2-functor $U:Topos^{coop}\to Cat$ is essentially determined by its component $\Phi_{Set}$. But which endofunctors of $Set$ induce such a $\Phi$? More generally, one can consider pseudonatural transformations $U^n \Rightarrow U$, which are determined by a functor $\mathsf{Set}^n \to \mathsf{Set}$.

Here

• $\mathsf{Topos}^\mathrm{coop}$ is the 2-category of Grothendieck toposes and left exact left adjoint functors -- doubly dual to the 2-category of toposes and geometric morphisms.

• The functor $U : \mathsf{Topos}^\mathrm{coop} \to \mathsf{Cat}$ sends a topos to its underyling category and a left exact left adjoint to its underlying functor.

• For $n \in \mathbb N$, the functor $U^n$ is the composite of $U$ with the $n$th power functor $\mathcal C \mapsto \mathcal C^n$.

So it's natural to define a 2-natural operation on toposes to be a pseudonatural transformation $U^n \to U$ for $n \in \mathbb N$. In this language, the question is:

Question: What are all the 2-natural operations on toposes?

To see that such an operation $\Phi: U^n \Rightarrow U$ is determined by the component $\Phi_\mathsf{Set}: \mathsf{Set}^n \to \mathsf{Set}$, first note that $\Phi$ is determined by its components on presheaf toposes, because left exact localizations are essentially split epimorphisms in $\mathsf{Cat}$. Then the components at presheaf toposes are determined by the components at $\mathsf{Set}$ because $U$ preserves $\mathsf{Cat}$-cotensors.

I would also be interested in the analogous question for $\infty$-toposes.

Answer 1

The "2-natural operations" $U^n \to U$ correspond to functors $\mathbf{FinSet}^n \to \mathbf{Set}$. (edit: As Simon points out, these correspond to the finitary functors $\mathbf{Set}^n \to \mathbf{Set}$.)

The 2-functor $U$ is birepresented by the object classifier $\mathbf{Set}[\mathbb{O}] = [\mathbf{FinSet},\mathbf{Set}]$ (the classifying topos for the theory of objects). Thus by the bicategorical Yoneda lemma, the category of pseudonatural transformations $U^n \to U$ is equivalent to the category $U(n\cdot \mathbf{Set}[\mathbb{O}])$ (where $ n \cdot {}$ denotes $n$-fold (bi)coproduct in $\mathrm{Topos}^\mathrm{coop}$), which is equivalent to the category $[\mathbf{FinSet}^n,\mathbf{Set}]$ (see Cole's paper The bicategory of topoi and spectra).

Answer 2

(For me the category of toposes is the opposite of the category of left exact left adjoint functors and natural transformations, so $Topos^{co}$ in your sense)

The functor $U$ is representable by the classifying topos of objects, i.e. the topos $S[\mathbb{O}]$ which as a category is the category of functors from finite sets to sets, i.e. :

$$ U( \mathcal{T}) = Hom(\mathcal{T}, S[ \mathbb{O}] ) $$ Similarly,

$$ U(\mathcal{T}) ^n = Hom(\mathcal{T}, S[\mathbb{O}]^n)$$

$S[\mathbb{O}]^n$ being the category of functors from (finite set)$^n$ to Set.

Now by the Yoneda lemma, and up to $2$-categorical details that I will totally ignore, a natural transformation from $U^n$ to $U$ is the same as a morphism from $S[\mathbb{O}]$ to $S[\mathbb{O}]^n$, i.e. it is given by an object of $S[\mathbb{O}]^n$, i.e. a functor from (finite set)$^n$ to Set.

Claim/exercice: given a functor from (finite set)$^n$ to $Set$ its actions $Set^n \rightarrow Set$ corresponds to the unique extension commuting to directed colimits. So in the end those "operations" are exactly the same as the operations $Sets^n \rightarrow Sets$ which commutes to directed colimits.

I believe everything works exactly the same for $\infty$-toposes, replacing sets and finite sets by "spaces" and "finitely generated spaces" (I mean finite CW-complexes), (Of course this will relies on a large amount of results from Lurie's books, although I think one can avoid manipulating $(\infty,2)$-categories by just forgeting the non-invertible $2$-cells at least in a first time...)