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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.

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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).

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(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...)

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  • $\begingroup$ I was considering 2-naturality rather than (2,1)-naturality mostly because only in that case did I know how to reduce to endofunctors of $\mathsf{Set}$, but it seems that this argument works in either setting -- it's probably actually more interesting to understand (2,1)-natural or $(\infty,1)$-natural transformations rather than 2-natural or $(\infty,2)$-natural transformations since they could a priori be more general. Thanks! Since these two answers are so similar, it's a coin flip to decide which to accept. $\endgroup$ – Tim Campion May 17 '18 at 15:18
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    $\begingroup$ @TimCampion : It is possible to see very abstractly why $2$-naturality and $(2,1)$-naturality are equivalent as a natural transformation $f \Rightarrow g$ between two geometric morphism $f,g : \mathcal{X} \rightrightarrows \mathcal{Y}$ is the same as a geometric morphism $\mathcal{X} \times \mathbb{S} \rightarrow \mathcal{Y}$ where $\mathbb{S}$ is the Sierpinski locale (topos). There is probably some abstract statement to make here in terms of the fact that the category of toposes is tensored over Cat... But I'm sure what it is. $\endgroup$ – Simon Henry May 18 '18 at 11:34
  • $\begingroup$ Ah, right! Really, under representability hypotheses, a 2-category structure on a $(2,1)$ category corresponds to a choice of some internal cocategory object. And in the context of topoi, I've even seen this explained before, so I should have known! $\endgroup$ – Tim Campion May 18 '18 at 13:59

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