Grothendieck's Galois Theory, as developed in SGA I, V.4, or very gently in Lenstra's notes, establishes an equivalence between profinite groups and Galois categories. We can put this into the following more general framework: Let $\mathsf{sets}$ denote the category of finite sets; for a topological group $\pi$ let $\pi\mathsf{-sets}$ denote the category of finite sets on which $\pi$ acts continuously. There are functors
$\mathsf{Cat}/\mathsf{sets} \leftrightarrow \mathsf{TopGrp}^{\mathrm{op}}$
given by $(F : C \to \mathsf{sets}) \mapsto \mathrm{Aut}(F)$ (which is topologized as a closed subgroup of $\prod_{X \in C} \mathrm{Aut}(F(X))$) and in the other direction by $\pi \mapsto \pi\mathsf{-sets}$. In order to avoid set-theoretic problems, $C$ and $\pi$ should be essentially small. There are canonical maps $\eta_C : C \to \mathrm{Aut}(F)\mathsf{-sets}$ and $\varepsilon_{\pi} : \pi \to \mathrm{Aut}(\pi\mathsf{-sets})$ which satisfy the triangular identities; hence we have an adjunction! And every adjunction restricts to an equivalence of categories between its fixed points. Now it's fairly easy to recognize profinite groups as those fixed points on the right hand side, but Grothendieck's nontrivial insight is the classification of the fixed points on the left hand side, which he calls Galois categories (in short: $C$ has finite limits and colimits, which $F$ preserves, there are mono-epi decompositions, monos split off, and $F$ is conservative).
Question 1. Does this point of view of Grothendieck's Galois theory have already appeared somewhere?
My real question is the following: Actually $\mathsf{Cat}/\mathsf{sets}$ is a $2$-category. A morphism $(C,F) \to (C',F')$ is a functor $P : C \to C'$ together with a chosen isomorphism $F \cong F' P$. A $2$-morphism between morphisms $P,Q : C \to C'$ is a natural transformation of the underyling functors, which is base-point preservering in the obvious sense.
Question 2. How can we endow $\mathsf{TopGrp}^{\mathrm{op}}$ with the structure of a $2$-category in such a way that the adjunction above becomes an $2$-adjunction?
– – Edit – – The comments + answers make me believe that it is just the "trivial" $2$-categorical enrichment with identities as $2$-morphisms. And since there is no reaction to Q1, I expect that the answer is "no, this is new"?
Best Answer
EDIT: answer is expanded on OP's request.
The category $\pi Set_{fin}$ of finite, continuous $\pi$-sets is just the functor category $TopCat(\mathbf{B}\pi,finSet)$ where $\mathbf{B}$ gives the one-object topological groupoid associated with a topological group, and $finSet$ is the category of finite sets, viewed as a topologically discrete category. The groupoid $\mathbf{B}\pi$ is canonically pointed, so induces the functor $\pi Set_{fin} \to finSet$ i.e. an object of $Cat/finSet$. If you replace $TopGrp$ with the equivalent category $TopGpd_{1obj}$ of one-object topological groupoids, then this naturally comes with a notion of 2-arrow, and it is, as John Baez points out, conjugation by an element of the codomain. Then 2-arrows in $\pi Set_{fin}$ are natural transformations of functors to $finSet$.
Now really you are working with $TopGrp^{op}$, so you have two choices as to the direction of the 2-arrows, if you are taking 2-arrows as specified above. This should fall out of the definitions. Also, since $TopGpd_{1obj}$ is a (2,1)-category, you need to restrict attention to the (2,1)-category $Cat/finSet_{(2,1)}$ underlying $Cat/finSet$. Here $Cat/finSet_{(2,1)}$ is the isocomma category: objects are categories over $finSet$, arrows are triangles commuting up to a natural isomorphism and 2-arrows are natural isomorphisms in $Cat$ that are compatible with the 2-arrows in the triangles.
In more detail: given two objects $F\colon C\to finSet$ and $G\colon D\to finSet$, an arrow $F\to G$ is a pair consisting of a functor $f\colon C\to D$ and a natural isomorphism $c\colon G\circ f \Rightarrow F$. Given two arrows, $(f,c),(g,d)\colon F\to G$, a 2-arrow between them is a natural isomorphism $a\colon f\Rightarrow g$ such that the obvious 3-dimensional diagram commutes. Since all 2-arrows are invertible, from this commuting 3-d diagram we can write down an invertible endo-arrow $c+(G\circ a)+d^{-1}$ of $F$ (here $+$ is vertical composition of natural transformations--from right to left--$G$ really denotes the identity transformation on the functor $G$ and $\circ$ is horizontal composition).
I'm fairly confident that this is a 2-arrow in $TopGpd_{1obj}$. If one doesn't restrict to the (2,1)-category $Cat/Set_{(2,1)}$ then this doesn't work. (So really you should be thinking about $Gpd$-enriched categories, not so much general 2-categories.) EDIT: Actually, it is easy to see that this is a 2-arrow in $TopGpd_{1obj}$, because the homomorphism induced by $(f,c)$ is conjugation by $c$ (with whiskering): $\alpha \mapsto c+(f\circ \alpha)+c^{-1}$.
Thus we have a 2-functor $Cat/finSet_{(2,1)}\to TopGpd_{1obj}^{op}$.
Now in going the other way, I think we actually need to use not just $TopGpd_{1obj}$, but the isococomma category $\ast/TopGpd_{1obj}$, where a functor between topological groupoids respects the basepoint up to a 2-arrow (which is an automorphism of the canonical basepoint). I haven't checked but this looks like the description of the functor in the preceeding paragraphs works better. Then the functor $\ast/TopGpd_{1obj}^{op} \to Cat/finSet_{(2,1)}$ is just exponentiation with $finSet$, and we don't have to think too hard about what the 2-arrows etc do.
Then you need to worry about the adjunction. But if you already have a 1-adjunction, half the work is done.