## Return to Question

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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"?

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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?

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# $2$-categorical structure in Grothendieck's Galois Theory

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: 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?