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Consider $Set_{f}$ the category of finite sets and functions and let $G$ be a finite group.

We can define the category $Fun(G,Set_f)$ of functors, that, is objects are finite sets with group actions and morphisms equivariant functions.

I would like to assemble all the group action of all the subgroups $H\leq G$. That is, i want a category which has

  • an object is a pair $(X,\phi)$ where $X$ is a finite set and $\phi:H\longrightarrow Bij(X)$ a group homomorphism from some subgroup $H\leq G$,
  • a morphism $(X,\phi)\longrightarrow (Y,\psi)$ is a function of the underlying sets such that it is equivariant with respect the actions (possibly $\psi:H^{\prime}\longrightarrow Bij(Y)$ with $H\leq H^{\prime}$)

Is that the category of functors $Fun(Sub(G),Set_f)$ where $Sub(G)$ is the category of subojects of $G$ ?

If not, could someone explain what would be a functor $Sub(G)\longrightarrow Set_f$

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  • $\begingroup$ It's a bit hard to say what is a functor $Sub(G)\to Set_f$, because it's not clear what the category of subgroups of $G$ is. Is it the subcategory of groups formed by the subgroups of $G$? The poset of sugbroups seen as a thin category? $\endgroup$ – Arnaud D. May 19 '17 at 10:47
  • $\begingroup$ Probably i mean the poset of subgroups... $\endgroup$ – Masterstudent May 19 '17 at 11:03
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To answer your question, no, that is not what that functor category is. $\mathsf{Sub}(G)$ is just a partially ordered set (poset) which can be viewed as a category, and any poset with the same structure would be equivalent. The fact that you labelled the elements of this poset with groups is irrelevant. A functor $F : \mathsf{Sub}(G) \to \mathbf{Set}_f$ is a collection of finite sets, one per subobject of $G$, with a function $F(H_1)\to F(H_2)$ whenever $H_1 \leq H_2$. The only equations that would need to be satisfied by these functions is if $A \leq X \leq B$ and $A \leq Y \leq B$ then $F(A) \to F(X) \to F(B)$ is the same function as $F(A)\to F(Y)\to F(B)$.

Two equivalent ways of organizing the your data are: indexed categories and fibered categories. For the indexed category perspective, let $\mathbf{Grp}$ the category of groups and $\iota : \mathbf{Grp}\to\mathbf{Cat}$ be the functor viewing each group as a one-object category. Write $\mathcal{D}^\mathcal{C}$ for the functor category from $\mathcal{C}$ to $\mathcal{D}$. The (pseudo)functor $\mathbf{Set}_f^{\iota(-)} : \mathbf{Grp}^{op} \to \mathbf{Cat}$ gives you an indexed category. You can now restrict this (pseudo)functor to the (not full) subcategory of $\mathbf{Grp}$ consisting of $G$ and its subobjects to get an indexed category corresponding to what you desire.

Alternatively, consider the category whose objects are pairs $(G, F_G)$ where $G$ is a group and $F_G : \iota G \to \mathbf{Set}_f$, and whose arrows $(f,\alpha) : (H,F_H) \to (G,F_G)$ consist of a group homomorphism $f : H \to G$ and a natural transformation $\alpha : F_H \to F_G \circ \iota(f)$. There is a projection $\pi(G,F_G) = G$ that takes this category to $\mathbf{Grp}$ and makes this a fibered category. (Indeed, this is just the Grothendieck construction applied to the indexed category above.) You can then change the base of the fibration to be the subcategory of $\mathbf{Grp}$ consisting of $G$ and its subobjects to get a fibered category over the subgroups. The total category of the resulting fibration will be exactly the category you describe.

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