## Quotient of a category by a group action

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Let a group $G$ act on a small category $C$. If $G$ acts freely on objects, there is a sensible construction of the quotient $C/G$ (this is briefly spelt out here)

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The appropriate 2-categorical construction is to replace $C$ by $C \times EG$ where $EG$ is the codiscrete category ($\mathrm{Hom}(x, y) = *$ for all $x$ and $y$) with object set $G$ and $G$ acting by left translation, and forming the quotient $(C \times EG)/G$ as described there. But, I guess your question is about the 1-categorical colimit? – Reid Barton Apr 27 2010 at 6:21
Indeed. Do you have a reference for the 2-categorical construction, anyways? – Mariano Suárez-Alvarez Apr 27 2010 at 6:31

## 2 Answers

From the view point of representations of finite dimensional algebras, there is a construction called the "orbit category" or "skew category" construction. The construction appears, for example, a paper by Cibils and Marcos, a paper by Keller, and a paper by Asashiba.

If you browse these papers, you will notice that their construction is a version of the Grothendieck construction. Here we regard an action of a group $G$ on a category $C$ as a functor $$G \longrightarrow Cats.$$

So it's related to Reid's comment.

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Thomason's thesis says that for any diagram $F:D\to Cat$, the Grothendieck wreath product of D by F (or F by D?) is a model for the homotopy colimit, in the sense that its geometric realization is homotopy colimit of the geometric realization of the diagram. So I guess that's what's going on here (hence the relationship with Reid's comment). – Dan Ramras Apr 27 2010 at 18:31
Yes. That’s exactly what I wanted to say in the last sentence. – Dai Tamaki Apr 27 2010 at 19:42

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Here is a particular special case that seems to be useful to me:

Let $F: D \to G$-Sets be a diagram of $G$-sets, indexed by some small category D. Then ignoring the G-actions, one can form the Grothendieck wreath product $D\wr F$. Objects are pairs (d, x) with $x\in F(d)$, and morphisms $(d,x)\to (d', x')$ are arrows $a:d\to d'$ such that $F(a)(x) = x'$. This category inherits an action of G from the actions on the sets in the diagram; $g\cdot (d,x) = (d, gx)$ (on morphisms, $g$ looks like the identity: $g\cdot (a:(d,x)\to(d',x')) = a:(d,gx)\to(d',gx')$, which makes since $F(a)$ is $G$-equivariant.

There is another diagram $F/G : D\to$ Sets, which takes $d\in D$ to $F(D)/G$, and there's a natural functor $D\wr F \to D\wr (F/G)$. I claim that this functor satisfies the universal property of the colimit, i.e. $(D\wr F)/G = D\wr (F/G)$.

This special case has the interesting property that the nerve of $D\wr (F/G)$ is precisely $N_\cdot (D\wr F)/G$. I don't think that will hold for arbitrary group actions on categories.

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