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Let Cat denote the 1-category of small categories. The functor Mor : Cat -> Set which assigns to a category its set of morphisms (aka Hom([• -> •], -)) does not commute with most colimits. Does it commute with quotients by free group actions? In other words, if C is a small category and G is a group acting on C such that the action of G on the objects of C is free, does Mor(C/G) = (Mor C)/G?

edited Oct 15 2009 at 22:53

asked Oct 15 2009 at 19:19

Reid Barton
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Reid Barton · Accepted Answer · 2009-10-16 06:20:50Z UTC

OK, I found a "high-tech" argument which is maybe more convincing.

The external input to the argument is that if X -> Z <- Y is a diagram of G-sets with G acting freely on Z (and hence on X and Y) then (X x_Z Y)/G = (X/G) x_(Z/G) (Y/G). This is pretty clear because we can always write our original X -> Z <- Y in the form (X/G) x G -> (Z/G) x G <- (Y/G) x G.

Now we use the presentation of Cat as the localization of the category of simplicial sets in which the local objects are the simplicial sets X such that X_n -> X₁ x_X₀ ... x_X₀ X₁ is an isomorphism for all n >= 2. By general nonsense about localizations, if C is a category (viewed as a simplicial set) with a G-action, we can compute the colimit C/G by taking the colimit levelwise and then localizing the resulting object. But the levelwise quotient is already local, by our previous observation. Since C₁ represents the set of morphisms of C, we're done.

S. Carnahan · Answer 2 · 2009-10-15 22:48:46Z UTC

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The answer is yes, provided your action has the following data that you didn't specify:

For any g in G and any f: a -> b, an assignment gf: ga -> gb, in a way that strictly respects multiplication in G (including 1f = f).

If you don't say what G does on morphisms, I don't see how the question makes sense.

answered Oct 15 2009 at 22:48

S. Carnahan♦
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	You're right, I meant to say that G acts on C as an object of Cat. Hopefully the new wording is better. That was the answer I expected, but why is it true? – Reid Barton Oct 15 2009 at 22:55
	I guess the short answer would be that the set of all morphisms is a disjoint union of morphism sets between pairs of objects (this is often an explicit axiom in the definition of category), so the freeness of the G-action on objects translates to a free action on morphisms. I may be missing something - are there subtleties in your definition of quotient category C/G? – S. Carnahan♦ Oct 15 2009 at 23:28
	I was defining C/G as the colimit of the diagram BG -> Cat given by the G-object C (where BG is the groupoid with one object which has automorphism group G). Usually when forming a colimit in Cat we have to throw in some compositions of morphisms (e.g. consider the quotient [• -> •] / [• ∐ •] which has infinitely many morphisms). But now you've convinced me that in my case I can just write down a category with objects (Ob C)/G and morphisms (Mor C)/G and check the universal property (functors out of it correspond to G-invariant functors from C). – Reid Barton Oct 16 2009 at 0:10

Dan Ramras · Answer 3 · 2010-05-02 20:06:03Z UTC

This came up in something I'm working on, and this question was the first hit on google.

It seems to me that this has nothing to do with G acting freely, and that the answer is always yes. One can always define a quotient category C/G with objects (Ob C)/G and morphisms (Mor C)/G, with composition and identities defined in a more-or-less obvious manner (use the action of G to match up two arrows $f: C\to D$ and $h: gD\to E$ whose domains are equal in C/G, and then compose; note that $h\circ g(f)$ and $g^{-1} (h) \circ f$ are equivalent morphisms in the quotient). I was a little surprised that this makes sense, but it seems to.

I think it is elementary to check that C/G satisfies the universal property defining the colimit of the functor $BG \to$ Cat.

Am I missing something? Edit: YES! See the comments below. If you care to see what I was actually thinking about (a version of this which is actually correct) look at my answer to this subsequent question.

Quotient of a category by a free group action

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