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?



Quotient of a category by a free group action
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 xZ 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 Xn -> X1 xX0 ... xX0 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.


