Dear Jim The answer is No, I believe, but the only thinking I have done about the question is as follows. In his paper on adjoint functors, Kan constructed a category %A for each category A. He used this to construct what we now call Kan extensions. It provides the domain of a diagram whose limit gives the end of a functor T : A^op x A --> X. At Bowdoin College in 1969, Mac Lane called %A the Kan subdivision category. It is a bit like barycentric subdivision in that each arrow f (edge) of A becomes an object (vertex) [f] of %A; the only non-identity arrows of %A are formally adjoined as in the situation [a] --> [f] <-- [b] where f : a --> b and a and b are identified with their identity arrows. There are not many composable pairs in %A so it seems that N%A is not the barycenric subdivision of the nerve of A. At this point I dug out my old copy of Kan's 1957 paper "On c.s.s. complexes" on which I scribbled some notes back in the late 1960s. If S is the category of finite sets and P : S --> Ord is the covariant powerset functor into ordered sets, we can define a functor D : S --> Simp into the category of simplicial sets (= css complexes) by (Ds)_q = Ord([q] , Ps). Kan's functor "Delta prime" is the restriction of D to the (topologists') simplicial category. Then Sd : Simp --> Simp is the extension along the Yoneda embedding of "Delta prime" to a colimit preserving functor. This yields the formula Sd(X)_q = coend^[n] X_n x Ord([q] , P[n]), but I see no way of using this when X is the nerve of a category A. Maybe there is more chance in the case of groupoids. (The left adjoint to Sd is Kan's functor Ex which starts a simplicial set on its way to becoming a Kan complex.) What made you suspect the existence of such a subdivision of categories? Ross On 16/08/2007, at 3:35 AM, James Stasheff wrote:
Is there a `barycentric' subdivision operator Sd on categories such that with N = nerve SdN = NSd ?