subdivision

subdivision

[James Stasheff]

Is there a `barycentric' subdivision operator Sd on categories
such that with N = nerve
SdN = NSd
?
and how many other notions of Sd are there?

jim



 	Jim Stasheff		jds@math.upenn.edu

 		Home page: www.math.unc.edu/Faculty/jds

As of July 1, 2002, I am Professor Emeritus at UNC and
I will be visiting U Penn but for hard copy
         the relevant address is:
         146 Woodland Dr
         Lansdale PA 19446       (215)822-6707

Re: subdivision

[Ross Street]

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
> ?