* Re: subdivision
@ 2007-08-25 7:27 Ross Street
0 siblings, 0 replies; 2+ messages in thread
From: Ross Street @ 2007-08-25 7:27 UTC (permalink / raw)
To: Categories
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
> ?
^ permalink raw reply [flat|nested] 2+ messages in thread
* subdivision
@ 2007-08-15 17:35 James Stasheff
0 siblings, 0 replies; 2+ messages in thread
From: James Stasheff @ 2007-08-15 17:35 UTC (permalink / raw)
To: categories
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
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