Subject: Re: question & response
Date: Fri, 5 Apr 2002 12:16:31 -0500
From: "Vidhyanath Rao" <rao.3@osu.edu>
To: "Don Davis" <dmd1@lehigh.edu>

Michael Barr wrote:

> I am not familiar with the Artin-Mazur codiagonal, but Jon Beck once
> showed me a proof (unpublished like most of his work) that the
geometric
> realization of the diagonal and of the whole thing were homotopic.

I think that there is a proof in Anderson's article in an old Bull AMS.
It lloks complicated because he was doing it for all model categories,
but my memory is that it becomes simple if applied only to sets/spaces.
He defined both realization and bi-realization explicitely as coends. It
would be interesting to know the pre-history of this cycle of ideas.

I liked this when I first saw it, because it gave a simple proof
Dowker's theorem about the equivalence of Cech and Vietrois complexes:
Given a relation R between X and Y, form the bicomplex of (x_0, \dots
x_n, y_0, \dots\,y_m) such that x_i R y_j. Vertical and horizontal
realizations, applied to X and an open cover, give complexes equivalent
to Vietrois and Cech complexes and both are equivalent to the
bi-realization.

PS: What is the Artin-Mazur codiagonal?