Subject: answer to the question about homotopy commutative diagram
From: Nathalie Wahl
Date: Fri, 27 Jul 2007 18:48:13 +0200 (CEST)
You can always rectify a homotopy commutative square, but if it's a cube,
you need one higher homotopy. Following Dwyer-Kan and Segal, here is a
simple condition for rectifying a diagram: Thinking of a strictly
commutative diagram of topological spaces as a functor from a discrete
category D to Top, a homotopy commutative version of the diagram is
rectifiable if it can be given as a functor from a topological category tD
with the same objects as D and with a contractible space of morphisms over
each morphism of D, i.e. there is a projection
p:tD -> D
taking path components on the spaces of morphisms which gives a homotopy
equivalence between tD(a,b) and D(a,b) for each pair of objects (a,b). In
the case of a homotopy commmutative square, tD is the same as D except
along the diagonal of the square, where it has one interval of morphisms
encoding the homotopy. For a cube, there are intervals of morphisms along
the diagonals of each sides, and a contractible space of morphisms along
the long diagonal which contains a hexagone given by the 6 homotopies
coming from the side diagonals. Boardmann and Vogt give more structure to
the contractible space sitting over the long diagonal, in particular
because a homotopy commutative cube usually comes with more than 6
homotopies between the 6 ways of going down the cube, and you can give it
as much structure as you like or as fits your example. All that matters is
that the space is contractible.
I wrote this up in a short section in my paper [Infinite loop space
structure(s) on the stable mapping class group, Topology 43 (2004),
343-368, section 2]. If you want something closer to Boardmann and Vogt,
you could also look at Markl-Blanc [Higher homotopy operations, Math. Z.
245 (2003), no. 1, 1--29.]
Nathalie
On Fri, 27 Jul 2007, Don Davis wrote:
> Subject: question about homotopy commutative diagram
> From: Gaucher Philippe
> Date: Thu, 26 Jul 2007 19:18:54 +0200
>
> Dear all,
>
> I'd like to prove that it is possible to rectify a homotopy commutative
cube
> of topological spaces. I know how to do that using the Boardman-Vogt
> construction but it's very complicated. Does anyone know a paper where
this
> fact is already proved ?
>
> Thanks in advance. pg.