Subject: excision and n-cubes
From: "Ronnie Brown"
Date: Mon, 24 Jul 2006 16:47:38 +0100
What I wrote on excision got too abbreviated!
Of course an n-cube in a category C may be regarded as a map of
(n-1)-cubes. However there is a functor from (r-cubes in C) to (r-cubes of
r-cubes in C). To see an example, a square
C A
B X
becomes a square of squares
CC CA
CC CA
CC CA
BB BX
and a pushout square becomes a pushout of squares of squares. This is
useful if you are given a functor \Pi: (squares) \to (some algebra).
So the excision map for an n-cube becomes a map of (n-1)-cubes of
(n-1)-cubes, or an n-cube of (n-1)-cubes!
Ronnie Brown
www.bangor.ac.uk/r.brown