Re: Name for a concept

["Ronald Brown" <ronnie@ll319dg.fsnet.co.uk>]

This kind of condition occurs in topology as a fibrant square - but where
all the maps are fibrations as is the map A ---> B x_D C . This can be
generalised to cubes. See a paper by R. Steiner on

`Resolutions of spaces by n-cubes of fibrations',  J. London Math. Soc.(2),
34, 169-176, 1986

used to build a complete (strict) algebraic model of homotopy n-types which
allows some computations.

This raises the spectre in algebra of

Resolutions of A by free  crossed n-cubes of A.

to give a more `nonabelian' homological algebra. Of course crossed n-cubes
of A should be equivalent to n-fold groupoids in A. This would presumably
bring in higher versions of nonabelian tensor products in A; a bibliography
of such a tensor, mainly for n=2, has 90 items.

This probably does not help to answer Mike's question on the name!

Ronnie
www.bangor.ac.uk/r.brown/nonabtens.html



----- Original Message -----
From: "Michael Barr" <mbarr@math.mcgill.ca>
To: "Categories list" <categories@mta.ca>
Sent: Thursday, December 01, 2005 1:48 AM
Subject: categories: Name for a concept


> Is there a standard name for a square
> A ----> B
> |       |
> |       |
> |       |
> v       v
> C ----> D
> in which the canonical map A ---> B x_D C is epic?  I had always called it
> a weak pullback, but Peter Freyd claims that that phrase is reserved for
> the case that it satisfies the existence, but not necessarily the
> uniqueness of the definition of pullback.  In fact, he claims it means
> that Hom(E,-) converts it to the kind of square I am talking about.
> What is interesting is that in an abelian category, it satisfies
> this condition iff it satisfies the dual condition iff the evident
> sequence A ---> B x C ---> D is exact.  Putting a zero at the left end
> characterizes a genuine pullback and at the other end a pushout.
>
> Michael
>
>
>