Subject: categories: Name for a concept

From: Michael Barr <[email-masked]>
Date: Wed, 30 Nov 2005 20:48:52 -0500 (EST)

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