Subject: categories: Re: name for a concept

From: "Clemens.BERGER" <[email-masked]>
Date: Thu, 08 Dec 2005 12:06:15 +0100

Let me suggest still another terminology:

For this, call a class S of maps in an arbitrary category *(co)stable*
iff S is closed under composition and under (co)base change. Then call a
commutative square *S-exact * (resp. *S-coexact*) iff the induced map to
the pullback (resp. from the pushout) belongs to S. It is then easy to
check that S-(co)exact squares compose for any (co)stable class S (which
I believe is the minimal condition to impose on any reasonable
distinguished class of commutative squares).

In an abelian category, the class M of monos (resp. the class E of epis)
is not only stable (resp. costable) but also costable (resp. stable).
With this terminology, Hilton's exact squares can either be identified
with the E-exact squares or with the M-coexact squares, which explains
why it is a self-dual concept, cf. the first message of Michael Barr and
the last message of Marco Grandis.

In homotopy theory, there is the important concept of a *homotopy
pullback* which is the ``homotopy invariant'' substitute for an
ordinary pullback. For those who are familiar with Quillen model
categories, it is very useful in practice that if a Quillen model
category is *right proper* (i.e. its class of fibrations is stable),
then a commutative square with two parallel fibrations is a homotopy
pullback *if and only if* the square is exact with respect to the class
of trivial fibrations (those fibrations which are also weak
equivalences). There is of course a dual statement for homotopy pushouts
in a left proper Quillen model category.

With best regards,

Clemens Berger.