Re: algebraic logic via arrows

[categories <cat-dist@mta.ca>]

Date: Mon, 30 Jun 1997 11:07:12 +0100
From: Marco Grandis <grandis@dima.unige.it>

This is a collateral remark, but I would be surprised if there were no
connections.

In an abelian category  C,  a square of epis and monos as considered by
Zinovy Diskin

       *  --m-->  A
        |                 |
        e'               e
        |                 |
       v                v
      X  --m'-->  *

is a pullback iff it is a pushout.  Such a bicartesian square represents a
"subquotient"  X  of  A  (a subobject  m'  of a quotient  e,  and a
quotient  e'  of a subobject  m);  and it is a subobject  X  >-+->  A  in
the category of relations  RelC.  Subquotients are a crucial tool in
homological algebra, where everything - from homology to the terms of
spectral sequences - is a subquotient of some "main object" (or an induced
morphism between subquotients). See MacLane, "Homology".

A categorical study of subquotients in abelian categories and their
extensions can be found in the following papers of mine. The last setting
("semiexact" and "homological" categories) is much more general than the
classical abelian one

M. Grandis, Sous-quotients et relations induites dans les categories
exactes, Cahiers Top. Geom. Diff. 22 (1981), 231-238.

-, On distributive homological algebra, I. RE-categories; II. Theories and
models; III. Homological theories. Cahiers Top. Geom. Diff. 25 (1984),
259-301; 353-379; 26 (1985), 169-213.

-, On the categorical foundations of homological and homotopical algebra,
Cahiers Top. Geom. Diff. Categ. 33 (1992), 135-175.


With best regards
Marco Grandis