algebraic logic via arrows

algebraic logic via arrows

[categories]

Date: Mon, 23 Jun 1997 17:39:12 +0000
From: Zinovy Diskin <diskin@fis.lv>

I'd be grateful for comments on the following question motivated by a problem 
in categorizing algebraic logic a la Tarski.

Let C be a  complete category with a factorization
system (E,M). Given an object A\in C, let us call a pair (m,e) with 
m\in M, cod(m)=A and e\in E, dom(e)=A, {\it compatible}, m~e, if the
square

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

is pull-back  (where (e',m') factorizes m;e ).            

In SET with the standard surjection-injection factorization, m~e iff 
for all a,b\in A, a\in A_m and e(a)=e(b)  entail b\in A_m where A_m 
is the subset of A corresponding to m.

Now let (e_i, i\in I) be a family of congruences compatible with
some m:*--->A,  e_i ~ m for all i\in I.  The question is what
properties of (C,E,M) are required to provide sup(e_i, i\in I) ~ m ?
(the collection CongA of e:A-->*, e\in E is a meet-complete semilattice due
to products, sup is join-via-meets in this lattice). 

If C is a category of finitary algebras over SET with the standard
epi-mono factorization, then sup(e_i) ~ m always holds due to
the finitary deduction property of taking sup in the congruence 
lattice: 

(a,b)\in sup(e_i) iff there exists a finite subfamily  e_1,...,e_k 
and c_0,...,c_k \in A s.t. a=c_0, b=c_k and e_j(c_{j-1}) =e_j(c_j) 
for  all j=1,...,k .

Zinovy Diskin

Re: algebraic logic via arrows

[categories]

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