From: categories <cat-dist@mta.ca>
To: categories <categories@mta.ca>
Subject: algebraic logic via arrows
Date: Sun, 29 Jun 1997 11:37:27 -0300 (ADT) [thread overview]
Message-ID: <Pine.OSF.3.90.970629113718.25190G-100000@mailserv.mta.ca> (raw)
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
next reply other threads:[~1997-06-29 14:37 UTC|newest]
Thread overview: 2+ messages / expand[flat|nested] mbox.gz Atom feed top
1997-06-29 14:37 categories [this message]
1997-07-01 2:42 categories
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