Subject: categories: When are all monos regular?

From: Andrej Bauer <[email-masked]>
Date: Mon, 13 Mar 2006 16:01:17 +0100

This might be an embarrassingly easy question, but I always get confused about
it. When are all monos in an algebraic category regular (or more generally,
when are all monos in a regular category regular)? What are some sensible
sufficient or necessary conditions?

For example, all monos in the category of groups are regular. How about the
category of lattices, or lattices with a top element?

Andrej Bauer

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From: "Walter Tholen" <[email-masked]>
Date: Tue, 14 Mar 2006 10:14:19 -0500

To say that monos are regular (in a variety or, more generally, in a general
category satisfying some very minor hypotheses) amounts to the so-called
Intersection Property of Amalgamations: for any two algebras A, B with a common
subalgebra C, if there are monomorphisms f : A --> D, g: B --> D that coincide
on C, then one can choose f, g with the additional property that C is their
pullback. References:

C.M. Ringel: JPAA 2 (1972) 341-42
W. Tholen: Algebra Univ. 14 (1982) 391-397
E.W. Kiss, L. Marki, P. Prohle, W. Tholen: Studia Sci. Math. Hungaricum 18
(1983) 79-141.

The last paper contains a large table of specific categories, including
lattices (the affirmative answer is attributed to Gratzer in this case), plus
an extensive list to the literature. For some categories, the question whether
monos are regular can get quite involved, for example in the category of
compact (Hausdorff) groups, for which an affirmative answer was provided by
Poguntke (Math. Z. 130 (1973) 107-117).

The property in question obviously implies "epimorphisms are surjective", but
examples witnessing failure of the converse statement are harder to find: see
again the four-author paper.

Hope this helps.

Walter Tholen.

On Mar 13,  4:01pm, Andrej Bauer wrote:
> Subject: categories: When are all monos regular?