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Gmane
From: Walter Tholen <tholen <at> pascal.math.yorku.ca>
Subject: Re: When are all monos regular?
Newsgroups: gmane.science.mathematics.categories
Date: Tuesday 14th March 2006 15:14:19 UTC (over 10 years ago)
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?
> 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
 
CD: 3ms