[FOM] CAT co-well-powered?

Todd Wilson twilson at csufresno.edu
Thu Sep 11 11:32:38 EDT 2003

On Thu, 11 Sep 2003, HAUCOURT emmanuel wrote:
> Is CAT (the category of small categories) co-well-powered?
> Is anyone has references about epic of CAT?

Yes, CAT is co-wellpowered.  This was first proved (implicitly) in

    J.R. Isbell, "Epimorphisms and dominions, III".  Amer. J. Math. 90
    (1968), 1025-1030.  [reviewed in Math Reviews 38 #5877]

which gives an explicit description of the "dominion" of a functor
f:A->B (i.e., the smallest subcategory of B containing f(A) that
appears as the equalizer of two functors out of B) extending a similar
description of dominions in semigroups (his so-called Zig-Zag theorem)
from the first paper in his four-paper series on epis and dominions:

    J.R. Isbell, "Epimorphisms and dominions", Proc. Conf. Categorical
    Algebra (La Jolla, CA, 1965), Springer, New York, 1966, 232-246.
    [Math Reviews 35 #105a]

Since epimorphisms are characterized by having dominions equal to
their ranges, this gives a description of epis in CAT, and shows, in
particular, that their codomains are bounded in cardinality in terms
of their domains: |B| <= |A| + Aleph_0.  Otherwise, epis in CAT are
not particularly well-behaved: for example, there are extremal epis
that are not regular, and regular epis that are not surjective.

For a very comprehensive survey of categorical and algebraic
properties of over 100 concrete categories, complete with a large
bibliography up to the early 1980s, see the paper:

    E.W. Kiss, L. Márki, P. Pröhle, W. Tholen, "Categorical algebraic
    properties:  A compendium on amalgamation, congruence extension,
    epimorphisms, residual smallness, and injectivity", Studia
    Scientiarum Mathematicarum Hungarica 18 (1983), 79-141.

Todd Wilson                               A smile is not an individual
Computer Science Department               product; it is a co-product.
California State University, Fresno                 -- Thich Nhat Hanh

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