#
Colimits of representable algebra-valued functors

##
George M. Bergman

If C and D are varieties of algebras in the sense
of general algebra, then by a representable functor
C --> D we understand a functor which, when
composed with the forgetful functor D --> Set, gives a
representable functor in the classical sense; Freyd showed that these
functors are determined by D-coalgebra objects of C.
Let Rep(C, D) denote the category of all
such functors, a full subcategory of Cat(C, D,
opposite to the category of D-coalgebras in C.

It is proved that Rep(C, D) has small
colimits, and in certain situations, explicit constructions for
the representing coalgebras are obtained.

In particular, Rep(C, D) always has an initial object.
This is shown to be ``trivial'' unless
C and D either both have *no* zeroary operations, or both
have *more than one* derived zeroary operation.
In those two cases, the functors in question may have
surprisingly opulent structures.
It is also shown that every set-valued representable functor
on C admits a universal morphism to a D-valued
representable functor.
Several examples are worked out in detail, and areas for further
investigation are noted.

Keywords:
representable functor among varieties of algebras,
initial representable functor, colimit of representable functors,
final coalgebra, limit of coalgebras;
binar (set with one binary operation), semigroup, monoid, group,
ring, Boolean ring, Stone topological algebra

2000 MSC:
Primary: 18A30, 18D35.
Secondary: 06E15, 08C05, 18C05, 20M50, 20N02

*Theory and Applications of Categories,*
Vol. 20, 2008,
No. 12, pp 334-404.

http://www.tac.mta.ca/tac/volumes/20/12/20-12.dvi

http://www.tac.mta.ca/tac/volumes/20/12/20-12.ps

http://www.tac.mta.ca/tac/volumes/20/12/20-12.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/20/12/20-12.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/20/12/20-12.ps

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