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.

TAC Home