Colimits of Representable Algebra-valued Functors

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 noted. In §§1-7 below we develop our general results, and in §§9-14, some examples. (One example is also worked in §5, to motivate the ideas of §6.) R. Paré has pointed out to me that my main result, Theorem 4.6, can be deduced from [17, Theorem 6.1.4, p.143, and following remark] and [17, Corollary 6.2.5, p.149]. However, as he observes, it is useful to have a direct proof.