Colimits of representable algebra-valued functors

TL;DR

This paper studies the category of algebra-valued functors that are representable, showing it has small colimits and initial objects, with explicit constructions in certain cases, and explores their structural properties.

Contribution

It proves that the category of representable algebra-valued functors has small colimits and initial objects, providing explicit constructions and analyzing their structural complexity.

Findings

Rep(C,D) has small colimits and an initial object.

Explicit constructions for representing coalgebras are provided in certain cases.

Representable functors can have complex structures depending on operations in C and D.

Abstract

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.