[Paper Review] Colimits of representable algebra-valued functors

George M. Bergman|arXiv (Cornell University)|Nov 5, 2007

Homotopy and Cohomology in Algebraic Topology15 references4 citations

TL;DR

This paper establishes that the category of representable functors from one variety of algebras to another admits small colimits, with explicit constructions for the representing coalgebras in certain cases. The key contribution is proving the existence of an initial representable functor, which is nontrivial only when both source and target varieties either lack zeroary operations or have more than one derived zeroary operation, revealing rich structural behavior in such cases.

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.

Motivation & Objective

• To investigate the existence and structure of colimits in the category of representable functors between varieties of algebras.
• To determine when the initial representable functor is nontrivial, particularly in relation to zeroary operations in the source and target varieties.
• To provide explicit constructions of representing coalgebras for colimits in specific cases, such as monoids, groups, and rings.
• To show that every set-valued representable functor on a variety admits a universal morphism to a D-valued representable functor.
• To explore structural properties of the category of representable functors, including non-isomorphic monic-epic morphisms and non-one-to-one equalizers.

Proposed method

• Uses the duality between representable functors and coalgebras in the source variety, showing that the category of representable functors is equivalent to the opposite of the category of D-coalgebras in C.
• Applies general category-theoretic results on colimits in categories of coalgebras, leveraging the fact that colimits in the category of coalgebras correspond to limits in the dual category.
• Employs term algebras and identities to define varieties of algebras, with a fixed regular infinite cardinal bounding arities to ensure smallness.
• Constructs explicit representations of colimits via universal properties, particularly using free algebras and quotient constructions in cases like monoids and rings.
• Analyzes specific examples—such as free monoids, groups, and tensor rings—to illustrate structural phenomena like non-isomorphic monic-epic maps and non-injective equalizers.
• Relies on known results from the literature (e.g., [9], [6]) to characterize coalgebras via bipartite graphs and to verify properties of morphisms.

Experimental results

Research questions

• RQ1 When does the initial representable functor between two varieties of algebras fail to be trivial?
• RQ2 Under what conditions on the varieties C and D can one explicitly construct the representing coalgebra for a colimit of representable functors?
• RQ3 Can every set-valued representable functor on a variety C be universally mapped to a D-valued representable functor for a given variety D?
• RQ4 What structural phenomena arise in the category of representable functors, such as monic-epic morphisms that are not isomorphisms?
• RQ5 How do properties like one-to-one-ness and equalizers behave under the duality between representable functors and coalgebras?

Key findings

• The category of representable functors from C to D has small colimits, and these colimits are represented by coalgebras in C.
• The initial representable functor is trivial unless both C and D either have no zeroary operations or have more than one derived zeroary operation.
• In the nontrivial case, representable functors can exhibit surprisingly rich and complex structures.
• Explicit constructions of representing coalgebras are obtained for key examples, including free monoids, groups, and tensor rings over abelian groups.
• There exist morphisms of coalgebras that are both monic and epic but not isomorphisms, demonstrating non-trivial behavior in the category of coalgebras.
• Equalizers in the category of D-coalgebras in C need not be injective on underlying sets, even when they are equalizers in the category of abelian groups.