A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on

TLDR

Algebraic structures such as free monads, monoids, cocomplete categories, and generalized sheaf categories motivate the study of their algebras. The paper aims to prove that the category of algebras is reflective in a suitable comma‑category, thereby guaranteeing free algebras, cocompleteness, and adjoints to algebraic functors. This is achieved by constructing the reflection as the colimit of a transfinite sequence in the comma‑category T/A, under cocompleteness of A and preservation of colimits or unions by the endofunctors. The resulting reflection is stronger than previous results and will be used in future enriched studies of structured categories.

Abstract

Many problems lead to the consideration of “algebras”, given by an object A of a category A together with “actions” T k A → A on A of one or more endofunctors of A, subjected to equational axioms. Such problems include those of free monads and free monoids, of cocompleteness in categories of monads and of monoids, of orthogonal subcategories (= generalized sheaf-categories), of categories of continuous functors, and so on; apart from problems involving the algebras for their own sake. Desirable properties of the category of algebras - existence of free ones, cocompleteness, existence of adjoints to algebraic functors - all follow if this category can be proved reflective in some well-behaved category: for which we choose a certain comma-category T/A We show that the reflexion exists and is given as the colimit of a simple transfinite sequence, if A is cocomplete and the T k preserve either colimits or unions of suitably-long chains of subobjects. The article draws heavily on the work of earlier authors, unifies and simplifies this, and extends it to new problems. Moreover the reflectivity in T/A is stronger than any earlier result, and will be applied in forthcoming articles, in an enriched version, to the study of categories with structure.