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

Published online by Cambridge University Press:  17 April 2009

G.M. Kelly
G.M. Kelly
Affiliation:
Department of Pure Mathematics, University of Sydney, Sydney, New South Wales 2006, Australia.
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Abstract

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Many problems lead to the consideration of “algebras”, given by an object A of a category A together with “actions” TkA → 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 Tk 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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 22 , Issue 1 , August 1980 , pp. 1 - 83
Copyright
Copyright © Australian Mathematical Society 1980

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