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Many problems lead to the consideration of "algebras", given by an object A of a category A together with "actions" T,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/k .We show that the reflexion exists and is given as the colimit of a simple transfinite sequence, if A is cocomplete and the T 1 preserve either colimits or unions of suitably-long chains of subobjects.

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