How complete are categories of algebras?

E-ISSN： 1755-1633|36|3|389-409

ISSN： 0004-9727

Source： Bulletin of the Australian Mathematical Society, Vol.36, Iss.3, 1987-12, pp. : 389-409

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Abstract

Completeness properties of (i) the category Alg(T) of T-algebras over a functor T: X → X and (ii) the subcategory X^T in the case where T = (T, μ, η) is a monad, are investigated. It is known that if X is compact, then each X^T is compact; we present a functor T: Set → Set such that Alg(T) is non-compact, although it is hypercomplete. If T either preserves epis or has a rank, we prove that Alg(T) and X^T are topologically algebraic over X provided X satisfies mild additional hypotheses. Nevertheless, a natural monad over the category of Δ-comp1ete posets is exhibited such that its category of algebras is solid, but not topologically algebraic, over Set.