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How complete are categories of algebras?

Published online by Cambridge University Press:  17 April 2009

Jiří Adámek
Jiří Adámek
Affiliation:
Faculty of Electrical Engineering, FEL CVUT Suchbátarova 2, Praha 6, Czechoslovakia
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Completeness properties of (i) the category Alg(T) of T-algebras over a functor T: X → X and (ii) the subcategory XT in the case where T = (T, μ, η) is a monad, are investigated. It is known that if X is compact, then each XT 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 XT 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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 36 , Issue 3 , December 1987 , pp. 389 - 409
Copyright
Copyright © Australian Mathematical Society 1987

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