Bulletin of the Australian Mathematical Society

Table of Contents - 1987 - Volume 36, Issue 03  

Research Article

How complete are categories of algebras?

Jiří Adámeka1

a1 Faculty of Electrical Engineering, FEL CVUT Suchbátarova 2, Praha 6, Czechoslovakia

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.

(Received November 13 1986)