Corrigendum: How complete are categories of algebras

Extract

Proposition 1.6 states that the category Alg(P*) is hypercomplete but non-compact. This is true, but the argument for non-compactness must be corrected as follows.

Let A be the category of algebras (X, x′, x″) where x′, x″: P*X → X are operations with x′(φ) = x″(φ), and homomorphisms are mappings which are P*-homomorphisms with respect to both operations. The embedding E: Alg(P*) → A with E(X, x) = (X, x, x) preserves colimits, although it is not a left adjoint. In fact, the preservation of colimits C = colim D is obvious in case C is finite, and for the infinite case the original argument presented in the paper is correct (namely, one of the colimit maps is onto). E is not a left adjoint because given A = (X, x′, x″) in A with X infinite and x′(M) ≠ x″(M) for any M ≠ φ, then there is no universal arrow into A with respect to E.