Canonical and Op-canonical Lax Algebras

The definition of a category of (T,V)-algebras, where V is a unital commutative quantale and T is a Set-monad, requires the existence of a certain lax extension of T. In this article, we present a general construction of such an extension. This leads to the formation of two categories of (T,V)-algebras: the category Alg(T,V) of canonical (T,V)-algebras, and the category Alg(T′,V) of op-canonical (T,V)-algebras. The usual topological-like examples of categories of (T,V)-algebras (preordered sets, topological, metric and approach spaces) are obtained in this way, and the category of closure spaces appears as a category of canonical (P,V)-algebras, where P is the powerset monad. This unified presentation allows us to study how these categories are related, and it is shown that under suitable hypotheses both Alg(T,V) and Alg(T′,V) embed coreflectively into Alg(P,V).