Canonical and op-canonical lax algebras
Gavin J. Seal · Theory and applications of categories · 2005
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).