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).

Keywords: V-matrix, (T,V)-algebra, ordered set, metric space, topological space, approach space, closure space, closeness space, topological category

2000 MSC: 18C20, 18B30, 54A05

*Theory and Applications of Categories,*
Vol. 14, 2005,
No. 10, pp 221-243.

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