Canonical and op-canonical lax algebras
By Gavin J. Seal. Theory and Applications of Categories, 2005
62 citations recorded by the source corpus.
The definition of a category of (T,V)-algebras, where V is a unital com- mutative 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 )o fcanonical (T,V)-algebras, and the category Alg(T,V )o fop-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 )a ndAlg(T,V) embed coreflectively into Alg(P,V).
Subfields: Computational Theory and Mathematics, Management Science and Operations Research, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/be160b1ef229e19f5b0a446318f075f80a23d4a9
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In Computational Theory and Mathematics
In Management Science and Operations Research
In Mathematical Physics
Citations in the corpus
- A unified framework for generalized multicategories (2009, unknown): “of the powerset monad, whose algebras would be closure spaces. To include this type of example, the requirement that TM was the same as T on functions needed to be removed. Seal’s definition, given in [Sea05], was the following: B.8. Definition. Suppose that V is a monoidal preorder. A set monad with”
- Monoidal topology : a categorical approach to order, metric, and topology (2014, methodology): “The fact that Barr’s two ultrafilter convergence axioms may be taken verbatim as filter convergence axioms to already fully characterize topological spaces appeared only in [Seal, 2005]; a slightly weaker version of this result had been established earlier in [Pisani, 1999].”