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Relative injectivity as cocompleteness for a class of distributors

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Maria Manuel Clementino and Dirk Hofmann

Notions and techniques of enriched category theory can be used to study
topological structures, like metric spaces, topological spaces and
approach spaces, in the context of topological theories. Recently in [D.
Hofmann, Injective spaces via adjunction, arXiv:math.CT/0804.0326] the
construction of a Yoneda embedding allowed to identify injectivity of
spaces as cocompleteness and to show monadicity of the category of
injective spaces and left adjoints over SET. In this paper we
generalise these results, studying cocompleteness with respect to a given
class of distributors. We show in particular that the description of
several semantic domains presented in [M. Escardo and B. Flagg, Semantic
domains, injective spaces and monads, Electronic Notes in Theoretical
Computer Science 20 (1999)] can be translated into the V-enriched
setting.

Keywords:
Quantale, V-category, monad, topological theory, distributor,
Yoneda lemma, weighted colimit

2000 MSC:
18A05, 18D15, 18D20, 18B35, 18C15, 54B30, 54A20

*Theory and Applications of Categories,*
Vol. 21, 2008,
No. 12, pp 210-230.

http://www.tac.mta.ca/tac/volumes/21/12/21-12.dvi

http://www.tac.mta.ca/tac/volumes/21/12/21-12.ps

http://www.tac.mta.ca/tac/volumes/21/12/21-12.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/21/12/21-12.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/21/12/21-12.ps

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