×

The closure of a class of colimits. (English) Zbl 0656.18004

The authors consider \({\mathcal V}\)-categories, where \({\mathcal V}\) is a symmetric monoidal closed category. Write \(\phi\) *T for the colimit of T: \({\mathcal K}\to {\mathcal A}\) indexed by \(\phi\) : \({\mathcal K}^{op}\to {\mathcal V}\). Suppose that \(\Phi\) is a class of such indexing types (\({\mathcal K},\phi)\) and write \(\Phi^*\) for the class of indexing types (\({\mathcal F},\psi)\) such that every \(\Phi\)-cocomplete \({\mathcal A}\) is \(\psi\)-cocomplete and every \(\Phi\)-cocontinuous functor is \(\psi\)-cocontinuous. The authors show that \(\psi\in [{\mathcal F}^{op},{\mathcal V}]\) lies in \(\Phi^*\) if and only if it lies in the \(\Phi\)-colimit closure of \({\mathcal F}\) in [\({\mathcal F}^{op},{\mathcal V}]\), and they characterize those \(\Phi\) for which \(\Phi^*=\Phi\). They also comment on a similar sort of questions and pose open problems.
Reviewer: P.Pták

MSC:

18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)

Citations:

Zbl 0656.18005
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Betti, R., Cocompleteness over coverings, J. Austral. Math. Soc. Ser. A, 39, 169-177 (1985) · Zbl 0574.18005
[2] Deleanu, A.; Hilton, P. J., Borsuk shape and a generalization of Grothendieck’s definition of pro-category, Math. Proc. Cambridge Philos. Soc., 79, 473-482 (1976) · Zbl 0327.18004
[3] Grothendieck, A.; Verdier, J. L., Préfaisceaux; Exposé I, (Théorie des Topos et Cohomologie Étale des Schemas, 269 (1972), Springer: Springer Berlin), SGA IV, Lecture Notes in Mathematics · Zbl 0249.18021
[4] Johnstone, P. T.; Joyal, A., Continuous categories and exponentiable toposes, J. Pure Appl. Algebra, 25, 255-296 (1982) · Zbl 0487.18003
[5] Kelly, G. M., Basic Concepts of Enriched Category Theory, (London Mathematical Society Lecture Notes Series, 64 (1982), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0709.18501
[6] Kock, A., Limit monads in categories, Aarhus Univ. Mat. Inst., 6 (1967), Preprint Series 1967/68
[7] Lindner, H., Morita equivalences of enriched categories, Cahiers Topologie Géom. Différentielle, 15, 377-397 (1974) · Zbl 0319.18006
[8] Street, R. H.; Walters, R. F.C., The comprehensive factorization of a functor, Bull. Amer. Math. Soc., 79, 936-941 (1973) · Zbl 0274.18001
[9] Tholen, W., Completions of categories and shape theory, Seminarberichte aus dem Fachbereich Mathematik und Informatik der Fernuniversität Hagen, 12, 125-142 (1982)
[10] Wood, R. J., Free colimits, J. Pure Appl. Algebra, 10, 73-80 (1977) · Zbl 0382.18002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.