Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T02:57:52.109Z Has data issue: false hasContentIssue false
  • English
  • Français

Note on monoidal localisation

Published online by Cambridge University Press:  17 April 2009

Brian Day
Brian Day
Affiliation:
University of Chicago, Chicago, Illinois, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

If a class Z of morphisms in a monoidal category A is closed under tensoring with the objects of A then the category obtained by inverting the morphisms in Z is monoidal. We note the immediate properties of this induced structure. The main application describes monoidal completions in terms of the ordinary category completions introduced by Applegate and Tierney. This application in turn suggests a “change-of-universe” procedure for category theory based on a given monoidal closed category. Several features of this procedure are discussed.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 8 , Issue 1 , February 1973 , pp. 1 - 16
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Applegate, H. and Tierney, M., “Categories with models”, Seminar on triples and categorical homology theory, 156–244 (Lecture Notes in Mathematics, 80. Springer-Verlag, Berlin, Heidelberg, New York, 1969).CrossRefGoogle Scholar
[2]Applegate, H. and Tierney, M., “Iterated cotriples”, Reports of the Midwest Category Seminar IV, 56–99 (Lecture Notes in Mathematics, 137. Springer-Verlag, Berlin, Heidelberg, New York, 1970).CrossRefGoogle Scholar
[3]Day, Brian, “On closed categories of functors”, Reports of the Midwest Category Seminar IV, 1–38 (Lecture Notes in Mathematics, 137. Springer-Verlag, Berlin, Heidelberg, New York, 1970).Google Scholar
[4]Day, Brian, “Construction of biclosed categories”, PhD thesis, University of New South Wales, 1970.Google Scholar
[5]Day, Brian, “A reflection theorem for closed categories”, J. Pure Appl. Algebra 2 (1972), 1–11.CrossRefGoogle Scholar
[6]Ellenberg, Samuel and Kelly, G. Max, “Closed categories”, Proc. Conf. Categorical Algebra, La Jolla, California, 1965, 421–562. (Springer-Verlag, Berlin, Heidelberg, New York, 1966.)Google Scholar
[7]Gabriel, Peter, Ulmer, Friedrich, Lokal präsentierbare Kategorien (Lecture Notes in Mathematics, 221. Springer-Verlag, Berlin, Heidelberg, New York, 1971).CrossRefGoogle Scholar
[8]Gabriel, P. and Zisman, M., Calculus of fractions and homotopy theory (Ergebnisse der Mathematik imd ihrer Grenzgebiete, Band 35. Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
[9]Wolff, H., “V-localisations and V-fractional categories”, (to appear).Google Scholar
[10]Wolff, H., “V-localisations and V-monads”, J. Algebra (to appear).Google Scholar