Skip to main content
×
×
Home

On the ordered set of reflective subcategories

  • G. M. Kelly (a1)
Abstract

Given a category A, we consider the (often large) set Ref A of its reflective (full, replete) subcategories, ordered by inclusion. It is known that, even when A is complete and cocomplete, wellpowered and cowellpowered, the intersection of two reflective subcategories need not be reflective. Supposing that A admits (i) small limits and (ii) arbitrary (even large) intersections of strong subobjects, we prove that an infimum ∧iCi in Ref A must necessarily be the intersection ∩iCi. Accordingly Ref A is not in general, even for good A, a complete lattice. We show, however, under the same conditions on A, that Ref A does admit small suprema ∨iCi, given by the closure in A of the union ∪iCi under the limits of type (i) and (ii) above.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      On the ordered set of reflective subcategories
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      On the ordered set of reflective subcategories
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      On the ordered set of reflective subcategories
      Available formats
      ×
Copyright
References
Hide All
[1]Adámek, J. and Rosický, J., “Intersections of reflective subcategories”, (in preparation).
[2]Borceux, F. and Kelly, G.M., “On locales of localizations”, J. Pure Appl. Algebra, 46 (1987), (to appear).
[3]Börger, R., Tholen, W., Wischnewsky, M.B. and Wolff, H., “Compact and hypercomplete categories”, J. Pure Appl. Algebra 21 (1981), 129144.
[4]Cassidy, C., Hébert, M. and Kelly, G.M., “Reflective subcategories, localizations and factorization systems”, J. Austral. Math. Soc. Ser. A. 38 (1985), 287329; Corrigenda Ibid 41 (1986), 286.
[5]Freyd, P.J. and Kelly, G.M., “Categories of continuous functors, I”, J. Pure Appl. Algebra 2 (1972), 169191.
[6]Im, G.M. and Kelly, G.M., “Some remarks on conservative functors with left adjoints”, J. Korean Math. Soc. 23 (1986), 1933.
[7]Kelly, G.M., “Monomorphisms, epimorphisms, and pull-backsJ. Austral. Math. Soc. 9 (1969), 124142.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed