Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2009
    • (edited Oct 16th 2012)

    edited reflective subcategory and expanded a bit the beginning

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeNov 26th 2009

    I added a query and done a bit more succinct sectioning: distinguishing characterizations from just properties.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 5th 2010
    • (edited May 5th 2010)

    at reflective subcategory

    • replied to Zoran’s old query box query

    • added the theorem about reflective subcategories of cartesian closed categories from cartesian closed category

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 16th 2012

    added at reflective subcategory a new subsection Accessible reflective subcategory with a quick remark on how accessible localizations of an accessible category are equivalently accessibe reflective subcategories.

    This is prop. in HTT. Can anyone give me the corresponding theorem number in Adamek-Rosicky’s book?

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeOct 16th 2012

    I can’t find it stated explicitly. But “only if” follows immediately from 2.53 (an accessibly embedded subcategory of an accessible category is accessible iff it is cone-reflective), while “if” follows immediately from 2.23 (any left or right adjoint between accessible categories is accessible).

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 16th 2012

    Thanks! I have put that into the entry here.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeOct 16th 2012

    I wonder whether it’s explicit in Makkai-Pare; I don’t have my copy of that with me in Princeton.

    • CommentRowNumber8.
    • CommentAuthorZhen Lin
    • CommentTimeSep 30th 2013

    I replaced statement (3) in Proposition 1 to make the statements equivalent. (Statement (3) previously only had the part about idempotent monads.)

  1. Added an example: The category of affine schemes is a reflective subcategory of the category of schemes, with the reflector given by XSpecΓ(X,𝒪 X)X \mapsto Spec \Gamma(X,\mathcal{O}_X).

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeOct 9th 2014

    Thanks! There is much discussion of the infinity-version of this example at function algebras on infinity-stacks (following what Toën called “affine stacks” and what Lurie now calls “coaffine stacks”).

    • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeOct 12th 2014

    Proposition 4.4.3 in

    • A.L. Rosenberg, Noncommutative schemes, Comp. Math. 112, 93–125 (1998)

    is a noncommutative analogue.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeOct 12th 2014

    I have now added both of these comments to the example.

    • CommentRowNumber13.
    • CommentAuthorIngoBlechschmidt
    • CommentTimeMay 9th 2016
    • (edited May 9th 2016)

    Added to reflective subcategory the observation that (assuming classical logic) SetSet has exactly three reflective subcategories. I learned this from Kelly’s and Lawvere’s article On the complete lattice of essential localizations.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeMay 9th 2016

    Thanks for all your additions recently!

    Here is a direct pointer to the edit that, I suppose, you are announcing. (Providing direct links like this within an entry of non-trivial length makes it easier for us all to spot what you are pointing us to.)

    By the way, we have an entry subterminal object. I have added cross-links with subsingleton.

    • CommentRowNumber15.
    • CommentAuthorIngoBlechschmidt
    • CommentTimeApr 8th 2017
    • (edited Apr 8th 2017)

    Added to reflective subcategory a reference by Adámek and Rosický about non-full reflective subcategories.

    • CommentRowNumber16.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 8th 2017

    Perhaps the link to the paper should not be to the pdf? And better would be a proper human-readable reference.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJun 29th 2018
    • (edited Jun 29th 2018)

    I gave the Prop. with “alternative characterizations” a proof, by pointing to the relevant sub-propositions proved in other entries.

    diff, v86, current

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)