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    • 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. 5.5.1.2 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

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