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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. 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 9th 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

  2. Added an idea section. Expanded the section regarding limits and colimits in a reflective subcategory (one sentence for the case when the ambient category is complete and complete) and added a reference.

    I also added two examples. The first example is that Cat is a reflective subcategory of sSet. The second is example is that for a Lawvere theory T, its category of models is a reflective subcategory of the category of all functors from T to Set. I added references for these examples.

    This is my first contribution to nLab so my code hygiene may not be the best. If you have any suggestions on anything please let me know and I’ll be happy to make fixes.

    Jade Master

    diff, v91, current

  3. Fixed typo in the Cat \to sSet example.

    Christopher James Stough-Brown

    diff, v92, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeJul 9th 2019
    • (edited Jul 9th 2019)

    Jade, thanks for contributing!

    I made just some minor cosmetic edits to the Idea-section here that you added, such as hyperlinking more of the keywords.

    Here some hints on editing:

    • Hyperlinks within the nLab are and should be obtained simply by including the word/phrase to be hyperlinked in double square brackets. For instance [[Grp]] produces Grp. Most entries have all possible variants of their titles declared as redirects. For instance [[category of groups]] produces category of groups which gives the same link as Grp. Should the redirect not exist, you can either a) add a redicrect (look at the very bottom of the source code of any page to see how) or b) force it on the spot with a vertical slash: [[Grp|Groups]] produces Groups, which once again links to the same page as the previous two cases.

    • Instiki maths is almost exactly like LaTeX maths, except for some small differences. One of them is the “feature” that consecutive letters in maths mode are rendered in roman font. This is useful when declaring category names such as in $AbGroups \to Groups$, because this gets rendered as intended AbGroupsGroupsAbGroups \to Groups. But it means that for operators applied to variables, such as in Tc \to Td get rendered as TcTdTc \to Td instead of as the intended TcTdT c \to T d. For the latter you need to introduce a whitespeace and type ‘T c \to T d“.

    • CommentRowNumber21.
    • CommentAuthorjademaster
    • CommentTimeJul 9th 2019
    Thanks Urs,

    These are helpful tips. I have put some names of categories in \mathsf font. I will change it to roman to match the rest of nLab.
    • CommentRowNumber22.
    • CommentAuthorvarkor
    • CommentTimeJul 20th 2021

    Mention that full faithfulness of a right adjoint is equivalent to density of the left adjoint.

    diff, v99, current

    • CommentRowNumber23.
    • CommentAuthorvarkor
    • CommentTimeOct 12th 2021
    • (edited Oct 12th 2021)

    Use LRL \dashv R rather than Q *R *Q^* \dashv R_*. (The latter can be confusing, because () *({-})_* is often used to denote left adjoints, whereas LL and RR are unambiguous).

    diff, v103, current

    • CommentRowNumber24.
    • CommentAuthormaxsnew
    • CommentTimeApr 19th 2022
    • (edited Apr 19th 2022)

    A typo for when we get editing back: In Theorem 4.1 “If X is a reflective subcategory…” should be “If C is a reflective subcategory…”

    • CommentRowNumber25.
    • CommentAuthormaxsnew
    • CommentTimeMay 4th 2022

    Fix aforementioned error

    diff, v105, current

    • CommentRowNumber26.
    • CommentAuthorJohn Baez
    • CommentTimeMay 13th 2023

    I added another equivalent characterization of adjunctions for which the counit ε:LR1 B\varepsilon : L R \to 1_B of the adjunction is a natural isomorphism. Namely, it’s enough that there exists some natural isomorphism LR1 BL R \to 1_B! I added a reference to Johnstone and a sketch of the proof.

    I find this a delightful simplification.

    diff, v108, current

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2023

    added pointer to:

    diff, v110, current

    • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeOct 9th 2023
    • (edited Oct 9th 2023)

    Following discussion in another thread (here)

    I have deleted the string


    (apparently introduced in revision 111, but you need to look into the source code of the revision, via “rollback”, to see this)

    which, in immediately following the closing =-- of an Example-environment, caused the whole remainder of the entry not to be rendered anymore.

    diff, v113, current

    • CommentRowNumber29.
    • CommentAuthorncfavier
    • CommentTimeOct 9th 2023

    Removed the other “apocalypticists” later down the page.

    diff, v114, current

    • CommentRowNumber30.
    • CommentAuthorm
    • CommentTimeMar 30th 2024

    What is the relation between the notions of a reflexive subcategory and a retract of a category?

    The definitions of the two notions look similar, and the page should clarify the relationship, I think. Is it the same as a retract of a category such that the retraction is fully faithful?

    • CommentRowNumber31.
    • CommentAuthorUrs
    • CommentTimeMar 30th 2024

    Are you thinking of ordinary retracts in the 1-category of categories, or did you see another definition of “retract of categories” anywhere?

    • CommentRowNumber32.
    • CommentAuthorm
    • CommentTimeMar 31st 2024
    The ordinary ones.

    That is, in the simplest form the claim (conjecture) is:

    A full subcategory A of C is reflexive
    in the 1-category of categories
    there is a retraction A->C->A such that C->A is fully faithful.
    • CommentRowNumber33.
    • CommentAuthorUrs
    • CommentTimeMar 31st 2024

    Besides not being invariant under equivalence of categories, this misses the structure in the adjunction between the functors ACA \to C and CAC \to A.

    • CommentRowNumber34.
    • CommentAuthorncfavier
    • CommentTimeMar 31st 2024

    I think you’d want the “inclusion” A → C to be fully faithful anyway, rather than the “reflector” C → A.

    • CommentRowNumber35.
    • CommentAuthorUrs
    • CommentTimeMar 31st 2024
    • (edited Mar 31st 2024)

    Let’s sum this up:

    A reflective subcategory is first and foremost a pair of adjoint functors

    𝒜ιL𝒞 \mathcal{A} \underoverset {\underset{\iota}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot} \mathcal{C}

    such that

    which means equivalently that

    This is probably where you make the connection to retracts:

    If we take 𝒜\mathcal{A} and 𝒞\mathcal{C} to be represented strict categories

    and if it so happens that with that presentation the components of the adjunction counit are not just isomorphisms but in fact equalities,

    (which are some strong “if”s)

    then the above reflection is in particular a retract in the 1-category of strict categories. It’s still a bit more, even in that case.