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nLab > Latest Changes: injective object

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  1.  
    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 4th 2012

    Added material to injective object, including a proof of Baer’s criterion for injective modules, and the result that for modules over Noetherian rings, direct sums of injective modules are injective.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 26th 2012

    Months later…

    Thanks, Todd! :-)

    I have added some hyperlinks and split off the Bass-Papp result as a separate numbered proposition.

    I’ll now copy (not move) this stuff over to the new entry injective module (to parallel projective module.)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2016

    Years later…

    I have made two little lemmas more explicit:

    1. right adjoints of left exacts preserves injective objects (here);

    2. right adjoints of faithful left exacts transfer enough injectives (here).

    Then I made the use of these two lemmas in the statement that RModR Mod has enough injectives (here) more explicit.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 17th 2016

    I added a few more examples (in toposes, in topological spaces).

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2016

    Thanks!

    By the way, I discovered that we had parts of the lemmas missing at injective object spelled out at injective module, and vice versa. I have tried to fix that.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 17th 2016

    And now also in Boolean algebras, mentioning Gleason’s theorem.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2016

    Thanks. I have added cross-links with Scott topology.

  8. For a right adjoint to preserve injective objects, it suffices for its left adjoint to be merely left exact (instead of exact; with the same proof). I strengthened the formulation of the lemma accordingly.

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 19th 2016

    Left adjoints are always right exact anyway though.

  10. Oh, of course. Sorry for the noise; the statement now reads: “Given a pair of additive adjoint functors between abelian categories such that the left adjoint L is a left exact functor (thus automatically exact), then the right adjoint preserves injective objects.” Also I added a remark that additivity of the left adjoint is given automatically (being exact, the functor preserves biproducts).

    • CommentRowNumber11.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 19th 2016

    (Not wishing to seem like I’m finding fault) right adjoints also preserve biproducts automatically, since they preserve products.

  12. Right. I think each of the following conditions is sufficient for guaranteeing that a functor 𝒜→ℬ\mathcal{A} \to \mathcal{B} preserves biproducts (where 𝒜\mathcal{A} and ℬ\mathcal{B} are categories with a zero object):

    1. The functor preserves finite products (for instance, because it’s a right adjoint) and any product in ℬ\mathcal{B} is a biproduct.
    2. The functor preserves finite coproducts (for instance, because it’s a left adjoint) and any coproduct in ℬ\mathcal{B} is a biproduct.
    3. The functor preserves finite products and coproducts.
    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMar 22nd 2016

    For completeness, I have finally created also an entry for injective morphisms (which had long been requested at cofibrantly generated model category)

  14. And I recorded the trivial observation in #12 to additive functor (with a crosslink from injective object).

    • CommentRowNumber15.
    • CommentAuthorIngoBlechschmidt
    • CommentTimeMar 26th 2016
    • (edited Mar 29th 2016)

    I added to injective object a couple of observations about internally injective objects in toposes. Somewhat surprisingly (to me), it turns out that external injectivity and internal injectivity actually coincide, in stark contrast to the situation with internally projective objects. I have checked this only for localic toposes, but believe it’s true in more generality; I’ll update the entry when I know.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeMar 29th 2016

    Thanks!!

    I just added some more hyperlinks to keywords.

    • CommentRowNumber17.
    • CommentAuthorThomas Holder
    • CommentTimeJun 3rd 2018

    I added the statement at injective object that right adjoints preserve injectives provided the left adjoint preserves monos which came to my awareness during yesterday’s edit at sufficiently cohesive topos.

    diff, v56, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJun 20th 2019

    added pointer to

    diff, v58, current

  19. Broken link

    Anonymous

    diff, v61, current

    • CommentRowNumber20.
    • CommentAuthorGuest
    • CommentTimeFeb 24th 2021
    I don't see a difference between 3.1 and 3.3 (despite unused extra hypotheses like abelian categories).

    Jochen Wengenroth
    • CommentRowNumber21.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 24th 2021

    Yup: 3.1 could be cited as a corollary of 3.3.

    • CommentRowNumber22.
    • CommentAuthorSeth Chaiken
    • CommentTimeAug 7th 2022

    Should “morphisms” at the end of the sentence below be “morphism”?

    Remark 1.3. If CC has a terminal object \ast, then the JJ-injective objects II according to def. 1.2 are those for which I→∃!I \stackrel{\exists!}{\to} \ast is called a JJ-injective morphisms.

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeAug 7th 2022

    Yes, thanks. I have fixed it (here).

    diff, v63, current

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