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

Site Tag Cloud

2-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology natural nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

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.
    • 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


    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.

  1. 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.

  2. 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.

  3. Right. I think each of the following conditions is sufficient for guaranteeing that a functor 𝒜\mathcal{A} \to \mathcal{B} preserves biproducts (where 𝒜\mathcal{A} and \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 \mathcal{B} is a biproduct.
    2. The functor preserves finite coproducts (for instance, because it’s a left adjoint) and any coproduct in \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)

  4. 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


    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

  5. Broken link


    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.

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)