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-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundle bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity 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 homology homotopy homotopy-theory homotopy-type-theory index-theory infinity integration integration-theory itex k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics planar 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 superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft 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.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 30th 2009
    Started injective hull.
    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 30th 2009
    There's a lot of terminology being used by the authors. I'm wary of starting up pages for 'extension' a kind of 'embedding', and 'injective' and 'essential embedding'. Do these only apply to concrete categories?
    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 1st 2009
    Added the fact that assigning an object its injective hull is not natural. Does this tell us something interesting?
    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeDec 4th 2009

    This has always been a strange thing to beginning category theorists (and maybe still for the rest of us). The usual example seems to be the algebraic closure of a field (although that's an injective hull only in a slightly unobvious category, now noted on our page). Any two algebraic closures are isomorphic, but not canonically so, and this applies to injective hulls in general.

    It should tell us something, but I don't know what.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeApr 8th 2015

    Added to injective hull a general definition in terms of a class of morphisms rather than a class of objects, taken from the paper Essential weak factorization systems by Tholen (which Patrick Schultz just told me about).

    I am somewhat confused by the current status of this page. It has an “Idea” section which puts itself in the context of a general concrete category, but then refers to the page essential embedding whose definition is stated only for modules. There is no “Definition” section and it then plunges into Examples, with a Generalization section afterwards. Can we put one or more general definitions at the top and then explain exactly how the examples are examples of this definition?

    Also, I wonder how are all the definitions related to each other? I think that an \mathcal{H}-injective hull in the sense of the second definition should also be a 𝒞\mathcal{C}-hull in the sense of the first definition where 𝒞\mathcal{C} is the class of \mathcal{H}-injectives, but I don’t immediately see any way to go the other direction. Which of these definitions is more directly a generalization of the more concrete one?

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeApr 8th 2015

    I wrote the original version of essential embedding. I am not sure what is the intrinsic characterization of the categories where this makes sense so I wrote in some categories of modules (say, over operator algebras). It seems that the categories with pullbacks and zero object suffice at least for the definition.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeApr 8th 2015

    Does it agree with the general definition in terms of a class of morphisms \mathcal{H}?

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)