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 book 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 gauge-theory gebra geometric-quantization geometry goodwillie-calculus 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 homology homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology newpage nforum nlab noncommutative noncommutative-geometry number-theory object 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 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.
    • CommentAuthoreparejatobes
    • CommentTimeMar 6th 2012
    • (edited Mar 6th 2012)

    added to Kan lift the def of absolute Kan lifts, and some more examples.

    Btw, in all of the sources I’ve read about this, Kan lifts are simply called (left, right) liftings. What do you think about renaming this to liftings? will it conflict with some other kind of “lifting”?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 6th 2012
    • (edited Mar 6th 2012)

    Btw, in all of the sources I’ve read about this, Kan lifts are simply called (left, right) liftings. What do you think about renaming this to liftings? will it conflict with some other kind of “lifting”?

    I wouldn’t quite want to rename the entry, as “Kan lifts” crucially differ from the usual default meaning of “lift” (in homotopy thory) by allowing the 2-cell to be non-invertible.

    But what I do very much agree with is that it would be good amplify the relation to other notions of lifts. One could create a table that lists various notions of “lift” and the contexts in which they apply.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeMar 6th 2012

    I think this is one of those things where once one is known to be working in a particular context, the adjective is unnecessary. So papers that are about, say, Yoneda structures, can say at the beginning “by a lifting we mean …” and then it is unambiguous for that paper. But there are enough other types of “liftings” in mathematics as a whole that the nLab page about this concept needs to distinguish itself from those other meanings somehow. But the page could helpfully remark that often the adjective is omitted.

    Which is basically to say that I agree with Urs, I guess.

    • CommentRowNumber4.
    • CommentAuthoreparejatobes
    • CommentTimeMar 7th 2012

    Now that I think about it, you’re right and guess I have to disagree with myself :)

    I’ve added a remark about the Kan-omitted terminology, and a redirect from Kan lifting.

    Urs,

    It’d certainly be nice to have “lift disambiguation” page, but I don’t feel qualified to that (my knowledge of homotopy theory is pretty shallow). Out of curiosity, what you were referring to with “allowing the 2-cell to be non-invertible” ??

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 7th 2012
    • (edited Mar 7th 2012)

    what you were referring to with “allowing the 2-cell to be non-invertible” ??

    Precisely what it says in the entry: the 2-cell called ε\epsilon there is in general not invertible.

    For a lift in the sense of homotopy theory it would be invertible.

    • CommentRowNumber6.
    • CommentAuthoreparejatobes
    • CommentTimeMar 7th 2012

    Precisely what it says in the entry: the 2-cell called ϵ there is in general not invertible.

    For a lift in the sense of homotopy theory it would be invertible.

    Ok I see, thanks.

    Maybe that would be expressible as a Kan lift in some locally groupoidal 2-category.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMar 7th 2012
    • (edited Mar 7th 2012)

    Yes. In a (2,1)-category we “have homotopy theory”.

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)