Not signed in (Sign In)

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 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 foundation foundations functional-analysis functor 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 limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad 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 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 sheaves simplicial space spin-geometry stable-homotopy-theory stack 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).

nLab > Latest Changes: Kan fibrant replacement

Bottom of Page

1 to 10 of 10

  1.  
    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 4th 2019

    Added more properties and applications.

    diff, v16, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 8th 2020

    Added redirects “Ex^∞” and “Ex-infinity”.

    diff, v19, current

    • CommentRowNumber3.
    • CommentAuthorHurkyl
    • CommentTimeApr 9th 2020
    • (edited Apr 9th 2020)
    The article proposes

    Ex^∞(f) is a simplicial weak equivalences if and only if f is a simplicial homotopy equivalence.

    Isn't that incorrect? It's clear that the statement

    Ex^∞(f) is a weak equivalence in the classical model structure if and only if f is a weak equivalence in the classical model structure

    is true. But, AFAIK, the class of homotopy equivalences expressed using the cylinder X+X -> X×Δ[1] does not coincide with the class of weak equivalences. (if these do coincide, it's surprisingly hard to find this fact explicitly stated...)

    Was the statement meant the other way around? i.e.

    Ex^∞(f) is a simplicial homotopy equivalences if and only if f is a simplicial weak equivalence.

    is also true, since Ex^∞(f) is a morphism between fibrant objects, so it is a homotopy equivalence iff it is a weak equivalence.


    I'm assuming "simplicial homotopy equivalence" means you have simplicial homotopies fg=>1 and 1=>gf.

    I'm assuming the term "simplicial weak equivalence" means a weak equivalence in the classical model structure, but AFAICT that term is barely used on the ncatlab and is never defined.
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 9th 2020

    Please feel invited to edit the entry to clarify/fix.

    Recently I made “simplicial weak equivalence” redirect to classical model structure on simplicial sets, but it would deserve it’s own entry.

    • CommentRowNumber5.
    • CommentAuthorHurkyl
    • CommentTimeApr 9th 2020
    I'll take that to mean I've said nothing factually inaccurate, in which case I'm happy to make the change.
    • CommentRowNumber6.
    • CommentAuthorHurkyl
    • CommentTimeApr 9th 2020

    Fixed an error in the phrasing of the property that Ex^\infty reflects weak equivalences.

    diff, v20, current

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 11th 2020

    It was meant to be the other way around, of course.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeOct 25th 2023

    added pointer to the analog version of ExEx for the (localization of) quasi-categories incarnated as marked simplicial sets:

    diff, v23, current

    • CommentRowNumber9.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 26th 2023

    Re #8: It seems to me these authors only develop an analogue of Ex, not Ex^∞.

    Their main theorem seems to say that if a marked quasicategory satisfies the calculus of left fractions, then the analogue of the functor Ex sends it to a quasicategory.

    As far as I can see, there is nothing about fibrantly replacing arbitrary marked simplicial sets, so there is no analogue of Ex^∞.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeOct 27th 2023

    Absolutely, it’s an “analogue version of ExEx”, as per #8.

    diff, v24, current

1 to 10 of 10

  1.