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 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 limit 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 simplicial space spin-geometry stable-homotopy-theory stack string string-theory subobject 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.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2010

    one more remark at relation between quasi-categories and simplicial categories

    (to be expanded...)

    • CommentRowNumber2.
    • CommentAuthorGuest
    • CommentTimeNov 24th 2020
    Hi,

    I am aware that this entry is quite old and not finished. However, I was wondering if it is a typo or a theorem (and in the latter case, I would be interested in a reference) that in the relation via the homotopy coherent nerve, neither the simplicial category C is assumed to be fibrant nor the unit is derived. I would be curious about any sort of insight here.

    Best,
    Viktoriya Ozornova
    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeNov 24th 2020

    This claim was added by Urs in Revision 7 on February 11, 2010.

    I am pretty sure that the homotopy coherent nerve functor must be derived, since it doesn’t preserve weak equivalence of arbitrary simplicial categories.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2020

    Thanks for the alert. Could you fix it?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2020

    added the qualifiers “derived” and “fibrant”, which were probably missing.

    But this entry still needs to cite actual facts anyways, where it currently says

    (…details/links…)

    diff, v21, current

    • CommentRowNumber6.
    • CommentAuthorDmitri Pavlov
    • CommentTimeNov 25th 2020
    • (edited Nov 25th 2020)

    Added more details and references. Changed the notation to the one used in Bergner’s book.

    diff, v22, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2020

    Thanks!

    • CommentRowNumber8.
    • CommentAuthorGuest
    • CommentTimeNov 26th 2020
    Thank you for the clarification!

    Best,
    Viktoriya
    • CommentRowNumber9.
    • CommentAuthorHurkyl
    • CommentTimeNov 5th 2021

    R is fibrant replacement for the Dwyer-Kan-Bergner model structure, not the Joyal one.

    diff, v25, current

    • CommentRowNumber10.
    • CommentAuthorHurkyl
    • CommentTimeNov 5th 2021

    Added in the calculation of \mathfrak{C} for some nice simplicial sets.

    diff, v26, current

    • CommentRowNumber11.
    • CommentAuthorHurkyl
    • CommentTimeNov 5th 2021
    • (edited Nov 5th 2021)

    I wanted to add the reason I chose the ordering convention I did on the hom-sets is by analogy with lax operations that have the direction [g][f][gf][g] \cdot [f] \to [gf].

    Also, this has analogy with the Duskin nerve, and it is the ordering used in Lurie’s Kerodon, 2.4.3.1 (simplicial path categories), which later uses this when discussing (,2)(\infty,2)-categories.

    • CommentRowNumber12.
    • CommentAuthorHurkyl
    • CommentTimeNov 5th 2021

    Somehow, I had completely missed the homotopy coherent nerve page had much of what I wrote (albeit with the opposite ordering convention on homspaces). So I’ve removed the additions, and will probably add in the additional examples over there.

    diff, v26, current

    • CommentRowNumber13.
    • CommentAuthorTim_Porter
    • CommentTimeJan 2nd 2023

    Lightened the wording slightly.

    diff, v28, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2023

    touched wording and formatting, added publication data to references.

    Finally hyperlinked the previously unnamed operation \mathfrak{C} of rigidification of quasi-categories (entry to be created now).

    diff, v30, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2023

    In particular, it used to say in the references section that the Quillen equivalence “is described in” Bergner (2007), while Lurie (2009) was only found at the end of the list. I have changed that.

    diff, v30, current

    • CommentRowNumber16.
    • CommentAuthorTim_Porter
    • CommentTimeMay 31st 2023

    Fixed a defunct link.

    diff, v34, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2023

    so let’s add the publication data proper:

    diff, v35, current