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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2023

    starting an entry on the left adjoint of the homotopy coherent nerve (which seems to have been missing all along)

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2023

    added the statement (here) about \mathfrak{C} preserving products up to natural DK-equivalence

    v1, current

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 29th 2023


    The Joyal rigidification functor

    The Joyal rigidification functor is defined as the left adjoint of the Cordier–Vogt homotopy coherent nerve functor.

    The Dugger–Spivak rigidification functor

    The Dugger–Spivak rigidification functor provides a more explicit model for the same (∞,1)-functor, by virtue of writing down an explicit model that does not use colimits.

    Specifically, given a simplicial set SS (not necessarily fibrant in the Joyal model structure), we construct the Dugger–Spivak rigidification necS\mathfrak{C}^{\mathrm{nec}}S as the following simplicial category. Objects are vertices of SS.

    The simplicial set of morphisms xyx\to y is the nerve of the category of necklaces in SS (introduced by Hans-Joachim Baues). A necklace is a simplicial map

    Δ n 1Δ n kS,\Delta^{n_1}\vee \cdots \vee \Delta^{n_k}\to S,

    where \vee glues the final vertex of the preceding simplex to the initial vertex of the following simplex. Morphisms are commutative triangles of simplicial maps that preserve the initial and final vertex of the entire necklace.

    Composition is defined by concatenating necklaces. The resulting functor from simplicial sets admits a zigzag of weak equivalences to the Joyal rigidification functor.

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2023

    added (here) statement of the comparison map to the Dwyer-Kan groupoid

    diff, v3, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2023

    In the proposition on weak respect for products (here) I have made more explicit the form of the comparison map (S×S)((pr S),(pr S))(S)×(S) \mathfrak{C}(S \times S') \overset{ \big( \mathfrak{C}(pr_S) ,\, \mathfrak{C}(pr_{S'}) \big) }{\longrightarrow} \mathfrak{C}(S) \times \mathfrak{C}(S')

    diff, v5, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2023

    added pointer to:

    diff, v6, current