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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 23rd 2022



    Suppose CC is a category that admits small coproducts.

    Given simplicial objects A,BC Δ opA,B\in C^{\Delta^{op}}, their function complex is a simplicial set


    whose set of nn-simplices is the set of maps

    Δ nAB,\Delta^n\otimes A\to B,

    where \otimes denotes the copowering of simplicial objects over simplicial sets given by

    (CD) n= iC nD n.(C\otimes D)_n=\coprod_{i\in C_n}D_n.

    Related concepts


    The original definition of a function complex in the generality stated above is due to Daniel M. Kan:

    • Daniel M. Kan, On c.s.s. categories, Boletín de la Sociedad Matemática Mexicana 2 (1957), 82–94. PDF.

    diff, v2, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 4th 2022

    added pointer to:

    diff, v3, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 4th 2022
    • (edited Dec 4th 2022)

    Looking over the entry now, I have adjusted the previous notation a little in order to avoid a couple of clashes:

    • the notation “Hom” for the simplicial hom-complex I have replaced with “Hom ΔHom_\Delta

    • the letter “CC” used to denote both the ambient category as well as a simplicial set tensor factor. I have changed to writing “𝒞\mathcal{C}” for the former and “SS” for the latter.

    Also expanded the presentation just a little.

    diff, v3, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 4th 2022

    added description (here) of the evaluation map on the function complex

    diff, v3, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 4th 2022

    added pointer to:

    • Charles Rezk, Section 15 of: Introduction to quasicategories (2022) [pdf]

    diff, v4, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 5th 2022

    I have spelled out in the detail the example (now here) of the evaluation map on simplicial function complexes which are nerves of inertia groupoids.

    diff, v6, current