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    • CommentRowNumber1.
    • CommentAuthorShamrock
    • CommentTimeSep 27th 2024
    • (edited Sep 27th 2024)

    On the page “ind-object in an (infinity,1)-category” there’s a section “In terms of formal colimits” which is left as a stub with “(… should be made more precise…)”. Does anyone know if this has been made precise anywhere, in any model of (,1)(\infty, 1)-categories? I’ve only ever seen colimit completions of (,1)(\infty, 1)-categories defined in terms of presheaves. In general it’s harder to give “objects & morphisms” definitions of (,1)(\infty, 1)-categories.

    • CommentRowNumber2.
    • CommentAuthorHurkyl
    • CommentTimeSep 28th 2024

    For what it’s worth, I consider the presheaf version to be precisely the way to make it more precise; you can easily verify that it has the right hom-spaces:

    C^(colim iC(,F i),colim jC(,G j))lim iC^(C(,F i),colim jC(,G j))lim icolim jC(F i,G j) \widehat{C}(colim_i C(-, F_i), colim_j C(-, G_j)) \simeq \lim_i \widehat{C}(C(-, F_i), colim_j C(-, G_j)) \simeq \lim_i \colim_j C(F_i, G_j)

    Maybe it would be more interesting to identify C^\widehat{C} as a localization of (,1)Cat /C(\infty,1)Cat_{/C} (namely the right fibrations) so that the objects have a more direct interpretation as diagrams?