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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 $(\infty, 1)$-categories? I’ve only ever seen colimit completions of $(\infty, 1)$-categories defined in terms of presheaves. In general it’s harder to give “objects & morphisms” definitions of $(\infty, 1)$-categories.
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:
$\widehat{C}(colim_i C(-, F_i), colim_j C(-, G_j)) \simeq \lim_i \widehat{C}(C(-, F_i), colim_j C(-, G_j)) \simeq \lim_i \colim_j C(F_i, G_j)$Maybe it would be more interesting to identify $\widehat{C}$ as a localization of $(\infty,1)Cat_{/C}$ (namely the right fibrations) so that the objects have a more direct interpretation as diagrams?
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