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Suppose I am given a homotopy coherent diagram of spaces of shape $I$ (This is a simplicial functor $F:\mathfrak{C}[I] \to Top$, where $\mathfrak{C}$ is the standard cofibrant replacement functor in simplicial categories as in Lurie's HTT). There are several (equivalent) formulations or constructions of the homotopy colimit of the diagram $F$.

In this paper, it says that you can define the homotopy colimit as follows. First, define the (ordinary) category of cones over $F$, whose objects are coherent cones over $F$ (i.e. extensions of $F$ to a coherent diagram $\hat{F}$ over the category obtained from $I$ by adding additional terminal object $*$) with morphisms being maps between the end points $f:\hat{F}(*)\to \hat{G}(*)$, such that the cone $\hat{G}$ is induced from $\hat{F}$ by "composing" with $f$ in the (hopefully) obvious way. Now, this category has an initial object and it is the homotopy colimit of the diagram $F$ [Definition 3.1]. It then proves that this coincides with the Bousfield-Kan style definition [Theorem 4.1].

My question is, where can I find a "standard" reference for this fact? I am aware of some of the standard literature on coherent diagrams (Vogt, Cordier, Porter,...), but I couldn't find this specific approach to the homotopy colimit.

Remark: I am actually more interested in the homotopy limit, but it should be completely analogues.

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    $\begingroup$ Dwyer-Spalinski is a good place to learn about homotopy (co)limits. Hirschhorn's book too. I'm not sure, but I'll bet there is a discussion in May-Ponto as well. $\endgroup$
    – David White
    Dec 30, 2015 at 15:16
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    $\begingroup$ I added a tag to clarify that I am looking for a reference for a specific result and not to learn about homotopy colimits in general. Dwyer-Spalinski treat only homotopy colmits of strict diagrams (by projective/injective model structure on the category of diagrams). Hirschhorn has some explicit simplicial formulas, but I can't see if any of them gives what I am looking for. May-Punto does not seem to cover coherent diagrams at all. $\endgroup$
    – KotelKanim
    Dec 30, 2015 at 20:45
  • $\begingroup$ What makes you think there is another reference for this fact in the first place? Steimle makes it pretty clear in the introduction (“the model for the homotopy colimit is in the spirit of the original definition of Vogt [12], with an emphasis on universal properties”) that this is his original contribution. $\endgroup$
    – Dmitri Pavlov
    Dec 31, 2015 at 8:57
  • $\begingroup$ Well, when you put it this way... I don't know. I wasn't sure which part exactly is an original contribution and which is just a cleaner presentation of classical stuff. I just assumed that something like this, is probably mentioned, even if in an obscure way, in the standard literature. But perhaps it isn't. $\endgroup$
    – KotelKanim
    Dec 31, 2015 at 12:37
  • $\begingroup$ Perhaps I could benefit from some clarification. What's the distinction between the result you'd like a reference to and theorem 5.12 in Vogt's "Homotopy limits and colimits," which asserts that the homotopy colimit functor is left adjoint to the "constant diagram" functor from spaces to coherent $I$-diagrams of spaces? $\endgroup$
    – Tyler Lawson
    Dec 31, 2015 at 15:13

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