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Consider a small category $I$. There exists a small diagram $D:I^{op}\to {\rm Top}$ where ${\rm Top}$ is a convenient category for doing algebraic topology such that for all small diagrams $X:I\to {\rm Top}$, the coend $\int^{i} X(i)\times D(i)$ has the homotopy type of ${\rm hocolim\ } X$ (e.g. see Model categories and their localizations Remark 18.5.4). For example $D(i)=|B(i\!\downarrow\! I)^{op}|$.

Is there a way to characterize these diagrams $D:I^{op}\to {\rm Top}$ ?

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    $\begingroup$ One guess could be something like: those D such that the augmentation to a point is a pointwise equivalence + such that the coend you wrote down is invariant under weak equivalences in the variable X. (i.e. something like "flat resolutions of the constant diagram at a point"). Possible proof: Induct over a 'cell' decomposition of X and use that Top is proper. $\endgroup$
    – Dylan Wilson
    Commented Nov 17, 2021 at 20:51
  • $\begingroup$ Can this be deduced from Theorem 8.2.1 of Riehl's Categorical Homotopy Theory book? $\endgroup$
    – David White
    Commented Nov 18, 2021 at 1:23
  • $\begingroup$ @DylanWilson I asked the question because I suspect that a coend I have behaves (at least sometimes) as a homotopy colimit and the diagram $D$ does not look at all like the diagram in the question, nor I can see how to realize it as a simplicial set. $\endgroup$
    – Philippe Gaucher
    Commented Nov 18, 2021 at 7:46
  • $\begingroup$ @DavidWhite That seems to only cover one case of D (the example in the question), unless I'm misunderstanding. Something like 11.5.1 seems closer, but there's something to check about the difference between 'cofibrant' diagrams (which are kinda like 'projective resolutions') and 'flat' diagrams (like flat resolutions). $\endgroup$
    – Dylan Wilson
    Commented Nov 18, 2021 at 13:44
  • $\begingroup$ @DylanWilson The question is badly formulated and badly abstracted from my situation. Consider a coend $\int^{i} X(i)\times D(i)$. What condition should satisfy the diagram $D$ so that by replacing $X$ by a weakly equivalent diagram $Y$, one obtains a weakly homotopy equivalent coend ? (this coend does not have to calculate the homotopy colimit of $X$). I think that something like Theorem 11.5.1 should help. $\endgroup$
    – Philippe Gaucher
    Commented Nov 18, 2021 at 19:30

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