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It is a follow-up of my question Calculation of the homotopy colimit of a diagram of spaces which was badly formulated.

Consider a fixed diagram $D:I^{op}\to {\rm Top}$ where ${\rm Top}$ is a convenient category for doing algebraic topology. Is there any general sufficient condition on $D$ for a colimit of the form $\int^{i} X(i)\times D(i)$ to give weakly homotopy equivalent spaces by replacing $X:I\to {\rm Top}$ by an objectwise weakly homotopy equivalent diagram $Y:I\to {\rm Top}$ ?

I can think of two situations, I believe quite different, where this happens.

  1. The first one is $D(i)=|B(i\!\downarrow\! I)^{op}|$ (for interested people, the argument is explained in Dugger and Isaksen's paper https://doi.org/10.1007/s00209-003-0607-y),
  2. The second one is when $X$ and $Y$ are enriched functors ($I$ must be enriched as well) and $D=I(-,k)$. Then the coend is equal to $X(k)$ by the enriched Yoneda lemma.
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    $\begingroup$ Perhaps your question is answered here: mathoverflow.net/questions/390900/… $\endgroup$
    – Niall Taggart
    Commented Nov 29, 2021 at 9:30
  • $\begingroup$ @NiallTaggart Indeed in the second case, $I(-,k)$ is cofibrant for the projective model structure so Case 2 is also related to homotopy theory and not only to enriched category theory. $\endgroup$
    – Philippe Gaucher
    Commented Nov 29, 2021 at 10:12

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