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Let $X:I\to M$ be a functor from a small category $I$ to a simplicial accessible (not combinatorial) proper model category $M$ which is already injective cofibrant. Let $W:I^{op}\to \mathrm{Set}$ be a weight. I consider the weighted colimit

$$ \int^{c\in I} W(c).X(c) = {\mathrm{coeq}}\bigg(\coprod_{c,c'}(W(c')\times I(c,c')).X(c)\rightrightarrows \coprod_{c} W(c).X(c)\bigg) $$

I cannot figure out whether there is a general fact behind what follows: in my case, if $X:I\to M$ is projective cofibrant, then the weighted colimit behaves like a homotopy weighted colimit, i.e. by replacing a projective cofibrant $X:I\to M$ by another one which is objectwise weakly equivalent, I obtain a weak equivalent weighted colimit. Taking a cofibrant replacement of $X:I\to M$ in the projective model structure is quite complicated. Now here is my question.

Is there a way to use the simplicial structure of $M$ to obtain a formula for the homotopy weighted colimit without having to find a projective cofibrant replacement of $X:I\to M$ ?

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    $\begingroup$ IIRC, if the weight is projective cofibrant, it suffices for the diagram to be pointwise cofibrant. $\endgroup$
    – Alexander Campbell
    Commented Aug 30, 2022 at 7:44
  • $\begingroup$ @AlexanderCampbell In my case, $X:I\to M$ is already injective cofibrant indeed. Could you explain the reason please ? $\endgroup$
    – Philippe Gaucher
    Commented Aug 30, 2022 at 8:43
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    $\begingroup$ This is Theorem 3.3 in the paper doi.org/10.1016/j.jpaa.2009.10.006. $\endgroup$
    – Alexander Campbell
    Commented Aug 30, 2022 at 9:18
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    $\begingroup$ You may also be interested in arxiv.org/abs/math/0610194. $\endgroup$
    – Mike Shulman
    Commented Sep 1, 2022 at 5:07

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