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Let $C$ be a model category and $B$ a direct category. By Theorem 5.1.3 in Mark Hovey's book Model categories, there is a model category structure on the diagram category $C^B$ such that weak equivalences and fibrations are defined pointwise.

Assume that $C\leftrightarrows D$ is a Quillen equivalence. By pointwise application, we obtain a pair of functors $C^B\leftrightarrows D^B$. Is this also a Quillen equivalence?

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  • $\begingroup$ (If the question is too basic, please feel free to migrate it over to MSE!) $\endgroup$
    – Rasmus
    Commented Jul 12, 2013 at 19:46
  • $\begingroup$ I think it follows easily from the definition and from the fact that a cofibrant object in $C^B$ is pointwise cofibrant. $\endgroup$
    – Fernando Muro
    Commented Jul 12, 2013 at 20:27
  • $\begingroup$ @FernandoMuro Do we want that fibrant objects in $D^B$ are pointwise fibrant, since the OP write that "weak equivalences and fibrations are defined pointwise"? $\endgroup$
    – Baby Dragon
    Commented Jul 12, 2013 at 20:30
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    $\begingroup$ I'm not saying that pointwise cofibrant diagrams are cofibrant, but that cofibrant diagrams are pointwise cofibrant, see for instance Corollary 15.3.12 in Hirschhorn's book. That's enough, I think. $\endgroup$
    – Fernando Muro
    Commented Jul 12, 2013 at 21:08
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    $\begingroup$ Anyway, I've just found out that it is Proposition 15.4.1 in Hirschhorn's book. $\endgroup$
    – Fernando Muro
    Commented Jul 12, 2013 at 21:35

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Edit: As Fernando pointed out, the claim can be found in Hirschhorn's book as Proposition 15.4.1.

Here is my own attempt. We write $(F,G)\colon C\leftrightarrows D$ for the given Quillen equivalence and $(F^X,G^X)\colon C^X\leftrightarrows D^X$ for the pointwise induced adjunction. First we check that $(F^X,G^X)$ is a Quillen adjunction. It suffices to show that $G^X$ preserves fibrations and trivial fibrations which is clear because of the pointwise definition of fibrations and weak equivalences. It remains to show that $(F^X,G^X)$ is a Quillen equivalence, that is, a map $F^X(c^X)\to d^X$ in $D^X$ with $c^X$ cofibrant and $d^X$ fibrant is a weak equivalence if and only if the adjoint map $c^X\to G^X(d^X)$ is a weak equivalence in $C^X$. This follows from the pointwise definition of weak equivalences and fibrations once we know that cofibrant objects in $C^X$ are pointwise cofibrant (which can be shown inductively similarly to the proof of Proposition 6.8 in DIAGRAM SPACES AND SYMMETRIC SPECTRA by STEFFEN SAGAVE AND CHRISTIAN SCHLICHTKRULL).

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  • $\begingroup$ (If this is correct, then it is easier than I thought.) $\endgroup$
    – Rasmus
    Commented Jul 12, 2013 at 21:52
  • $\begingroup$ $c^X$ must be cofibrant and $d^X$ fibrant. $\endgroup$
    – Fernando Muro
    Commented Jul 13, 2013 at 11:53
  • $\begingroup$ @FernandoMuro: Indeed. That's seems to be missing in the wikipedia article. $\endgroup$
    – Rasmus
    Commented Jul 13, 2013 at 13:30
  • $\begingroup$ (I added it here and on wikipedia.) $\endgroup$
    – Rasmus
    Commented Jul 13, 2013 at 13:37
  • $\begingroup$ This is why it is relevant that cofibrant diagrams are pointwise cofibrant $\endgroup$
    – Fernando Muro
    Commented Jul 13, 2013 at 14:13

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