Let $\mathcal{C}$ be a Reedy category, $\mathcal{M}$ be a model category. Then consider $Fun(\mathcal{C}, \mathcal{M})$, the category of diagrams from $\mathcal{C}$ to $\mathcal{M}$ equipped with the Reedy model structure. I wonder if there is a simple characterization of fibrant, cofibrant objects in this category. Let $0,1$ be the initial and terminal object in $Fun(\mathcal{C}, \mathcal{M})$. It seems to me that for a map $0\to X$ its relative latching map at $c\in \mathcal{C}$ is isomorphic to the usual latching map $$ L_{c}X \to X $$ at $c\in\mathcal{M}$. On the other hand the relative matching map $X\to 1$ at $c\in \mathcal{C}$ is isomorphic to the usual matching map $$ X\to M_{c}X $$ Thus it follows that an object $X$ is Reedy fibrant (cofibrant) iff its matching (latching) maps are fibant (cofibrant).

a) Am I right? This in particular implies that the (co)simplicial frames defines in the Hovey's book: model categories(definition 5.2.7) are (co)simplicial resolutions in the sense of Hirschhorn (see definition 16.1.1 in model categories and their localizations).

b)Do you know more simpler characterization when $\mathcal{C}$ is the simplex category and $\mathcal{M}$ is a model category with some conditions (for example cofibrantly generated,...)?

  • $\begingroup$ There are some mild technical differences between Hovey's definition and Hirschhorn's definition, as I recall, particularly regarding what happens with non-(co)fibrant objects. $\endgroup$ – Zhen Lin Sep 12 '15 at 16:13
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    $\begingroup$ a) A Reedy diagram $X\colon\mathcal C\to \mathcal M$ is cofibrant iff all latching maps $L_cX\to X_c$ are cofibrations, $c\in Ob\mathcal C$. Dually for fibrant objects. b) No, this is the usual characterization also in the simplicial case. $\endgroup$ – Fernando Muro Sep 14 '15 at 13:30

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