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Let $\mathcal{C}$, $\mathcal{D}$ be two small categories. Let $f\: : \: \mathcal{C}\to \mathcal{D}$ be a functor. Then it induces a functor $$ f^{*}\: : \: sPsh(\mathcal{D})\to sPsh(\mathcal{C}) $$ via $f^{*}(X)=X\circ f$. Now consider $sPsh(\mathcal{D}), sPsh(\mathcal{C})$ equipped with the projective model structure. Then $f^{*}$ is a right Quillen functor and its left adjoint is given by the left kan extension $$ f_{!}\: : \: sPsh(\mathcal{C}) \to sPsh(\mathcal{D}). $$ Assume that $X$ is a cofibrant replacement of the final functor $*\in sPsh(\mathcal{C})$. Then $f_{!}(X)$ is again cofibrant.

Q: When is $f_{!}(X)$ again a cofibrant replacement of $*$?

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  • $\begingroup$ May we assume that $sPsh(\mathcal D)$ means simplicial presheaves on $\mathcal D$? $\endgroup$
    – ACL
    Commented Mar 16, 2015 at 13:51
  • $\begingroup$ yes $sPsh(\mathcal{D})$ is the category of simplicial presheaves. $\endgroup$
    – Cepu
    Commented Mar 16, 2015 at 14:00

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$f_!(X)$ is a cofibrant replacement of $*$ if and only if the comma category $f/d$ has a weakly contractible classifying space, for all $d\in D$.

Indeed, the value of $f_!(X)$ on $d$ is weakly equivalent to the homotopy colimit of the constant diagram with value $*$ on the comma category $f/d$, because the restriction functor $sPsh(C)\to sPsh(f/d)$ is left Quillen. Such a homotopy colimit is weakly equivalent to the nerve of the indexing category.

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