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If $F$ is a left adjoint between $C$ and $D$, and $D$ has a model structure; We can define cofibrations and equivalences in $C$ to be those that are so after applying $F$. What are criterions for this to define a model structure? Where can I find a discussion? Also, the dual question about a right adjoint.

Thank you, Sasha

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For the first question (less standard, cofibration and weak equivalence) you can check section 1.2 theorem 1.13 of The dual version is in Hope it will help you – Ilias Amrani Mar 29 '12 at 8:49
up vote 8 down vote accepted

Since in practice the dual situation occurs more often than the one you stated I will dualize your question. Given categories $\bf {D}$ and $\bf {E}$ and adjoint functors $F:\bf {D}\to \bf {E}$ and $G:\bf {E}\to \bf {D}$, with $F$ left adjoint to $G$, if $D$ is a model category then define weak equivalences and fibrations in $E$ as follows. Declare an arrow $f$ in $E$ a weak equivalence (resp. a fibration) if $G(f)$ is a weak equivalence (resp. fibration).

Then it is guaranteed that with these fibrations and weak equivalences $E$ is a cofibrantly generated model category provided the following conditions hold:

1) $D$ is cofibrantly generated. 2) The left adjoint $F$ preserves small objects. 3) The weak equivalences in $E$ contain any sequential colimit of pushouts of images $F(g)$, where $g$ is allowed to vary over the generating trivial cofibrations in $D$.

Of course condition 3 is difficult to check so it is good to know that it is satisfied automatically if:

$E$ has a functorial fibrant replacement and $E$ has functorial path objects for fibrant objects.

First appearance in the literature of such transfer principles seems to be S. E. Crans – Quillen closed model structures for sheaves, J. Pure Appl. Alg. 101 (1995), 35–57.

The more recent (and where I learned of the answer) article "Axiomatic homotopy theory for operads", by Berger and Moerdijk can be found here: where the result is used to establish a model structure on (one-object) operads.

Of course a dual answer will provide a transfer principle for fibrantly generated model categories. However, as interesting/important fibrantly generated model categories are very rare such a transfer result is hardly useful (showing that mathematics is not self-dual).

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Thank you! So if I assume that my model categories come with functorial factorizations (triv cofib, fib) , (cofib, triv fib) I am OK with (3). What about (2), is there some easy general widely satisfied criterion for that? – Sasha Mar 29 '12 at 16:54
Ah, I am sorry. You mean that $E$, which we still do not know to be a model category, should be shown to have functorial factorization, in order not to check (3). – Sasha Mar 29 '12 at 18:56
One useful criterion for (2) is that the right adjoint $G$ preserves (sufficiently highly) filtered colimits. – Mike Shulman Mar 29 '12 at 18:58
Downvote me if I am wrong, but it seems that I don't need $E$ to have "functorial path objects for fibrant objects" as written above, but just "path objects for fibrant objects" (all the rest the same). – Sasha Apr 1 '12 at 16:28

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