Hello,


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



Transporting model structures via adjunctions
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:<http://arxiv.org/abs/math/0206094> 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). 


