When does a fibration $f:X\rightarrow Y$ in a model category admit a section?.

If we have a fibration $$f:X\rightarrow Y$$ in a model category $$C$$, where $$Y$$ is cofibrant and both $$X, Y$$ are fibrant.

Does f admit a section (right inverse)?. If it does not work in general, actually I saw this claimed when $$C$$ is the category of $$\mathbb{k}$$-dg algebras, where fibrations are morphisms such that $$f^n$$ are epimorphisms for all $$n\in\mathbb{Z}$$ (so, every object is fibrant) and $$Y$$ is (forgetting differentials) a quasi-free graded algebra, these restrictions may be helpful but I cannot conclude yet.

Thank you.

Edit: I realized this not work in a general model category, please focus only on the particular case of dg algebras. In this case, the morphism takes the form $$f:A\rightarrow\Omega D$$, I obtained a section (only as a morphism of graded algebras) $$s:\Omega D\rightarrow A$$, but I do not know why it should respect the differentials.

• This shall not work in every model category, for example, Set has several (two) model structures where every map is a fibration. In your case I don't know how it could be solved. – elidiot Apr 23 at 16:02
• Indeed, a counterexample in Top is a non trivial fiber bundle with CW spaces as base and total space. I just realised it. Thanks. – Victor TC Apr 23 at 17:03