I was wondering if there exists a model structure on the category of non-negative differential graded algebras with homological grading. To be more precise: Let $Ch\_{k}$ the model category of non-negative chain complexes over a field $k$ (any characteristic). An object is of the form $$M\_{0}\leftarrow M\_{1}\leftarrow \dots$$
Weak equivalences are quasi-isomorphisms and fibrations are morphisms of chain complexes $f\_{\bullet}:M\_{\bullet}\rightarrow N\_{\bullet} $ such that $f\_{\bullet}$ is surjective for $\bullet>0$. Is there a model structure on the category of differential graded algebras, and more generally is there a model structure on the category of $P$-algebras for some cofibrant operad $P$?
References are welcome. 
Thank you very much. 



Model structure on non-negative differential graded algebras with homological grading
Yes, this is possible.


Because of the way you worded your question, I'm going to assume you already know the existence of the model structure for unbounded $P$-algebras and use that in the proof. I also want to assume that your operad is concentrated in non-negative degrees so that the free bounded algebra functor behaves as we expect it to.


Theorem 3.3 of Crans, *Quillen closed model structures for sheaves* gives criteria for transfer of a cofibrantly generated model category structure along an adjoint that are easily checkable in this case.


Let $C$ be a cofibrantly generated model category with $I$ (respectively $J$) the set of generating (trivial) cofibrations and let $R:D\to C$ be a right adjoint with left adjoint $L$. You want a model category on $D$ where the fibrations and weak equivalences are created by $R$.


The conditions you have to verify are that


1. $D$ has small limits and colimits,
2. a smallness criterion on $L(I)$ and $L(J)$, and
3. relative $L(J)$-cell complexes become $C$ weak equivalences under $R$.


In this case $C$ is bounded chains, $D$ is bounded $P$-algebras, $R$ is the forgetful functor, and $L$ is the free algebra functor. We will exploit the fact that we already know something about *unbounded* algebras to show that we have condition (3.)


You presumably know (1.) or you wouldn't even be trying. (2.) is easy in algebraic categories and I'll skip it. You basically already know it anyway from the existence of the transferred model structure in the unbounded case.


For (3.), note that the inclusion $i$ of bounded algebras into unbounded algebras reflects weak equivalences: $R\_{\text{unbounded}}i(f)$ is a weak equivalence implies $R\_{\text{bounded}}(f)$ is a weak equivalence. So it suffices to check that $i$ applied to an $L\_\text{bounded}(J\_\text{bounded})$-relative cell complex is a weak equivalence under $R\_\text{unbounded}$.


The inclusion $i$ is a left adjoint and so preserves colimits, so such a morphism is an $iL\_\text{bounded}(J\_\text{bounded})$-relative cell complex. But $iL\_\text{bounded}$ is the restriction of $L\_\text{unbounded}$ and $J\_\text{bounded}\subset J\_\text{unbounded}$ so any such morphism is an $L\_\text{unbounded}(J\_\text{unbounded})$-relative cell complex, and thus a trivial cofibration in the model structure on unbounded algebras, and thus a weak equivalence under $R\_\text{unbounded}$.


