1)Let $Cd_{\geq 0}ga$ be the category of non negatively commutative cochain dg algebra over a field $\Bbbk$ of charachteristic zero. Let $w\: : \: Cd_{\geq 0}ga\to dg_{\geq 0}Mod$ be the forgethfull functor, where $dg_{\geq 0}Mod$ is the category of positively graded cochain modules over $\Bbbk$ (equipped with its standard model structure where weak equivalences are the quasi isomorphism and the fibrations are the degreewise surjections). Then $Cd_{\geq 0}ga$ is a model category where

a) $f$ is weak equivalences if so is $w(f)$ ,

b) the fibrations are the degree wise surjections,

2)On the other hand let $Cdga$ be the category of unbounded commutative cochain dg algebra over a field $\Bbbk$ of charachteristic zero. It is a model category with the "same" model category structure, it may be obtained in a similar way: from the model category of (unbounded) dg module via the forgethful functor.

Let $\Bbbk$ be a commutative ring. This argument can be extended to any operad $P$ over the category of unbounded (cochain) dg $\Bbbk$-module $dgMod$ such that $P$ is "$\Sigma$-split" (See Vladimir Hinich, http://arxiv.org/abs/q-alg/9702015, or http://ncatlab.org/nlab/show/model+structure+on+dg-algebras+over+an+operad).

3)Let $dga(P)$ be the category of unbounded cochain dg algebras over a $\Sigma$-split operad $P$, let $w\: : \: dga(P)\to dgMod$ be the forgethful functor. The a map $f$ in is a a weak equivalence if so is $w(f)$ and a map is a fibrations if it is a degrewise surjections.

In particular the operad $Ass$ is always $\Sigma$-split and any operad is $\Sigma$-split if $\Bbbk$ contains $\mathbb{Q}$.

Q1: Let $P$ an operad over the category of non negatively cochain dg modules over a field of char zero. Does the above free-forgetful adjunction work for non negatively cochain dg modules? I know that this is true when $P=Comm_{+}$ the unitary commutative operad.

Q2: What is the main difference (relation?) between the category of bounded (non negative cochain) dg algebras and unbounded (cochain) dg algebras?

I know for examples that in the category of bounded cochain dg algebras only the connected objects admit a nice cofibrant resolution (minimal model), what happen in the unbounded case?



The free-forgetful adjunction still works in the non-negatively graded setting and induces a cofibrantly generated model category structure in caracteristic zero.

This follows actually from a very general statement about the transfer of such model structures for algebras over operads in a symmetric monoidal model category, namely Theorem 12.3.A in Fresse's book Modules over operads and functors. In general one only gets a semi-model category for algebras over a $\Sigma$-cofibrant operad, but in your special case you have a full model category for algebras over any operad.

In the unbounded setting, cofibrant resolutions of algebras over an operad can be built explicitly as quasi-free resolutions over a bar construction. This is explained in section 4.2 of B. Fresse, Operadic cobar constructions, cylinder objects and homotopy morphisms of algebras over operads.


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