1
$\begingroup$

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?

Thanks

$\endgroup$

1 Answer 1

1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.