Contents

model category

for ∞-groupoids

# Contents

## Idea

This entry discusses model category structures on categories of algebras over an operad in the category of chain complexes (unbounded).

This is a special case of the general discussion at model structure on algebras over an operad, but the category of (unbounded) chain complexes warrents some special attention.

## Definition

Let $k$ be a commutative ring. Write $Ch_\bullet(k)$ for the category of unbounded chain complexes of $k$-modules.

###### Definition

An operad $P$ over $Ch_\bullet(k)$ is called $\Sigma$-split if (…)

A quasi-isomorphism between such operads $P_1 \to P_2$ is said to be compatible with $\Sigma$-splitting if (…)

###### Proposition

If $k$ contains the ring of rational numbers, $\mathbb{Q} \hookrightarrow k$, then every $Ch_\bullet(k)$-operad is $\Sigma$-split and every quasi-isomorphism of operads is compatible with $\Sigma$-splitting.

The associative operad $Assoc_k$ is $\Sigma$-split for all $k$.

This is (Hinich, example 4.2.5).

###### Theorem

Let $P$ be a $\Sigma$-split operad, def. , in $Ch_\bullet(k)$. Then the category $Alg_{Ch_\bullet(k)}(P)$ of algebras over the operad admits a model category structure whose

• weak equivalences are the underlying quasi-isomorphisms

• whose fibrations are the degreewise surjections

in $Ch_\bullet(k)$.

$Alg_{Ch_\bullet(k)}(P) \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} Ch_\bullet(k)$

is therefore a Quillen adjunction (see at transferred model structure).

This appears as (Hinich, theorem 4.1.1).

## Properties

### Invariance under equivalence and rectification

###### Theorem

If $P_1 \to P_2$ is a quasi-isomorphism of $\Sigma$-split operads compatible with splittings, then there is an induced Quillen equivalence

$Alg(P_1) \stackrel{\overset{}{\leftarrow}}{\underset{}{\rightarrow}} Alg(P_2)$

between the corresponding model structures on their algebras, as above.

This is (Hinich, theorem 4.7.4).

###### Remark

Theorem in particular provides rectification results for homotopy algebras: if $P$ is some operad and $\tilde P \stackrel{\simeq}{\to} P$ a cofibrant resolution in the suitable model structure on operads, then the theorem says that $P$-homotopy algebras have the same homotopy theory as the plain $P$-algebras.

Famous examples include the Quillen equivalence between the model structure on dg-Lie algebras and the model structure for L-infinity algebras.

### Simplicial enrichment

We discuss how the above model structure on $Alg_{Ch_\bullet(k)}(P)$ is almost enhanced to a simplicial model category structure.

First we recall the standard definition of polynomial differential forms on simplices:

###### Definition

For $n \in \mathbb{N}$ let $\Omega_{poly}^\bullet(\Delta^n)$ be the commutative dg-algebra of polynomial differential forms on the $n$-simplex:

as a graded algebra it is

$\Omega_{poly}^{\bullet}(\Delta^n) := k[t_0, \cdots, t_n, d t_0, \cdots, d t_n]/(\sum t_i -1, \sum d t_i)$

with the differential the usual de Rham differential under the embedding $\Omega^\bullet_{poly}(\Delta^n) \hookrightarrow \Omega^\bullet(\Delta^n)$.

For $f : [k] \to [l]$ a morphism in the simplex category let

$\Omega^\bullet_{poly}(f) : \Omega^\bullet_{poly}(\Delta^l) \to \Omega^\bullet_{poly}(\Delta^k)$

be the morphism of dg-algebras given on generators by

$\Omega^\bullet_{poly}(f) : t_i \mapsto \sum_{f(j) = i} t_j \,.$

This yields a simplicial commutative dg-algebra

$\Omega^\bullet_{poly}(\Delta^{(-)}) : \Delta^{op} \to cdgAlg_k$

or equivalently a cosimplicial object in the opposite category $cdgAlg_k^{op}$.

By the general definition of differential forms on presheaves this extends by left Kan extension to a functor

$\Omega^\bullet_{poly} : sSet \to cdgAlg_k^{op}$

given by

$\Omega^\bullet_{poly}(S) = \int^{[k]\in \Delta} S_k \cdot \Omega^\bullet_{poly}(\Delta^k) \,,$

where on the right be have the coend over the copowering of $cdgAlg_k^{op}$ over Set.

###### Definition

For $P$ a dg-operad as above, define sSet-hom-objects between objects $A,B \in Alg(P)$ by

$Alg_P(A,B) := ([n] \mapsto Hom_{Alg_P}(A, B \otimes \Omega^\bullet_{poly}(\Delta^n)) \in sSet \,.$
###### Proposition

These simplicial hom-objects satisfy the dual of the pushout-product axiom (see enriched model category).

This is (Hinich, lemma 4.8.4).

###### Proposition

For $S$ a degreewise finite simplicial set, we have a natural isomorphism

$Hom_{Alg_P}(A, B \otimes \Omega^\bullet_{poly}(S)) \simeq Hom_{sSet}(S, Alg_P(A,B)) \,.$

This is (Hinich, lemma 4.8.3).

###### Proposition

The homotopy category of $Alg_P$ is given by

$Ho(Alg_P)(A,B) \simeq \pi_0 Alg_P(Q A,B) \,,$

where $Q A$ is a cofibrant resolution of $A$.

This appears as (Hinich, section 4.8.10).

## Examples

The model structure on dg-algebras over an operad is discussed in

based on results in

• A.K. Bousfield, V.K.A.M. Gugenheim, On PL de Rham theory and rational homotopy type , Memoirs AMS, t.8, 179(1976)