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 ${\mathrm{Ch}}_{•}\left(k\right)$ for the category of unbounded chain complexes of $k$-modules.

###### Definition

An operad $P$ over ${\mathrm{Ch}}_{•}\left(k\right)$ 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, $ℚ↪k$, then every ${\mathrm{Ch}}_{•}\left(k\right)$-operad is $\Sigma$-split and every quasi-isomorphism of operads is compatible with $\Sigma$-splitting.

The associative operad ${\mathrm{Assoc}}_{k}$ is $\Sigma$-split for all $k$.

This is (Hinich, example 4.2.5).

###### Theorem

Let $P$ be a $\Sigma$-split operad, def. 1, in ${\mathrm{Ch}}_{•}\left(k\right)$. Then the category ${\mathrm{Alg}}_{{\mathrm{Ch}}_{•}\left(k\right)}\left(P\right)$ 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 ${\mathrm{Ch}}_{•}\left(k\right)$.

${\mathrm{Alg}}_{{\mathrm{Ch}}_{•}\left(k\right)}\left(P\right)\stackrel{\stackrel{F}{←}}{\underset{U}{\to }}{\mathrm{Ch}}_{•}\left(k\right)$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

$\mathrm{Alg}\left({P}_{1}\right)\stackrel{\stackrel{}{←}}{\underset{}{\to }}\mathrm{Alg}\left({P}_{2}\right)$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 2 in particular provides rectification results for homotopy algebras: if $P$ is some operad and $\stackrel{˜}{P}\stackrel{\simeq }{\to }P$ 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 ${\mathrm{Alg}}_{{\mathrm{Ch}}_{•}\left(k\right)}\left(P\right)$ is almost enhanced to a simplicial model category structure.

we have the standard definition of polynomial differential forms on simplices.

###### Definition

For $n\in ℕ$ let ${\Omega }_{\mathrm{poly}}^{•}\left({\Delta }^{n}\right)$ be the commutative dg-algebra of polynomial differential forms on the $n$-simplex:

as a graded algebra it is

${\Omega }_{\mathrm{poly}}^{•}\left({\Delta }^{n}\right):=k\left[{t}_{0},\cdots ,{t}_{n},d{t}_{0},\cdots ,d{t}_{n}\right]/\left(\sum {t}_{i}-1,\sum d{t}_{i}\right)$\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 ${\Omega }_{\mathrm{poly}}^{•}\left({\Delta }^{n}\right)↪{\Omega }^{•}\left({\Delta }^{n}\right)$.

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

${\Omega }_{\mathrm{poly}}^{•}\left(f\right):{\Omega }_{\mathrm{poly}}^{•}\left({\Delta }^{l}\right)\to {\Omega }_{\mathrm{poly}}^{•}\left({\Delta }^{k}\right)$\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

${\Omega }_{\mathrm{poly}}^{•}\left(f\right):{t}_{i}↦\sum _{f\left(j\right)=i}{t}_{j}\phantom{\rule{thinmathspace}{0ex}}.$\Omega^\bullet_{poly}(f) : t_i \mapsto \sum_{f(j) = i} t_j \,.

This yields a simplicial commutative dg-algebra

${\Omega }_{\mathrm{poly}}^{•}\left({\Delta }^{\left(-\right)}\right):{\Delta }^{\mathrm{op}}\to {\mathrm{cdgAlg}}_{k}$\Omega^\bullet_{poly}(\Delta^{(-)}) : \Delta^{op} \to cdgAlg_k

or equivalently a cosimplicial object in the opposite category ${\mathrm{cdgAlg}}_{k}^{\mathrm{op}}$.

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

${\Omega }_{\mathrm{poly}}^{•}:\mathrm{sSet}\to {\mathrm{cdgAlg}}_{k}^{\mathrm{op}}$\Omega^\bullet_{poly} : sSet \to cdgAlg_k^{op}

given by

${\Omega }_{\mathrm{poly}}^{•}\left(S\right)={\int }^{\left[k\right]\in \Delta }{S}_{k}\cdot {\Omega }_{\mathrm{poly}}^{•}\left({\Delta }^{k}\right)\phantom{\rule{thinmathspace}{0ex}},$\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 ${\mathrm{cdgAlg}}_{k}^{\mathrm{op}}$ over Set.

###### Definition

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

${\mathrm{Alg}}_{P}\left(A,B\right):=\left(\left[n\right]↦{\mathrm{Hom}}_{{\mathrm{Alg}}_{P}}\left(A,B\otimes {\Omega }_{\mathrm{poly}}^{•}\left({\Delta }^{n}\right)\right)\in \mathrm{sSet}\phantom{\rule{thinmathspace}{0ex}}.$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

${\mathrm{Hom}}_{{\mathrm{Alg}}_{P}}\left(A,B\otimes {\Omega }_{\mathrm{poly}}^{•}\left(S\right)\right)\simeq {\mathrm{Hom}}_{\mathrm{sSet}}\left(S,{\mathrm{Alg}}_{P}\left(A,B\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$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 ${\mathrm{Alg}}_{P}$ is given by

$\mathrm{Ho}\left({\mathrm{Alg}}_{P}\right)\left(A,B\right)\simeq {\pi }_{0}{\mathrm{Alg}}_{P}\left(QA,B\right)\phantom{\rule{thinmathspace}{0ex}},$Ho(Alg_P)(A,B) \simeq \pi_0 Alg_P(Q A,B) \,,

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

This appears as (Hinich, section 4.8.10).

## References

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)
Revised on March 14, 2013 13:15:07 by Urs Schreiber (82.169.65.155)