on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
symmetric monoidal (∞,1)-category of spectra
A model category structure on a category of algebras over an operad enriched in some suitable homotopical category $\mathcal{E}$ is supposed to be a presentation of the (∞,1)-category of ∞-algebras over an (∞,1)-operad.
Let $\mathcal{E}$ be a category equipped with the structure of
such that
the model structure is cofibrantly generated;
the tensor unit $I$ is cofibrant.
Under these conditions there is for each finite group $G$ the structure of a monoidal model category on the category $\mathcal{E}^{\mathbf{B}G}$ of objects in $\mathcal{E}$ equipped with a $G$-action, for which the forgetful functor
preserves and reflects fibrations and weak equivalences.
This is discussed in the examples at monoidal model category.
For $C \in$ Set a set of colours and $P$ a $C$-coloured operad in $\mathcal{E}$ we write $Alg_{\mathcal{E}}(P)$ for the category of $P$-algebras over an operad. There is a forgetful functor
from the category of algebras over the operad in $\mathcal{E}$ to the underlying $C$-colored objects of $\mathcal{E}$.
A $C$-coloured operad $P$ is called admissible if the transferred model structure on $Alg_{\mathcal{E}}(P)$ along the forgetful functor
exists.
So if $P$ is admissible, then $Alg_{\mathcal{E}}(P)$ carries the model structure where a morphism of $P$ algebras $f : A \to B$ is a fibration or weak equivalence if the underlying morphism in $\mathcal{E}$ is, respectively.
Below we discuss general properties of $P$ under which this model structure indeed exists.
The above transferred model structure on algebras over an operad exists if there is a suitable interval object in $\mathcal{E}$.
A cocommutative coalgebra interval object $H\in \mathcal{E}$ is
a cocommutative co-unital comonoid in $\mathcal{E}$
equipped with a factorization
of the codiagonal on $I$ into two homomorphisms of comonoids with the first a cofibration and the second a weak equivalence in $\mathcal{E}$.
Such cocommutative coalgebra intervals exist in
In
there is a coalgebra interval.
If $\mathcal{E}$ has a symmetric monoidal fibrant replacement functor and a coalgebra interval object $H$ then every non-symmetric coloured operad in $\mathcal{E}$ is admissible, def. 2: the transferred model structure on algebras exists.
If the interval is moreover cocommutative, then the same is true for every symmetric coloured operad.
This is (BergerMoerdijk, theorem 2.1), following (BergerMoerdijk-Homotopy, theorem 3.2). For more details see at model structure on operads.
Since the coalgebra interval in the category of chain complexes is not cocommutative, this case requires special discussion, as some of the statements below will not apply to it. For more on this case see model structure on dg-algebras over an operad.
Recall the notion of resolutions of operads and of the Boardman-Vogt resolution $W(H,P)$ from model structure on operads.
We now discuss conditions under which model categories of algebras over a resolved operad is Quillen equivalent to that over the original operad. This yields general rectification results for homotopy-algebras over an operad (see also the Examples below.)
Let $\mathcal{E}$ be in addition a left proper model category.
Then for $\phi : P \to Q$ a weak equivalence between admissible $\Sigma$-cofibrant well-pointed $C$-coloured operads in $\mathcal{E}$, the adjunction
is a Quillen equivalence.
This is (BergerMoerdijk, theorem 4.1).
(rectification of homotopy $T$-algebras)
Let still $\mathcal{E}$ be left proper.
Let $P$ be an admissible $\Sigma$-cofibrant operad in $\mathcal{E}$ such that also $W(H,P)$ is admissible.
Then
is a Quillen equivalence.
For $P = Assoc$ the associative operad it category of algebras $Alg_{\mathcal{E}} P$ is the category of monoids in $\mathcal{E}$. The above model structure on $Alg_{\mathcal{E}} P$ is the standard model structure on monoids in a monoidal model category.
Let $Assoc$ be the associative operad in Set regarded as an operad in Top under the discrete space embedding $Disc : Set \to Top$.
Let $I_*$ be the operad whose algebras are pointed objects. There is a canonical morphism $i : I_* \to Assoc$.
The relative Boardman-Vogt resolution
produces precisely Stasheff’s A-∞ operad.
This is (BergerMoerdijk, page 13)
This follows from the rectification theorem, using that by the above algebras over $W([0,1], I_* \to Assoc)$ are precisely A-∞ spaces.
This is a classical statement. See A-∞ algebra for background and references.
Let $Lie$ be the Lie operad.
A cofibrant resolution is $L_\infty$, the operad whose algebras in chain complexes are L-infinity algebras.
Now (…)
Let $C$ be a small $\mathcal{E}$-enriched category with set of objects $Obj(C)$. There is an operad $Diag_{C}$
whose algebras are enriched functors
hence diagrams in $\mathcal{E}$. Then the Boardman-Vogt resolution
is the operad for homotopy coherent diagrams over $C$ in $\mathcal{E}$.
The rectification theorem above now says that every homotopy coherent diagram is equivalent to an ordinary $\mathcal{E}$-diagram. For $\mathcal{E} =$ Top this is known as Vogt's theorem.
The constuction $Alg_{\mathcal{E}}(P)$ of a category of algebras over an operad is contravariantly functorial in $P$. Therefore if $P^\bullet$ is a cosimplicial object in the category of operads, we have that $Alg_{\mathcal{E}}(P^\bullet)$ is a (large) simplicial category of algebras. Moreover, the Boardman-Vogt resolution $W(P)$ is functorial in $P$.
These two facts together allow us to construct simplicial categories of homotopy algebras.
Specifically, there is a cosimplicial operad $Assoc^\bullet$ which
in degree 0 is the usual associative operad $Assoc^0 = Assoc$,
in degree 1 is the operad whose algebras are triples consisting of two associative monoids and one bimodule between them;
in degree 2 it is the operad whose algebras are tuples consisting of three associative algebras $A_0, A_1, A_2$ as well as one $A_i$-$A_j$-bimodule $N_{ i j}$ for each $0 \leq i \lt j \leq 2$ and a homomorphism of bimodules
and so on.
The simplicial category of algebras over $Assoc^\bullet$ is one incarnation of the 2-category of algebras, bimodules and bimodules homomorphisms.
We can pass to the corresponding $\infty$-algebras by applying the Boardman-Vogt resolution to the entire cosimplicial diagram of operads, to obtain the cosimplicial A-∞ operad
The simplicial category of algebras over this has as objects A-∞ algebras, as morphism bimodules between these, and so on.
This is discussed in (BergerMoerdijkAlgebras, section 6).
(∞,1)-operad, model structure on operads
algebra over an (∞,1)-operad, model structure on algebras over an operad
A general discussion of the model structure on operads is in
The concrete construction of the specific cofibrant resolutions in these structures going by the name Boardman-Vogt resolution is in
The discussion of the model structure on algebras over a suitable operad is in
More discussion on the transport of operad algebra structures along Quillen adjunctions/Bousfield localizations between the underlying model categories is in
Carles Casacuberta, Javier Gutiérrez, Ieke Moerdijk, Rainer Vogt, Localization of algebras over coloured operads, Proceedings of the London Mathematical Society (3) 101 (2010), no. 1, 105-136 (arXiv:0806.3983)
Javier Gutiérrez, Transfer of algebras over operads along derived Quillen adjunctions, Journal of the London Mathematical Society 86 (2012), 607-625 (arXiv:1104.0584)