related by the Dold-Kan correspondence
symmetric monoidal (∞,1)-category of spectra
A model category structure on a category of algebras over an operad enriched in some suitable homotopical category is supposed to be a presentation of the (∞,1)-category of ∞-algebras over an (∞,1)-operad.
Let be a cofibrantly generated symmetric monoidal model category. Let be a cofibrant operad. If satisfies the monoid axiom in a monoidal model category, then there is an induced model structure on the category of algebras over an operad.
See (Spitzweck 01, Theorem 4).
Let be a category equipped with the structure of
the model structure is cofibrantly generated;
the tensor unit is cofibrant.
preserves and reflects fibrations and weak equivalences.
This is discussed in the examples at monoidal model category.
from the category of algebras over the operad in to the underlying -colored objects of .
So if is admissible, then carries the model structure where a morphism of algebras is a fibration or weak equivalence if the underlying morphism in is, respectively.
Below we discuss general properties of under which this model structure indeed exists.
A cocommutative coalgebra interval object is
Such cocommutative coalgebra intervals exist in
there is a coalgebra interval.
If has a symmetric monoidal fibrant replacement functor and a coalgebra interval object then every non-symmetric coloured operad in 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.
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.
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 be in addition a left proper model category.
Then for a weak equivalence between admissible -cofibrant well-pointed -coloured operads in , the adjunction
is a Quillen equivalence.
This is (BergerMoerdijk, theorem 4.1).
(rectification of homotopy -algebras)
Let still be left proper.
Let be an admissible -cofibrant operad in such that also is admissible.
is a Quillen equivalence.
For the associative operad it category of algebras is the category of monoids in . The above model structure on is the standard model structure on monoids in a monoidal model category.
Let be the operad whose algebras are pointed objects. There is a canonical morphism .
This is (BergerMoerdijk, page 13)
This is a classical statement. See A-∞ algebra for background and references.
Let be the Lie operad.
Let be a small -enriched category with set of objects . There is an operad
whose algebras are enriched functors
is the operad for homotopy coherent diagrams over in .
The constuction of a category of algebras over an operad is contravariantly functorial in . Therefore if is a cosimplicial object in the category of operads, we have that is a (large) simplicial category of algebras. Moreover, the Boardman-Vogt resolution is functorial in .
These two facts together allow us to construct simplicial categories of homotopy algebras.
Specifically, there is a cosimplicial operad which
in degree 0 is the usual associative operad ,
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 as well as one --bimodule for each and a homomorphism of bimodules
and so on.
The simplicial category of algebras over is one incarnation of the 2-category of algebras, bimodules and bimodules homomorphisms.
This is discussed in (BergerMoerdijkAlgebras, section 6).
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
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)