related by the Dold-Kan correspondence
symmetric monoidal (∞,1)-category of spectra
We first consider the collections of operations underlying a symmetric operad (with no notion of composition of operations yet).
For a discrete group write for the delooping groupoid: the category with a single object and as its set of morphisms. Then for any other category, write for the functor category, consisting of functors . This is the category of actions of on objects in (the category of representations).
Write for the symmetric group on elements. Take and both to be the trivial group.
The category of collections (of potential operations) in is the product
A collection is a tuple of objects
each equipped with an action by the respective .
that factor the codiagonal on by a cofibration followed by a weak equivalence.
Such cocommutative coalgebra intervals exist in
there is a coalgebra interval.
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-operads.
Assume now that is moreover equipped with a compatible structure of a monoidal model category.
It follows that in this case the category of collections is a cofibrantly generated model category where a morphisms is a fibration or weak equivalence if it is so degreewise in , respectively.
A -operad is called -cofibrant if its underlying collection is cofibrant in the above model stucture
A -operad is called reduced if is the tensor unit, . A morphism of reduced operads is one that is the identity on the 0-component.
admits a commutative Hopf interval object.
This is BergerMoerdijk, theorem 3.1.
If is even a cartesian closed category, a stronger statement is possible:
Let be a cartesian closed category, such that
has a symmetric monoidal fibrant replacement functor.
Then there exists a cofibrantly generated model structure on the category of -operads, in which a morphism is a weak equivalence (resp. fibration) precisely if for all the morphisms are weak equivalences (resp. fibrations) in .
The conditions of the above theorems are satisfied for
The homotopy algebras over a simplicial/topological operad as defined by Boardman and Vogt (see references below), are algebras for cofibrant replacements of these operads in this model structure. This is essentially the statement of theorem 4.1 in (Vogt)
In these contexts,
the associative operad is admissible -cofibrant
the commutative operad is far from being -cofibrant.
Every cofibrant operad is also -cofibrant.
This is (BergerMoerdijk, prop. 4.3).
The relevance of this is in section Homotopy algebras: this property enters the proof of the statement that the model structure on algebras over an operad over a -cofibrant resolution is already Quillen equivalent to that of a full cofibrant resolution.
Many resolutions of operads that appear in the literature are in fact just -cofibrant.
We now discuss the construction and properties of cofibrant resolutions of operads and their algebras.
(assumptions now as at model structure on algebras over an operad)
The category of -coloured operads is itself the category of algebras over a non-symmetric operad. See coloured operad for more. Thus the above theorem provides conditions under which -coloured operads carry a model structure in which fibrations and weak equivalences are those morphisms of operads that are degreewise fibrations and weak equivalences in .
We shall from now on call an operad cofibrant if the morphism from the initial -coloured operad has the left lifting property against degreewise acyclic fibrations of operads (irrespective of whether the above conditions for the existence of the model structure hold).
This is (BergerMoerdijk, theorem 3.2).
For each well-pointed -cofibrant -coloured operad , the -counit factors as a cofibration followed by a weak equivalence
of -coloured operads, naturally in and .
If is a -cofibration between well-pointed -cofibrant -coloured operads, then the induced map is a cofibration of cofibrant -coloured operads.
This is (BergerMoerdijk, theorem 3.5).
Here is also called the coloured Boardman-Vogt resolution of .
An algebra over an operad over is called a -algebra up to homotopy.
We discuss model structures on algebras over resolutions of operads. A more detailed treatment is at model structure on algebras over an operad.
With as above, say
Under mild assumptions on , cofibrant operads are admissible.
For an arbirtrary -operad , the category of homotopy -algebras is the category of -algebras for some cofibrant replacement of .
Indeed, this is well defined up to Quillen equivalence:
Moreover, for this it is sufficient that be -cofibrant .
If is a left proper model category with cofibrant unit, then for a -cofibrant resolution of (not necessarily fully cofibrant!) the category of algebras is Quillen equivalent to that of homotopy -algebras.
For instance the associative operad is -cofibrant, so that by the above every -algebra may be rectified to an ordinary monoid.
See around BerMor03, remark 4.6.
For more see model structure on algebras over an operad.
For enrichment in Top or sSet, the dendroidal homotopy coherent nerve induces a Quillen equivalence between the model structure on coloured topological operads/simplicial operads and the model structure on dendroidal sets. (See there for more details.)
(∞,1)-operad, model structure on operads
An influential article in which many of the homotopical and -categorical aspects of operad theory originate is
Cofibrant Boardman-Vogt resolutions of operads are discussed in
and independently in
And the generalization to colored operads over more general suitable enriching categories is in
(generalizing acorresponding model structure on enriched categories).
An explicit construction of cofibrant resolution in this model structure and its relation to the original constructon of the Boardman-Vogt resolution is in
The induced model structures and their properties on algebras over operads are discussed in
The model structure on dg-operads is discussed in