model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
symmetric monoidal (∞,1)-category of spectra
If $V$ is a symmetric monoidal category that is also a monoidal model category, then under suitable conditions there is also the structure of a model category on the category of symmetric $V$-operads.
This is important for the notion of homotopy algebra over an operad, such as A-∞ algebras and E-∞ algebras.
Throughout, let $V$ be a symmetric closed monoidal model category with all small colimits and finite limits.
We first consider the collections of operations underlying a symmetric operad (with no notion of composition of operations yet).
For $G$ a discrete group write $\mathbf{B}G$ for the delooping groupoid: the category with a single object and $G$ as its set of morphisms. Then for $V$ any other category, write $V^{\mathbf{B}G}$ for the functor category, consisting of functors $\mathbf{B}G \to V$. This is the category of actions of $G$ on objects in $V$ (the category of representations).
For $G$ a finite group also $V^{\mathbf{B}G}$ inherits the structure of a closed symmetric monoidal category with small colimits and finite limits. There is a forgetful functor/free functor adjunction
Write $\Sigma_n$ for the symmetric group on $n \in \mathbb{N}$ elements. Take $\Sigma_0$ and $\Sigma_1$ both to be the trivial group.
The category of collections (of potential operations) in $V$ is the product
A collection $P$ is a tuple of objects
each equipped with an action by the respective $\Sigma_n$.
An object $H \in V$ is a Hopf algebra object if it is equipped with the structure of a monoid, that of a comonoid such that product and coproduct preserve each other.
If $V$ is equipped with a compatible structure of a monoidal model category we say that a a Hopf algebra object is an Hopf interval object if it is equipped with morphisms
that factor the codiagonal on $I$ by a cofibration followed by a weak equivalence.
Such cocommutative coalgebra intervals exist in
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 $V$ is moreover equipped with a compatible structure of a monoidal model category.
If $V$ is a cofibrantly generated model category, then for each finite group $G$ the transferred model structure on $V^{\mathbf{B}G}$ along the forgetful functor
exists.
It follows that in this case the category of collections $Coll(V)$ is a cofibrantly generated model category where a morphisms is a fibration or weak equivalence if it is so degreewise in $V$, respectively.
A $V$-operad is called $\Sigma$-cofibrant if its underlying collection is cofibrant in the above model stucture
A $V$-operad $P$ is called reduced if $P(0)$ is the tensor unit, $P(0) = I$. A morphism of reduced operads is one that is the identity on the 0-component.
If
$V$ is cofibrantly generated
$I$ is cofibrant;
the model structure on the overcategory $V/I$ has a symmetric monoidal fibrant replacement functor;
$V$ admits a commutative Hopf interval object.
Then there exists a cofibrantly generated model category structure on the category of reduced $V$-operads, in which
This is BergerMoerdijk, theorem 3.1.
If $V$ is even a cartesian closed category, a stronger statement is possible:
Let $V$ be a cartesian closed category, such that
$V$ is cofibrantly generated and the terminal object is cofibrant;
$V$ has a symmetric monoidal fibrant replacement functor.
Then there exists a cofibrantly generated model structure on the category of $V$-operads, in which a morphism $P \to Q$ is a weak equivalence (resp. fibration) precisely if for all $n \geq 0$ the morphisms $P(n) \to Q(n)$ are weak equivalences (resp. fibrations) in $V$.
The conditions of the above theorems are satisfied for
$V =$ sSet with the classical model structure on simplicial sets.
The induced model structure on $sSet$-operads is Quillen equivalent to the model structure on dendroidal sets.
$V =$ Top the equivalent classical model structure on topological spaces (compactly generated topological spaces);
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)
$V = Ch_\bullet$, the model structure on chain complexes;
$V = sSh(C)$ the model structure on simplicial sheaves on some site $C$.
In these contexts,
the associative operad is admissible $\Sigma$-cofibrant
the commutative operad is far from being $\Sigma$-cofibrant.
This means we have rectification theorems for A-∞ algebras but not for E-∞ algebras. See model structure on algebras over an operad for more.
Every cofibrant operad is also $\Sigma$-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 $\Sigma$-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 $\Sigma$-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)
First we describe free operads, and then Boardman-Vogt resolutions of operads, obtained from the construction of the free ones by adding labels for lengths in an interval object
The category of $C$-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 $C$-coloured operads carry a model structure in which fibrations and weak equivalences are those morphisms of operads $P \to Q$ that are degreewise fibrations and weak equivalences in $\mathcal{E}$.
We shall from now on call an operad $P$ cofibrant if the morphism $I_C \to P$ from the initial $C$-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).
The forgetful functor from $C$-colored operads to pointed $C$-colored collections has a left adjoint
This is (BergerMoerdijk, theorem 3.2).
For each well-pointed $\Sigma$-cofibrant $C$-coloured operad $P$, the $(F^*_C \dashv U_C)$-counit factors as a cofibration followed by a weak equivalence
of $C$-coloured operads, naturally in $P$ and $H$.
If $P \to Q$ is a $\Sigma$-cofibration between well-pointed $\Sigma$-cofibrant $C$-coloured operads, then the induced map $W(H,P) \to W(H,Q)$ is a cofibration of cofibrant $C$-coloured operads.
This is (BergerMoerdijk, theorem 3.5).
Here $W(H,P)$ is also called the coloured Boardman-Vogt resolution of $P$.
An algebra over an operad over $W(H,P)$ is called a $P$-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 $V$ as above, say
A $V$-operad $P$ is admissible if the category of $P$-algebras carries a transferred model structure from the free functor/forgetful functor adjunction
Under mild assumptions on $V$, cofibrant operads are admissible.
For an arbirtrary $V$-operad $P$, the category of homotopy $P$-algebras is the category of $\hat P$-algebras for some cofibrant replacement $\hat P$ of $P$.
Indeed, this is well defined up to Quillen equivalence:
Moreover, for this it is sufficient that $\hat P$ be $\Sigma$-cofibrant .
If $V$ is a left proper model category with cofibrant unit, then for $\hat P$ a $\Sigma$-cofibrant resolution of $P$ (not necessarily fully cofibrant!) the category of $\hat P$ algebras is Quillen equivalent to that of homotopy $P$-algebras.
For instance the associative operad is $\Sigma$-cofibrant, so that by the above every $A-\infty$-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 $\mathcal{E} =$ 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 $(\infty,1)$-categorical aspects of operad theory originate is
An early notion of resolution of operads in chain complexes is given in section 3 of
Cofibrant Boardman-Vogt resolutions of operads are discussed in
Bielefeld SB 343 (1999), to appear in Topology Appl.
A systematic study of model category structures on monochromatic symmetric operads and their algebras is in
The generalization to a model structure on coloured symmetric operads (symmetric multicategories) is discussed in
and independently in
And the generalization to colored operads over more general suitable enriching categories is in
(generalizing a corresponding 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