Homotopy theory of nonsymmetric operads
Abstract.
We endow categories of nonsymmetric operads with natural model structures. We work with no restriction on our operads and only assume the usual hypotheses for model categories with a symmetric monoidal structure. We also study categories of algebras over these operads in enriched nonsymmetric monoidal model categories.
Key words and phrases:
operad, algebra, model category, enriched category2010 Mathematics Subject Classification:
18D50, 55U35, 18D10, 18D35, 18D20Contents
1. Introduction
Operads are wellknown devices encoding the laws of algebras defined by multilinear operations and relations, e.g. there are operads Ass, Com and Lie whose algebras are associative, commutative and Lie algebras, respectively. Morphisms of operads codify relations between different kinds of algebras, e.g. there are morphisms telling us that any commutative algebra is an associative alegbra, and that commutators in an associative algebra yield a Lie algebra.
There are two kinds of operads: symmetric and nonsymmetric operads. Symmetric operads are needed whenever it is necessary to permute variables in order to describe the laws of the corresponding algebras, e.g. Com and Lie. Nonsymmetric operads are specially useful to deal with algebras in nonsymmetric monoidal categories, e.g. given a commutative ring and a set which is not a singleton, the category of modules with object set , which are collections of modules indexed by , , has a nonsymmetric tensor product,
whose associative algebras, i.e. algebras over the operad Ass, are linear categories with object set .
Any object in a symmetric monoidal category , such as the category of modules, has an endomorphism symmetric operad in such that, if is another symmetric operad in , the set of algebra structures on is the set of symmetric operad morphisms . If belongs to a nonsymmetric monoidal category enriched over , such as the category of modules with object set , then there is a nonsymmetric operad in such that the set of algebra structures on over a nonsymmetric operad in is the set of nonsymmetric operad morphisms (Definition LABEL:eop).
When the underlying symmetric monoidal category carries homotopical information, e.g. if we replace modules with differential graded modules, one is often more interested in a space of algebra structures on rather than a plain set. Such a space can be constructed by using the powerful machinery developed by Dwyer and Kan [slc, csl, fcha] provided we can place the operads and in an appropriate model category of operads.
Model categories of operads were first cosidered by Hinich in the differential graded context [haha, ehaha], and by Berger and Moerdijk in a more general setting [ahto]. They dealt with symmetric operads and showed that restrictive hypotheses are necessary to endow the category of all operads with an appropriate model category structure, e.g. when is a algebra or when the symmetric monoidal structure in is cartesian closed and there is a symmetric monoidal fibrant replacement functor.
Motivated by our interest in spaces of differential graded category structures, we consider the nonsymmetric case, which surprisingly enough does not need any restrictive hypothesys, just usual hypotheses for model categories with a monoidal structure [ammmc].
Theorem 1.1.
Let be a cofibrantly generated closed symmetric monoidal model category. Assume that satisfies the monoid axiom. Moreover, suppose that there are sets of generating cofibrations and generating trivial cofibrations in with presentable sources. Then the category of nonsymmetric operads in is a cofibrantly generated model category such that a morphism in is a weak equivalence (resp. fibration) if and only if is a weak equivalence (resp. fibration) in for all . Moreover, if is right proper then so is . Furthermore, if is combinatorial then is also combinatorial.
This theorem can be applied to all examples in [ammmc], see also the references therein:

Complexes of modules over a commutative ring with the usual tensor product of complexes.

Simplicial modules with the levelwise tensor product .

Modules over a finitedimensional Hopf algebra over a field with the tensor product over , e.g. the groupring of a finite group .

Symmetric spectra with their smash product, and more generally modules over a commutative ring spectrum.

spaces with Lydakis’ smash product.

Simplicial functors with their smash product.

modules with their smash product.
In particular, Theorem 1.1 will also be useful to study spaces of spectral category structures.
Remark 1.2.
Recall from [adamekrosicky, Definition 1.13 (2)] that an object of is presentable if there exists a cardinal such that the representable functor commutes with filtered colimits in . Presentable objects are also called small or compact in some references. All objects are presentable in many categories of interest, e.g. in all combinatorial model categories. Actually, up to set theoretical principles any cofibrantly generated model category is Quillen equivalent to a combinatorial model category [cgmc].
Categories of algebras over symmetric operads do not always have a model structure with fibrations and weak equivalences defined in the underlying category. Sufficient conditions can be found in [ahto]. In the framework of nonsymmetric operads they do. When both algebras and operads live in the same ambient symmetric monoidal model category , satisfying the monoid axiom, this has been recently proved by J. E. Harper [htmommc, Theorem 1.2]. We here extend this result to algebras in a monoidal model category satisfying the monoid axiom and appropriately enriched over . This is necessary, for instance, to construct model categories of enrieched categories, of enriched categories, or of any other categorified algebraic structure, see Section LABEL:appl.
Theorem 1.3.
Let and be cofibrantly generated biclosed monoidal model categories. Suppose is symmetric and has a algebra structure given by a strong braided monoidal functor to the center of , , such that the composite functor
is a left Quillen functor. Moreover, assume that and satisfy the monoid axiom (see Definitions LABEL:max and LABEL:nsmax). Furthermore, suppose that has sets of generating cofibrations and generating trivial cofibrations with presentable source. Let be a nonsymmetric operad in . The category of algebras in is a cofibrantly generated model category such that an algebra morphism is a weak equivalence (resp. fibration) if and only if is a weak equivalence (resp. fibration) in . Moreover, if is right proper then so is . Furthermore, if is combinatorial then is also combinatorial.
The notion of monoidal model category in [ammmc, Definition 3.1] makes sense with no modification in the nonsymmetric context, see Definition LABEL:defm.
Any operad morphism induces a change of operad functor
by restricting the action of to along . This functor is the identity on underlying objects in , hence it preserves fibrations and weak equivalences. Moreover, the functor has a left adjoint , therefore we have a Quillen adjunction,
(1.4) 