The Model Category of Operads in Simplicial Sets

In §1.4, we briefly explained the definition of a natural model structure for simplicial (and topological) operads. In what follows, we also refer to this model structure as the projective model structure. The weak-equivalences (respectively, fibrations) are, according to this definition, the morphisms of operads which form a weak-equivalence (respectively, a fibration) in the base category of simplicial sets (or topological spaces) in each arity. To summarize this definition, we also say that the weak-equivalences and fibrations of operads are created in the base category aritywise. The cofibrations are characterized by the left lifting property with respect to the acyclic fibrations. The first purpose of this chapter is to provide the proof that this definition of weak-equivalences, fibrations, and cofibrations in §1.4 does give a model structure. We devote the first and second section of the chapter (§§8.1-8.2) to this verification. We define a model structure at the level of symmetric sequences in a first step (in §8.1), and we use the free operad adjunction of §I.1.2 to extend this model structure to operads in a second step (in §8.2). We focus on the simplicial operad case, which is sufficient for our purpose. Indeed, we mentioned in §1.4 that the model category of simplicial and topological operads are Quillen equivalent and, in principle, we get equivalent results when we use simplicial and topological models for the study of the homotopy of operads. One may actually use the general scheme of the proof of §8.2 to define a model structure for operads in other instances of base model categories than the category of simplicial sets. We just give a few hints on the generalizations of our statement in §8.2. The definition of the projective model structure (and our arguments) is valid for the whole category of operads Op = sSet Op (at least, when we work in the category simplicial sets). But in practice we only use the plain (projective) model structure for non-unitary operads, and we therefore restrict ourselves to this subcategory Op∅ ⊂ Op in our constructions. In fact, we are going to introduce another specific model structure for the category of unitary operads, and we devote the third and fourth section of the chapter (§§8.3-8.4) to this subject. The general idea of model categories is to rely on the existence of solutions in certain lifting problems in order to handle the definition of morphisms in homotopy categories (see §1.1.1). In the operad context, the solutions of a lifting problem can often be effectively constructed by induction on the arity grading. In the case of unitary operads, we observed in §I.3.2.1 that the composition products with the arity zero term of a unitary operad P+ are equivalent to restriction morphisms u ∗ : P+(n) → P+(m) such that n ≥ m. Intuitively, these arity decreasing operations can be used to provide some initial value conditions in our inductive construction