Admissibility and rectification of colored symmetric operads

Abstract

We establish a highly flexible condition that guarantees that all colored symmetric operads in a symmetric monoidal model category are admissible, that is, the category of algebras over any operad admits a model structure transferred from the original model category. We also give a necessary and sufficient criterion that ensures that a given weak equivalence of admissible operads admits rectification, that is, the corresponding Quillen adjunction between the categories of algebras is a Quillen equivalence. In addition, we show that Quillen equivalences of underlying symmetric monoidal model categories yield Quillen equivalences of model categories of algebras over operads. Applications of these results include enriched categories, colored operads, prefactorization algebras, and commutative symmetric ring spectra. Contents

Homotopical Algebra for Lie Algebroids

“…Since the vertical forgetful functors detect equivalences and preserve sifted homotopy colimits (see [23,Proposition 7.8] for the case of Lie algebras), it suffices to check the right derived functor of ker : Mod dg A /T A → Mod dg A has these properties as well. But it follows immediately from the fact that Mod dg A is a stable model category that taking homotopy pullbacks along 0 → T A detects equivalences and preserves all homotopy colimits indexed by contractible categories.…”

Enriched ∞-operads

“…Proof. The operad Op S is Σ-cofibrant (which here just means that the Σ n -actions are all free), so this is a special case of [39,Theorem 7.10]. (As stated, this result requires a simplicial model category, but since our operad Op S is merely an operad in sets the same proof goes through without this assumption.)…”

Tangent categories of algebras over operads

“…Following [BM09] we will refer to P A as the enveloping operad of A, and refer to the M-enriched category P A 1 as the enveloping category of A. The category of algebras over P A is equivalent to the category (Alg P ) A of P-algebras under A (see [PS14,Proposition 4.4(iv)]). When A = P 0 is the initial P-algebra the natural map P → P A is an isomorphism ([PS14, Proposition 4.4(i)]).…”

“…While Theorem 1.0.1 pertains to model categories, it can also be used to obtain results in the ∞-categorical setting, using the rectification results of [PS14] and [NS15]. This is worked out in §4.3, where the following ∞-categorical analogue of the above result is established (see Theorem 4.3.3):…”