infinity-algebra over an (infinity,1)-operad


Higher algebra

(,1)(\infty,1)-Category theory



An \infty-algebra over an (,1)(\infty,1)-operad is an ∞-groupoid equipped with higher algebraic operations as encoded by an (∞,1)-operad. Since there is not really any other sensible notion of algebra for an (,1)(\infty,1)-operad, we feel free to drop the prefix (although in other cases it can be helpful to disambiguate).

This is the (∞,1)-category theory-analog of the notion of algebra over an operad. Notice that in the literature one frequently sees model category presentations of (,1)(\infty,1)-operads by ordinary operads enriched in a suitable monoidal model category. In these models \infty-algebras are be presented by ordinary algebras over cofibrant resolutions of ordinary enriched operads. This is directly analogous to how (∞,1)-categories may be presented by simplicially enriched categories.

Also notice that the enrichment used in these models is not necessarily over Top / sSet (the standard presentations of ∞Grpd) but often notably over a category of chain complexes. But at least for connective chain complexes, the Dold-Kan correspondence says that these, too, are in turn models for certain ∞-groupoids. This, in turn, is in direct analogy to how a stable (∞,1)-category may be presented by a dg-category.


In terms of (,1)(\infty,1)-categories of operators

We discuss \infty-algebras with (∞,1)-operads viewed in terms of their (∞,1)-categories of operators as in (Lurie).

In full generality we have:


For 𝒞 𝒪 \mathcal{C}^\otimes \to \mathcal{O}^\otimes a fibration of (∞,1)-operads, then for 𝒫 𝒪 \mathcal{P}^\otimes \to \mathcal{O}^\otimes any other homomorphism, an (∞,1)-algebra over 𝒫 \mathcal{P}^\otimes in 𝒞 \mathcal{C}^\otimes is a homomorphism of (∞,1)-operads from 𝒫\mathcal{P} to 𝒞\mathcal{C} over 𝒪\mathcal{O}

Specifically if 𝒞 𝒪 \mathcal{C}^\otimes \to \mathcal{O}^\otimes is a coCartesian fibration of (∞,1)-operads then this exhibits 𝒞\mathcal{C} as equipped with the structure of an 𝒪\mathcal{O}-monoidal (∞,1)-category. Then a section A:𝒪 𝒞 A \colon \mathcal{O}^\otimes \to \mathcal{C}^{\otimes} is a 𝒪\mathcal{O}-algebra in 𝒞\mathcal{C} with respect to this structure. (The “microcosm principle”).

Model category presentations

We discuss presentations of (∞,1)-categories of \infty-algebras over (∞,1)-operads by model category structures on categories of algebras over an operad enriched in some suitable monoidal model category.


For the moment see


See also


Model category structures for \infty-algebras are discussed in

Section 2.1.3 of

Revised on February 21, 2014 02:04:50 by Tim Porter (