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

Idea

An $\infty$-algebra over an $(\infty,1)$-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 $(\infty,1)$-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 $(\infty,1)$-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.

Definition

In terms of $(\infty,1)$-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:

Definition

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

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

• model structure on algebras over an operad.

Examples

See also

• homotopy algebra.

Related concepts

• algebra over a monad

  ∞-algebra over an (∞,1)-monad

• algebra over an algebraic theory

  ∞-algebra over an (∞,1)-algebraic theory

  • homotopy T-algebra / model structure on simplicial T-algebras

• (∞,1)-operad, model structure on operads

  • algebra over an operad, algebra over an (∞,1)-operad, model structure on algebras over an operad

    • module over an algebra over an (∞,1)-operad, model structure on modules over an algebra over an operad

    • rectification

References

• Gijs Heuts: algebras over ∞-operads, Arxiv:math/1110.1776

Model category structures for $\infty$-algebras are discussed in

• Clemens Berger, Ieke Moerdijk, Resolution of coloured operads and rectification of homotopy algebras (arXiv:math/0512576)

Section 2.1.3 of

• Jacob Lurie, Higher Algebra