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

Context

Higher algebra

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

Contents

Idea

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.

Definition

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:

Definition

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

Examples

See also

References

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 (2.26.41.0)