symmetric monoidal (∞,1)-category of spectra
An -algebra over an -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 -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 -operads by ordinary operads enriched in a suitable monoidal model category. In these models -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 full generality we have:
Specifically if is a coCartesian fibration of (∞,1)-operads then this exhibits as equipped with the structure of an -monoidal (∞,1)-category. Then a section is a -algebra in with respect to this structure. (The “microcosm principle”).
For the moment see
Model category structures for -algebras are discussed in
Section 2.1.3 of