nLab very commutative object

under constrution

Given a symmetric monoidal (∞,1)-category $\mathcal{C}$ then one may consider E-∞ space-objects inside it.

If the monoidal structure happens to be Cartesian then one may ask for “more” commutativity than that. Jacob Lurie called that “very commutative” (MO comment)

(should come down to whether one interprets commutative monoids as a 1-algebraic theory or an (∞,1)-algebraic theory, i.e. (2,1)-algebraic theory of E-infinity algebras)