category with duals (list of them)
dualizable object (what they have)
symmetric monoidal (∞,1)-category of spectra
The notion of commutative monoid (or commutative monoid object, commutative algebra, commutative algebra object) in a symmetric monoidal (infinity,1)-category is the (infinity,1)-categorical generalization of the notion of commutative monoid in a symmetric monoidal category. It is the commutative version of monoid in a monoidal (infinity,1)-category.
Note that commutative here really means , in the sense of E-infinity operad.
A commutative monoid in a symmetric monoidal (infinity,1)-category is a lax symmetric monoidal -functor
In more detail, this means the following:
a commutative monoid in is a section
such that carries collapsing morphisms in to coCartesian morphisms in .
For a symmetric monoidal (∞,1)-category write for the -category of commutative monoids in .
-Colimits over simplicial diagrams exists in and are computed in if they exist in and a preserved by tensor products.
An equivalent reformulation of commutative monoids in terms (∞,1)-algebraic theories is in