equivalences in/of $(\infty,1)$-categories
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
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 $E_\infty$, in the sense of E-infinity operad.
A commutative monoid in a symmetric monoidal (infinity,1)-category $C$ is a lax symmetric monoidal $(\infty,1)$-functor
In more detail, this means the following:
Given a symmetric monoidal (infinity,1)-category in its quasi-categorical incarnation as a coCartesian fibration of simplicial sets
a commutative monoid in $C$ is a section
such that $A$ carries collapsing morphisms in $FinSet_*$ to coCartesian morphisms in $C^\otimes$.
For $C$ a symmetric monoidal (∞,1)-category write $CMon(C)$ for the $(\infty,1)$-category of commutative monoids in $C$.
$CMon(C)$ has all (∞,1)-coproducts and these are computed as tensor products in $C$.
For $K$ a sifted (infinity,1)-category , (∞,1)-colimits of shape $K$ exist in $CMon(C)$ and are computed in $C$ if $K$-colimits exist in $C$ are preserved by tensor product with any object.
$CMon(C)$ has all (∞,1)-limits and these are computed in $C$.
This is (Lurie DAG III, section 4) or (Lurie HA, sections 3.2.2 and 3.2.3).
$(\infty,1)$-Colimits over simplicial diagrams exists in $CMon(C)$ and are computed in $C$ if they exist in $C$ and a preserved by tensor products.
Because the simplex category is a sifted (infinity,1)-category (as discussed there).
A commutative monoid in the stable (infinity,1)-category of spectra is a commutative ring spectrum or E-infinity ring.
An equivalent reformulation of commutative monoids in terms (∞,1)-algebraic theories is in