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?
A symmetric monoidal $(\infty,1)$-category is
which is “$\infty$-tuply monoidal”, or “stably monoidal”.
This means that it is
for which the tensor product is commutative up to infinite coherent homotopy.
This can be understood as a special case of an (∞,1)-operad (…to be expanded on…)
Equivalently, a symmetric monoidal $(\infty,1)$-category is a commutative algebra in an (infinity,1)-category in the (infinity,1)-category of (infinity,1)-categories.
Recall that in terms of quasi-categories a general monoidal (infinity,1)-category is conceived as a coCartesian fibration $C^\otimes \to N(\Delta)^{op}$ of simplicial sets over the (opposite of) the nerve $N(\Delta)^{op}$ of the simplex category satisfying a certain property.
The fiber of this fibration over the 1-simplex $[1]$ is the monoidal (infinity,1)-category $C$ itself, its value over a map $[n] \to [1]$ encodes the tensor product of $n$ factors of $C$ with itself.
The following definition encodes the commutativity of all these operations by replacing $\Delta$ with the category $FinSet_*$ of pointed finite sets.
A symmetric monoidal $(\infty,1)$-category is a coCartesian fibration of simplicial sets
such that
In other words, a symmetric monoidal $(\infty,1)$-category is an $\mathcal{O}$-monoidal (∞,1)-category for
the commutative (∞,1)-operad.
See (Lurie, def. 2.0.0.7).
The homotopy category of a symmetric monoidal $(\infty,1)$-category is an ordinary symmetric monoidal category.
There is a functor $\varphi : \Delta^{op} \to FinSet_*$ such that the monoidal (infinity,1)-category underlying a symmetric monoidal $(\infty,1)$-category $p : C^\otimes \to N(FinSet_*)$ is the (infinity,1)-pullback of $p$ along $\varphi$.
A presentation of the (∞,1)-category of all symmetric monoidal $(\infty,1)$-categories is provided by the model structure for dendroidal coCartesian fibrations.
See commutative monoid in a symmetric monoidal (∞,1)-category.
symmetric monoidal category, symmetric monoidal $(\infty,1)$-category, symmetric monoidal (∞,n)-category
prime spectrum of a symmetric monoidal stable (∞,1)-category
The defintion of symmetric monoidal quasi-category is definition 1.2 in
and definition 2.0.0.7 in
A concise treatment is also available in
Relation to monoidal model categories is discussed in