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 (∞,1)-category $(C,\otimes)$ is closed if for each object $X \in C$ the (∞,1)-functor
given by forming the tensor product with $C$ has a right adjoint (∞,1)-functor
(In cases where the monoidal structure is not assumed symmetric, the property of possessing a right adjoint to tensoring on the left (resp. right) is called left (resp. right) closed, while closed is used for the properties jointly (Def. Definition 4.1.1.17 of Higher Algebra).)
Every (∞,1)-topos with its structure of a cartesian monoidal (∞,1)-category is closed. See there for details.
The (∞,1)-category of (∞,1)-modules over an E-∞ ring is closed.
symmetric monoidal category, symmetric monoidal (∞,1)-category
closed monoidal category, closed monoidal $(\infty,1)$-category