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?
An (∞,1)-category with (∞,1)-products $\times$ which is a closed monoidal (∞,1)-category with respect to $\otimes := \times$ is called a cartesian closed $(\infty,1)$-category.
Every (∞,1)-topos is a cartesian closed $(\infty,1)$-category. See the section Closed monoidal structure.
cartesian closed category, locally cartesian closed category
cartesian closed model category, locally cartesian closed model category
cartesian closed (∞,1)-category, locally cartesian closed (∞,1)-category