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
(…) symmetric monoidal (∞,1)-category (…)
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, section 4).
$(\infty,1)$-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).