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 cocartesian monoidal (∞,1)-category is a symmetric monoidal (∞,1)-category whose tensor product is given by the categorical coproduct. This is dual to the notion of cartesian monoidal (∞,1)-category.
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If $C$ is an (infinity,1)-category admitting finite coproducts, then the cocartesian symmetric monoidal structure on $C$ is equivalent to the symmetric monoidal structure obtained as the opposite of the cartesian monoidal structure on the opposite (infinity,1)-category $C^{op}$.
If $C$ is an (infinity,1)-category with finite coproducts, then the cocartesian model structure on $C$ is universal among symmetric monoidal (infinity,1)-categories $D$ for which there exists a functor $C \to CAlg(D)$, to the symmetric monoidal (infinity,1)-category of commutative algebra objects in $D$.
See (Lurie, Theorme 2.4.3.18).
Section 2.4 of
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