With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
The Grothendieck construction gives a way of gluing together the constituent categories $F(x)$ of an indexed category $F \colon X^{op} \to \mathsf{Cat}$ to get a category $\int F$ which admits an obvious fibration over the base $\int F \to X$. This gives an equivalence between the 2-category of indexed categories and the 2-category of fibrations.
Two ways monoidal structures can join this story is on the total category $\int F$, or on the fibres $F(x)$. The 2-equivalence then lifts to two monoidal variants, one where the fibres are equipped with a monoidal structure, and one where the total category is equipped with a monoidal structure. Under certain conditions on the base category, these two settings are equivalent to each other as well. That is to say, under the right conditions, one can glue together the monoidal structures on the fibres to get a monoidal structure on the total category.
Mike Shulman, Framed bicategories and monoidal fibrations, Theory and Applications of Categories, Vol. 20, No. 18, 2008, pp. 650–738. TAC
Joe Moeller, Christina Vasilakopoulou, Monoidal Grothendieck Construction, Theory and Applications of Categories, 35 31 (2020) 1159-1207 $[$arXiv:1809.00727, tac:35-31$]$
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