category with duals (list of them)
dualizable object (what they have)
An indexed monoidal category is a kind of indexed category, consisting of a base category and a pseudofunctor to the 2-category of monoidal categories and strong monoidal functors between them. We write this as .
By the usual Grothendieck construction, this pseudofunctor can be regarded as a fibration. And if has finite products, then the “fiberwise” monoidal structures can also be “Grothendieckified” into an “external tensor product”
defined by . This makes the total category of the fibration a monoidal category and the fibration itself a strong monoidal functor (where is regarded as equipped with its cartesian monoidal structure); this is called a monoidal fibration. Moreover, we can recover from via , so the two structures have the same information. (Shulman 08).
If all the fibers are not just monoidal but closed monoidal categories and the base change morphisms are not just strong monoidal but also strong closed monoidal functors, then the indexed monoidal category is an indexed closed monoidal category (Shulman 08, def. 13.1, Shulman 12, theorem 2.14).
If in addition all fibers are symmetric monoidal one might also call this a system of Wirthmüller contexts of six operations. If furthermore all fibers have all duals, then this is also what should be called categorical semantics for dependent linear type theory.
In many cases, the reindexing functors induced by a morphism in all have left adjoints . If these left adjoints satisfy the Beck-Chevalley condition for all pullback squares in , then the indexed category is traditionally said to have indexed coproducts. For many applications, though, we only need this condition for a few pullback squares, which coincidentally (?) happen to be those that are pullbacks in any category with finite products (whether or not it even has all pullbacks).
The definition is due to
Generalization to enriched categories is in