category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
An indexed monoidal category is a kind of indexed category, consisting of a base category $S$ and a pseudofunctor $S^{op} \to MonCat$ to the 2-category of monoidal categories and strong monoidal functors between them. We write this as $A\mapsto (C^A, \otimes_A, I_A)$.
By the usual Grothendieck construction, this pseudofunctor can be regarded as a fibration. And if $S$ has finite products, then the “fiberwise” monoidal structures $\otimes_A$ can also be “Grothendieckified” into an “external tensor product”
defined by $M\boxtimes N = \pi_2^\ast M \otimes_{A\times B} \pi_1^\ast N$. This makes the total category of the fibration a monoidal category and the fibration itself a strong monoidal functor (where $S$ is regarded as equipped with its cartesian monoidal structure); this is called a monoidal fibration. Moreover, we can recover $\otimes_A$ from $\boxtimes$ via $M\otimes_A N = \Delta_A^\ast (M\boxtimes N)$, 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 $f^\ast\colon C^B \to C^A$ induced by a morphism $f\colon A\to B$ in $S$ all have left adjoints $f_!$. If these left adjoints satisfy the Beck-Chevalley condition for all pullback squares in $S$, 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 notion of indexed monoidal category has been rediscovered many times by different people in different contexts. Some references include:
M. F. Gouzou and R. Grunig, Fibrations relatives, Seminaire de Theorie des Categories, dirige par J. Benabou, November 1976
A. Asperti and A. Corradini. A categorical modal for logic programs: indexed monoidal categories. In:J.W. de Bakker and W.P. de Roever, Ed. REX Workshop, 110–137, Lecture Notes in ComputerScience, Vol. 666. Springer Verlag, New York-Berlin, 1993.
G. Amato and J. Lipton, Indexed categories and bottom-up semantics of logic programs. Lecture Notesin Computer Science, Vol. 2250. Springer Verlag, New York-Berlin, 2001
Pieter Hofstra, Federico De Marchi, Descent for Monads, in Theory and Applications of Categories, 16 24 (2006) 668–699. (tac:16-24)
Mike Shulman, Framed bicategories and monoidal fibrations, in Theory and Applications of Categories, Vol. 20, 2008, No. 18, pp 650-738. (TAC)
Mike Shulman, Enriched indexed categories (arXiv:1212.3914)
Discussion of traces and of dual objects in indexed monoidal categories is in
This also presents a string diagram calculus for indexed monoidal categories, extending that for monoidal hyperdoctrines in
which is made rigorous in:
On the monoidal Grothendieck construction:
Last revised on May 29, 2022 at 14:31:31. See the history of this page for a list of all contributions to it.