nLab indexed monoidal category

Definition

Indexed monoidal 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”

$$\boxtimes\colon C^A \times C^B \to C^{A\times B}$$

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).

Indexed closed monoidal category

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.

Examples

• $S=Sets$, $C^A=$ $A$-indexed families of objects of $C$, for any monoidal category $C$.
• $S=Gpd$, $C^A=$ $A$-diagrams of objects of $C$
• $S=$ any category with pullbacks, $C^A = S/A$
• $S=Top$, $C^A=$ parametrized spectra over $A$
• $S=Grp$ or $TopGrp$, $C^A=$ sets or spaces with an action by $A$
• The homotopy category of any of the above equipped with a homotopy 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).

Related concepts

• external tensor product

• dependent linear type theory, Wirthmüller context

• monoidal Grothendieck construction

• indexed monoidal (infinity,1)-category

References

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, Theory Appl. Categ. 28 (2013) 616-695 [arXiv:1212.3914, tac:28-21]

Discussion of traces and of dual objects in indexed monoidal categories is in

• Kate Ponto, Mike Shulman, Duality and traces in indexed monoidal categories, Theory and Applications of Categories 26 23 (2012) [arXiv:1211.1555, tac:26-23, blog]

This also presents a string diagram calculus for indexed monoidal categories, extending that for monoidal hyperdoctrines in

• Geraldine Brady, Todd Trimble, A string diagram calculus for predicate logic (1998)

which is made rigorous in:

• Cary Malkiewich, Kate Ponto, Coherence for indexed symmetric monoidal categories $[$arxiv:1811.12873$]$

On the monoidal Grothendieck construction:

• Joe Moeller, Christina Vasilakopoulou, Monoidal Grothendieck Construction, Theory and Applications of Categories, 35 31 (2020) 1159-1207 $[$arXiv:1809.00727, tac:35-31$]$