indexed monoidal category


Monoidal categories

Category theory

Indexed monoidal categories


Index monoidal category

An indexed monoidal category is a kind of indexed category, consisting of a base category SS and a pseudofunctor S opMonCatS^{op} \to MonCat to the 2-category of monoidal categories and strong monoidal functors between them. We write this as A(C A, A,I A)A\mapsto (C^A, \otimes_A, I_A).

By the usual Grothendieck construction, this pseudofunctor can be regarded as a fibration. And if SS has finite products, then the “fiberwise” monoidal structures A\otimes_A can also be “Grothendieckified” into an “external tensor product

:C A×C BC A×B \boxtimes\colon C^A \times C^B \to C^{A\times B}

defined by MN=π 2 *M A×Bπ 1 *NM\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 SS is regarded as equipped with its cartesian monoidal structure); this is called a monoidal fibration. Moreover, we can recover A\otimes_A from \boxtimes via M AN=Δ A *(MN)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.


  • S=SetsS=Sets, C A=C^A= AA-indexed families of objects of CC, for any monoidal category CC.
  • S=GpdS=Gpd, C A=C^A= AA-diagrams of objects of CC
  • S=S= any category with pullbacks, C A=S/AC^A = S/A
  • S=TopS=Top, C A=C^A= parametrized spectra over AA
  • S=GrpS=Grp or TopGrpTopGrp, C A=C^A= sets or spaces with an action by AA
  • The homotopy category of any of the above equipped with a homotopy theory

In many cases, the reindexing functors f *:C BC Af^\ast\colon C^B \to C^A induced by a morphism f:ABf\colon A\to B in SS all have left adjoints f !f_!. If these left adjoints satisfy the Beck-Chevalley condition for all pullback squares in SS, 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

  • Mike Shulman, Framed bicategories and monoidal fibrations, in Theory and Applications of Categories, Vol. 20, 2008, No. 18, pp 650-738. (TAC)

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

Generalization to enriched categories is in

Revised on January 6, 2016 17:24:51 by Urs Schreiber (