nLab
indexed monoidal category

Definition

An indexed monoidal category is a kind of indexed category, consisting of a category $S$ and a pseudofunctor $S^{\mathrm{op}}\to \mathrm{MonCat}$, which we write 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 product”

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

defined by $M⊠N={\pi }_2^{*}M{\otimes }_{A\times B}{\pi }_1^{*}N$. This makes the total category of the fibration a monoidal category and the fibration itself a strict monoidal functor (when $S$ has its cartesian monoidal structure); this is called a monoidal fibration. Moreover, we can recover ${\otimes }_A$ from $⊠$ via $M{\otimes }_{A}N={\Delta }_A^{*}(M⊠N)$, so the two structures have the same information.

Examples

• $S=\mathrm{Sets}$, $C^A=$ $A$-indexed families of objects of $C$, for any monoidal category $C$.
• $S=\mathrm{Gpd}$, $C^A=$ $A$-diagrams of objects of $C$
• $S=$ any category with pullbacks, $C^A=S/A$
• $S=\mathrm{Top}$, $C^A=$ spectra parametrized over $A$
• $S=\mathrm{Grp}$ or $\mathrm{TopGrp}$, $C^A=$ 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 $f^{*}:C^B\to C^A$ induced by a morphism $f:A\to B$ 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 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).

References

• Mike Shulman, “Framed bicategories and monoidal fibrations”. Theory and Applications of Categories Vol. 20, 2008, No. 18, pp 650-738. Free online

• Kate Ponto and Mike Shulman, Duality and traces in indexed monoidal categories, (web and blog)