nLab
monoidal fibration

Definition

A monoidal fibration is a functor $\Phi :E\to B$ such that

• $\Phi$ is a Grothendieck fibration
• $E$ and $B$ are monoidal categories and $\Phi$ is a strict monoidal functor, and
• the tensor product of $E$ preserves cartesian arrows.

If $B$ is cartesian monoidal, then monoidal fibrations over $B$ are equivalent to pseudofunctors $B^{\mathrm{op}}\to \mathrm{MonCat}$, which are called indexed monoidal categories.

References

• Mike Shulman, “Framed bicategories and monoidal fibrations”, TAC