# nLab monoidal fibration

## Definition

A monoidal fibration is a functor $\Phi\colon E\to B$ 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 $B^{op} \to MonCat$, which are called indexed monoidal categories.

## References

Revised on September 6, 2012 18:05:03 by Mike Shulman (71.136.235.154)