# nLab bilax monoidal functor

### Context

#### Monoidal categories

monoidal categories

## With traces

• trace

• traced monoidal category?

# Contents

## Definition

A bilax monoidal functor is a functor $F:C\to D$ between categories equipped with the structure of braided monoidal categories that is both a lax monoidal functor as well as an oplax monoidal functor with natural transformations

$F\left(x\right)\otimes F\left(y\right)\stackrel{\stackrel{{\Delta }_{x,y}}{←}}{\underset{{\nabla }_{x,y}}{\to }}F\left(x\otimes y\right)$F(x) \otimes F(y) \stackrel{\overset{\Delta_{x,y}}{\leftarrow}}{\underset{\nabla_{x,y}}{\to}} F(x \otimes y)

satisfying two compatibility conditions:

• braiding For all $a,b,c,d\in C$ the following diagram commutes

$\begin{array}{ccc}& & F\left(a\otimes b\right)\otimes F\left(c\otimes d\right)\\ & ↙& & ↘\\ F\left(a\otimes b\otimes c\otimes d\right)& & & & F\left(a\right)\otimes F\left(b\right)\otimes F\left(c\right)\otimes F\left(d\right)\\ ↓& & & & ↓\\ F\left(a\otimes c\otimes b\otimes d\right)& & & & F\left(a\right)\otimes F\left(c\right)\otimes F\left(b\right)\otimes F\left(d\right)\\ & ↘& & ↙\\ & & F\left(a\otimes c\right)\otimes F\left(b\otimes d\right)\end{array}$\array{ && F(a \otimes b) \otimes F(c \otimes d) \\ & \swarrow && \searrow \\ F(a \otimes b \otimes c \otimes d) &&&& F(a) \otimes F(b) \otimes F(c) \otimes F(d) \\ \downarrow &&&& \downarrow \\ F(a \otimes c \otimes b \otimes d) &&&& F(a) \otimes F(c) \otimes F(b) \otimes F(d) \\ & \searrow && \swarrow \\ && F(a \otimes c) \otimes F(b \otimes d) }
• unitality (…)

## References

Definition 3.3 in

Revised on November 3, 2010 16:20:36 by Urs Schreiber (131.211.232.76)