# nLab symmetric monoidal functor

### Context

#### Monoidal categories

monoidal categories

## With traces

• trace

• traced monoidal category?

# Contents

## Idea

A symmetric monoidal functor is a functor $F:C\to D$ between symmetric monoidal categories that is a monoidal functor which respects the symmetry on both sides.

## Defnition

A monoidal functor $F:\left(C,\otimes \right)\to \left(D,\otimes \right)$ between symmetric monoidal categories is symmetric of for all $A,B\in C$ the diagram

$\begin{array}{ccc}FA\otimes FB& \stackrel{\sigma }{\to }& FB\otimes FA\\ {}^{{\nabla }_{A,B}}↓& & {↓}^{{\nabla }_{B,A}}\\ F\left(A\otimes B\right)& \stackrel{F\left(\sigma \right)}{\to }& F\left(B\otimes A\right)\end{array}$\array{ F A \otimes F B &\stackrel{\sigma}{\to}& F B \otimes F A \\ {}^{\mathllap{\nabla_{A,B}}}\downarrow && \downarrow^{\mathrlap{\nabla_{B,A}}} \\ F(A\otimes B) &\stackrel{F(\sigma)}{\to}& F(B \otimes A) }

commutes, where $\sigma$ denotes the symmetry isomorphism both of $C$ and $D$.

## Properties

As long as it goes between symmetric monoidal categories a symmetric monoidal functor is the same as a braided monoidal functor.

## References

An exposition is in

Revised on November 3, 2010 16:55:23 by Urs Schreiber (131.211.232.76)