nLab
symmetric monoidal functor

Contents

Idea

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

Defnition

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

FAFB σ FBFA A,B B,A F(AB) F(σ) F(BA) \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 σ\sigma denotes the symmetry isomorphism both of CC and DD.

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)