nLab
symmetric monoidal functor

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Context

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

  • trace

  • traced monoidal category?

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Examples

Theorems

In higher category theory

Contents

Idea

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

Defnition

A monoidal functor F:(C,⊗)→(D,⊗) between symmetric monoidal categories is symmetric of for all A,B∈C the diagram

FA⊗FB →σ FB⊗FA ∇ A,B↓ ↓ ∇ B,A F(A⊗B) →F(σ) F(B⊗A)\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 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)
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