nLab
bilax monoidal functor

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Definition

A bilax monoidal functor is a functor F:CDF : 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(x)F(y) x,yΔ x,yF(xy) 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,dCa,b,c,d \in C the following diagram commutes

    F(ab)F(cd) F(abcd) F(a)F(b)F(c)F(d) F(acbd) F(a)F(c)F(b)F(d) F(ac)F(bd) \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)