nLab
bilax monoidal functor

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

$$F(x)\otimes F(y)\stackrel{\stackrel{{\Delta }_{x,y}}{\leftarrow }}{\underset{{\nabla }_{x,y}}{\to }}F(x\otimes y)$$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

  $$\begin{array}{ccc}& & F(a\otimes b)\otimes F(c\otimes d)\\ & \swarrow & & \searrow \\ F(a\otimes b\otimes c\otimes d)& & & & F(a)\otimes F(b)\otimes F(c)\otimes F(d)\\ \downarrow & & & & \downarrow \\ F(a\otimes c\otimes b\otimes d)& & & & F(a)\otimes F(c)\otimes F(b)\otimes F(d)\\ & \searrow & & \swarrow \\ & & F(a\otimes c)\otimes F(b\otimes d)\end{array}$$\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 (…)

Related concepts

• monoidal functor

• strong monoidal functor

• lax monoidal functor

• oplax monoidal functor

• bilax monoidal functor

• Frobenius monoidal functor

References

Definition 3.3 in

• Marcelo Aguiar, Swapneel Mahajan, Monoidal Functors, Species and Hopf Algebras