nLab
Frobenius monoidal functor

Definition

A functor $F:C\to D$ between monoidal categories which is both a lax monoidal functor and an oplax monoidal functor is called Frobenius if the structure morphisms for the lax monoidal structure

$${\nabla }_{x,y}:F(x)\otimes F(y)\to F(x\otimes y)$$\nabla_{x,y} : F(x)\otimes F(y) \to F(x \otimes y)

and for the oplax monoidal structure

$${\Delta }_{x,y}:F(x\otimes y)\to F(x)\otimes F(y)$$\Delta_{x,y} : F(x \otimes y) \to F(x) \otimes F(y)

satisfy the axioms of a Frobenius algebra in that (in string diagram notation) for all objects $x,y,z$ in $C$ we have

$$\left(\begin{array}{ccccc}F(x)& & & & F(y\otimes z)\\ \downarrow & & & \swarrow & \downarrow \\ F(x)& & F(y)& & F(z)\\ \downarrow & \swarrow & & & \downarrow \\ F(x\otimes y)& & & & F(z)\end{array}\right)=\left(\begin{array}{ccccc}F(x)& & & & F(y\otimes z)\\ & \searrow & & \swarrow \\ & & F(x\otimes y\otimes z)\\ & \swarrow & & \searrow \\ F(x\otimes y)& & & & F(y)\end{array}\right)$$\left( \array{ F(x) &&&& F(y \otimes z) \\ \downarrow &&& \swarrow & \downarrow \\ F(x) && F(y)& & F(z) \\ \downarrow & \swarrow &&& \downarrow \\ F(x \otimes y) &&&& F(z) } \right) = \left( \array{ F(x) &&&& F(y \otimes z) \\ & \searrow && \swarrow \\ && F(x \otimes y \otimes z) \\ & \swarrow && \searrow \\ F(x \otimes y) &&&& F(y) } \right)

and

$$\left(\begin{array}{ccccc}F(x\otimes y)& & & & F(z)\\ \downarrow & \searrow & & & \downarrow \\ F(x)& & F(y)& & F(z)\\ \downarrow & & & \searrow & \downarrow \\ F(x)& & & & F(y\otimes z)\end{array}\right)=\left(\begin{array}{ccccc}F(x\otimes y)& & & & F(z)\\ & \searrow & & \swarrow \\ & & F(x\otimes y\otimes z)\\ & \swarrow & & \searrow \\ F(x)& & & & F(y\otimes z)\end{array}\right)$$\left( \array{ F(x \otimes y) &&&& F(z) \\ \downarrow & \searrow && & \downarrow \\ F(x) && F(y) & & F(z) \\ \downarrow & && \searrow & \downarrow \\ F(x ) &&&& F(y \otimes z) } \right) = \left( \array{ F(x \otimes y) &&&& F(z) \\ & \searrow && \swarrow \\ && F(x \otimes y \otimes z) \\ & \swarrow && \searrow \\ F(x) &&&& F(y \otimes z) } \right)

Examples

The Moore complex functor

$$C:\mathrm{sAb}\to {\mathrm{Ch}}_{•}^{+}$$C : sAb \to Ch_\bullet^+

from abeliam simplicial groups to connective chain complexes is Frobenius, as is the normalzed chains complex functor

$$N:\mathrm{sAb}\to {\mathrm{Ch}}_{•}^{+}.$$N : sAb \to Ch_\bullet^+ \,.

For more on this see monoidal Dold-Kan correspondence.

Related concepts

• monoidal functor

• strong monoidal functor

• lax monoidal functor

• oplax monoidal functor

• bilax monoidal functor

• Frobenius monoidal functor

References

Equation (3.26), (3.27) in p. 81 of

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

• M. B. McCurdy, R. Street, What separable Frobenius monoidal functors preserve, arxiv/0904.3449 and Cahiers TGDC, 51 (2010)p. 29 - 50.