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

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

and for the oplax monoidal structure

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

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

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

and

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

Examples

The Moore complex functor

$$C : sAb \to Ch_\bullet^+$$

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

$$N : sAb \to Ch_\bullet^+ \,.$$

For more on this see monoidal Dold-Kan correspondence.

Generalizations

Frobenius monoidal functors are a special case of Frobenius linear functors between linearly distributive categories.

Related concepts

• monoidal functor

• strong monoidal functor

• lax monoidal functor

• oplax monoidal functor

• bilax monoidal functor

• Frobenius monoidal functor

The concept of Frobenius monoidal functors is related to that of

• Frobenius functors,

albeit in a nontrivial way. See Theorem 3.18 of (Flake et al 2024) for more on this.

References

• Marcelo Aguiar, Swapneel Mahajan, (3.26), (3.27) on p. 81 of: Monoidal Functors, Species and Hopf Algebras, CRM Monograph Series 29, Amer. Math. Soc. (2010) [ISBN:978-0-8218-4776-3, pdf]

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

Relation to Drinfeld centers:

• Johannes Flake, Robert Laugwitz, Sebastian Posur: Frobenius monoidal functors from ambiadjunctions and their lifts to Drinfeld centers [arXiv:2410.08702]