nLab
Frobenius monoidal functor

Contents

Definition

A functor F:CDF : 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

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

and for the oplax monoidal structure

Δ x,y:F(xy)F(x)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,zx,y,z in CC we have

(F(x) F(yz) F(x) F(y) F(z) F(xy) F(z))=(F(x) F(yz) F(xyz) F(xy) F(y)) \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

(F(xy) F(z) F(x) F(y) F(z) F(x) F(yz))=(F(xy) F(z) F(xyz) F(x) F(yz)) \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:sAbCh + C : sAb \to Ch_\bullet^+

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

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

For more on this see monoidal Dold-Kan correspondence.

References

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

  • M. B. McCurdy, R. Street, What separable Frobenius monoidal functors preserve, arxiv/0904.3449 and Cahiers TGDC, 51 (2010)p. 29 - 50.
Revised on April 13, 2011 18:21:28 by Tim Porter (95.147.237.233)