# nLab Frobenius monoidal functor

### Context

#### Monoidal categories

monoidal categories

## With traces

• trace

• traced monoidal category?

# Contents

## 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

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

and for the oplax monoidal structure

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

$\left(\begin{array}{ccccc}F\left(x\right)& & & & F\left(y\otimes z\right)\\ ↓& & & ↙& ↓\\ F\left(x\right)& & F\left(y\right)& & F\left(z\right)\\ ↓& ↙& & & ↓\\ F\left(x\otimes y\right)& & & & F\left(z\right)\end{array}\right)=\left(\begin{array}{ccccc}F\left(x\right)& & & & F\left(y\otimes z\right)\\ & ↘& & ↙\\ & & F\left(x\otimes y\otimes z\right)\\ & ↙& & ↘\\ F\left(x\otimes y\right)& & & & F\left(y\right)\end{array}\right)$\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

$\left(\begin{array}{ccccc}F\left(x\otimes y\right)& & & & F\left(z\right)\\ ↓& ↘& & & ↓\\ F\left(x\right)& & F\left(y\right)& & F\left(z\right)\\ ↓& & & ↘& ↓\\ F\left(x\right)& & & & F\left(y\otimes z\right)\end{array}\right)=\left(\begin{array}{ccccc}F\left(x\otimes y\right)& & & & F\left(z\right)\\ & ↘& & ↙\\ & & F\left(x\otimes y\otimes z\right)\\ & ↙& & ↘\\ F\left(x\right)& & & & F\left(y\otimes z\right)\end{array}\right)$\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

$N:\mathrm{sAb}\to {\mathrm{Ch}}_{•}^{+}\phantom{\rule{thinmathspace}{0ex}}.$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)