# nLab Frobenius monoidal functor

Contents

### Context

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# 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

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

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.

Beware of the un-related terminoilogy:

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.