# nLab monoidal structure map

### Context

#### Monoidal categories

monoidal categories

## With traces

• trace

• traced monoidal category?

## In higher category theory

For $F:C\to D$ a functor between categories that are equipped with the structure of monoidal categories $\left(C,\otimes \right)$, $\left(D,\otimes \right)$, a lax monoidal structure map is a natural transformation

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

that equips $F$ with the structure of a lax monoidal functor.

Similarly, an oplax monoidal structure map, or lax comonoidal structure map is a natural transformation

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

that equips $F$ with the structure of an oplax monoidal functor.

An (op)lax (co)monoidal structure map is sometimes called an (op)lax (co)monoidal transformation; however, this is not a laxification (a directed weakening) of any strong notion of monoidal natural transformation (which has nothing to laxify).

Revised on September 7, 2011 16:34:06 by Toby Bartels (75.88.82.16)