nLab
monoidal structure map

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

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

that equips FF with the structure of a lax monoidal functor.

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

Δ x,y:F(xy)F(x)F(y) \Delta_{x,y}\colon F(x \otimes y) \to F(x) \otimes F(y)

that equips FF 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)