nLab monoidal structure map

Context

Monoidal categories

monoidal categories

With symmetry

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

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).

Last revised on September 7, 2011 at 16:34:06. See the history of this page for a list of all contributions to it.