Higher category theory
higher category theory
Extra properties and structure
The Godement product of two natural transformations between appropriate functors is their horizontal composition as 2-cells in the 2-category Cat of categories, functors and natural transformations:
For categories , if and are natural transformations of functors, the components of the Godement product (or ) are defined by any of the two equivalent formulas:
The interchange law in (general) -categories (which in the case of boils down to assertion that the two formulas above are equivalent) is also sometimes called Godement interchange law.
The definition above is for the Godement product of natural transformations, but we can generalise from to any natural number. The Godement product of natural transformations is the identity natural transformation on an identity functor.
The Godement product is strictly associative (so that Cat is a strict 2-category).