nLab Godement product

category theory

Applications

Higher category theory

higher category theory

Contents

Idea

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:

Definition

For categories $A,B,C$, if $\alpha\colon F_1\to G_1\colon A\to B$ and $\beta\colon F_2\to G_2\colon B\to C$ are natural transformations of functors, the components $(\beta \circ \alpha)_M$ of the Godement product $\beta \circ \alpha\colon F_2\circ F_1\to G_2\circ G_1\colon A\to C$ (or $\alpha \ast \beta\colon F_1 ; F_2 \to G_1 ; G_2\colon A\to C$) are defined by any of the two equivalent formulas:

$(\beta\circ\alpha)_M = \beta_{G_1(M)}\circ F_2(\alpha_M)$
$(\beta\circ\alpha)_M = G_2(\alpha_M)\circ \beta_{F_1(M)}$

that can be rewritten using the morphismwise notation into:

$(\beta\circ\alpha)_M = \beta(\alpha_M)$

that is:

$\array{ F_2(F_1(M)) & \stackrel{F_2(\alpha_M)}{\to} & F_2(G_1(M)) \\ \beta_{F_1(M)}\downarrow & \searrow^{(\beta\circ\alpha)_M} & \downarrow \beta_{G_1(M)} \\ G_2(F_1(M)) & \stackrel{G_2(\alpha_M)}{\to} & G_2(G_1(M)) } \,.$

The interchange law in (general) $2$-categories (which in the case of $Cat$ 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 $2$ natural transformations, but we can generalise from $2$ to any natural number. The Godement product of $0$ natural transformations is the identity natural transformation on an identity functor.

Properties

The Godement product is strictly associative (so that Cat is a strict 2-category).

Name after Roger Godement.