nLab
Godement product

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:

$$A\underset{F_1}{\overset{G_1}{\begin{array}{c} \end{array}}}\Downarrow \alpha B\underset{F_2}{\overset{G_2}{\begin{array}{c} Layer 1 \end{array}}}\Downarrow \beta C\mapsto A\underset{F_1:F_2}{\overset{G_1:G_2}{\begin{array}{c} Layer 1 \end{array}}}\Downarrow \alpha *\beta C$$A\mathrlap{\underoverset{\textsize{F_1}}{\textsize{G_1}}{\begin{matrix}\begin{svg} <svg width="76" height="39" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <use xlink:href="#curvearrows3466"/> </svg> \end{svg}\includegraphics[width=53]{curvearrows3466}\end{matrix}}} \qquad\Downarrow\mathrlap{\alpha}\qquad B \mathrlap{\underoverset{\textsize{F_2}}{\textsize{G_2}}{\begin{matrix}\begin{svg} <svg width="76" id="curvearrows3466" height="39" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="3466"> <g> <title>Layer 1</title> <path marker-end="url(#se_marker_end_svg_3466_2)" id="svg_3466_2" d="m1,15.75c23.958326,-15 51.865845,-15 71.875,0" stroke="#000000" fill="none"/> <path marker-end="url(#se_marker_end_svg_3466_2)" id="svg_3466_3" d="m1,26c23.874994,14.33334 44.37941,15.66666 71.625,1" stroke="#000000" fill="none"/> </g> <defs> <marker refY="50" refX="50" markerHeight="5" markerWidth="5" viewBox="0 0 100 100" orient="auto" markerUnits="strokeWidth" id="se_marker_end_svg_3466_2"> <path stroke-width="10" stroke="#000000" fill="#000000" d="m100,50l-100,40l30,-40l-30,-40l100,40z" id="svg_3466_1"/> </marker> </defs> </svg> \end{svg}\includegraphics[width=53]{curvearrows3466}\end{matrix}}} \qquad\Downarrow\mathrlap{\beta}\qquad C \mapsto A\mathrlap{\underoverset{\textsize{F_1\colon F_2}}{\textsize{G_1\colon G_2}}{\begin{matrix}\begin{svg} <svg width="86" height="39" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="3467"> <g> <title>Layer 1</title> <path fill="none" stroke="#000000" d="m1,15.75c27.249996,-15 58.991756,-15 81.75,0" id="svg_3467_2" marker-end="url(#se_marker_end_svg_3467_2)"/> <path fill="none" stroke="#000000" d="m1,26c26.999989,14.33334 50.188232,15.66666 81,1" id="svg_3467_3" marker-end="url(#se_marker_end_svg_3467_2)"/> </g> <defs> <marker id="se_marker_end_svg_3467_2" markerUnits="strokeWidth" orient="auto" viewBox="0 0 100 100" markerWidth="5" markerHeight="5" refX="50" refY="50"> <path id="svg_3467_1" d="m100,50l-100,40l30,-40l-30,-40l100,40z" fill="#000000" stroke="#000000" stroke-width="10"/> </marker> </defs> </svg> \end{svg}\includegraphics[width=65]{curvearrows3467}\end{matrix}}} \qquad\Downarrow\mathrlap{\alpha\ast\beta}\space{0}{0}{25} C

Definition

For categories $A,B,C$, if $\alpha :F_1\to G_1:A\to B$ and $\beta :F_2\to G_2:B\to C$ are natural transformations of functors, the components $(\alpha *\beta {)}_M$ of the Godement product $\alpha *\beta :F_1;F_2\to G_1;G_2$ (or $\beta \circ \alpha :F_2\circ F_1\to G_2\circ G_1$) 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 = \beta_{G_1(M)}\circ F_2(\alpha_M)

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

that is:

$$\begin{array}{ccc}{F}_2(F_1(M))& \stackrel{F_2({\alpha }_M)}{\to }& F_2(G_1(M))\\ {\beta }_{F_1(M)}\downarrow & {\searrow }^{(\beta \circ \alpha {)}_M}& \downarrow {\beta }_{G_1(M)}\\ G_2(F_1(M))& \stackrel{G_2({\alpha }_M)}{\to }& G_2(G_1(M))\end{array}.$$\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 $\mathrm{Cat}$ 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).