# 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:

$A\underset{{F}_{1}}{\overset{{G}_{1}}{\begin{array}{c} \end{array}}}\phantom{\rule{2em}{0ex}}⇓\alpha \phantom{\rule{2em}{0ex}}B\underset{{F}_{2}}{\overset{{G}_{2}}{\begin{array}{c} Layer 1 \end{array}}}\phantom{\rule{2em}{0ex}}⇓\beta \phantom{\rule{2em}{0ex}}C↦A\underset{{F}_{1}:{F}_{2}}{\overset{{G}_{1}:{G}_{2}}{\begin{array}{c} Layer 1 \end{array}}}\phantom{\rule{2em}{0ex}}⇓\alpha *\beta \phantom{\rule[-.0ex]{2.5em}{.0ex}}C$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 $\left(\alpha *\beta {\right)}_{M}$ 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:

$\left(\beta \circ \alpha {\right)}_{M}={\beta }_{{G}_{1}\left(M\right)}\circ {F}_{2}\left({\alpha }_{M}\right)$(\beta\circ\alpha)_M = \beta_{G_1(M)}\circ F_2(\alpha_M)
$\left(\beta \circ \alpha {\right)}_{M}={G}_{2}\left({\alpha }_{M}\right)\circ {\beta }_{{F}_{1}\left(M\right)}$(\beta\circ\alpha)_M = G_2(\alpha_M)\circ \beta_{F_1(M)}

that is:

$\begin{array}{ccc}{F}_{2}\left({F}_{1}\left(M\right)\right)& \stackrel{{F}_{2}\left({\alpha }_{M}\right)}{\to }& {F}_{2}\left({G}_{1}\left(M\right)\right)\\ {\beta }_{{F}_{1}\left(M\right)}↓& {↘}^{\left(\beta \circ \alpha {\right)}_{M}}& ↓{\beta }_{{G}_{1}\left(M\right)}\\ {G}_{2}\left({F}_{1}\left(M\right)\right)& \stackrel{{G}_{2}\left({\alpha }_{M}\right)}{\to }& {G}_{2}\left({G}_{1}\left(M\right)\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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).

Revised on October 8, 2010 04:16:42 by Anonymous Coward (72.179.54.121)