Godement product


Category theory

Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations



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 F 1G 1αB Layer 1 F 2G 2βCA Layer 1 F 1:F 2G 1:G 2α*βC A\mathrlap{\underoverset{\textsize{F_1}}{\textsize{G_1}}{\begin{matrix}\begin{svg} <svg width="76" height="39" xmlns="" xmlns: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="" xmlns:se="" 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="" xmlns:se="" 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


For categories A,B,CA,B,C, if α:F 1G 1:AB\alpha\colon F_1\to G_1\colon A\to B and β:F 2G 2:BC\beta\colon F_2\to G_2\colon B\to C are natural transformations of functors, the components (α*β) M(\alpha * \beta)_M of the Godement product α*β:F 1;F 2G 1;G 2\alpha * \beta\colon F_1 ; F_2 \to G_1 ; G_2 (or βα:F 2F 1G 2G 1\beta \circ \alpha\colon F_2\circ F_1\to G_2\circ G_1) are defined by any of the two equivalent formulas:

(βα) M=β G 1(M)F 2(α M) (\beta\circ\alpha)_M = \beta_{G_1(M)}\circ F_2(\alpha_M)
(βα) M=G 2(α M)β F 1(M) (\beta\circ\alpha)_M = G_2(\alpha_M)\circ \beta_{F_1(M)}

that is:

F 2(F 1(M)) F 2(α M) F 2(G 1(M)) β F 1(M) (βα) M β G 1(M) G 2(F 1(M)) G 2(α M) G 2(G 1(M)). \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) 22-categories (which in the case of CatCat 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 22 natural transformations, but we can generalise from 22 to any natural number. The Godement product of 00 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).

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