nLab
Godement product

Context

Category theory

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

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 G 1F 1aαaB Layer 1 G 2F 2aβaCA Layer 1 G 1:G 2F 1:F 2aα*βaC A\mathrlap{\underoverset{\textsize{G_1}}{\textsize{F_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}}} {\phantom{a}\space{0}{0}{12}\Downarrow\mathrlap{\alpha}\space{0}{0}{12}\phantom{a}} B \mathrlap{\underoverset{\textsize{G_2}}{\textsize{F_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="m1,16c24,-15 52,-15 72,0" stroke="#000000" fill="none"/> <path marker-end="url(#se_marker_end_svg_3466_2)" id="svg_3466_3" d="m1,26c24,15 44,15 72,0" 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}}} {\phantom{a}\space{0}{0}{12}\Downarrow\mathrlap{\beta}\space{0}{0}{12}\phantom{a}} C \mapsto A \mathrlap{\underoverset{\textsize{G_1\colon G_2}}{\textsize{F_1\colon F_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="m1,16c27,-15 59,-15 82,0" id="svg_3467_2" marker-end="url(#se_marker_end_svg_3467_2)"/> <path fill="none" stroke="#000000" d="m1,26c27,15 50,15 82,1" 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}}} {\phantom{a}\quad\Downarrow\mathrlap{\alpha\ast\beta}\space{0}{0}{20}\phantom{a}} C

Definition

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.

Properties

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

Revised on May 26, 2016 00:36:49 by Anonymous Coward (72.182.43.120)