nLab Godement product



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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:


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 (βα) M(\beta \circ \alpha)_M of the Godement product βα:F 2F 1G 2G 1:AC\beta \circ \alpha\colon F_2\circ F_1\to G_2\circ G_1\colon A\to C (or α*β:F 1;F 2G 1;G 2:AC\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:

(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 can be rewritten using the morphismwise notation into:

(βα) M=β(α M) (\beta\circ\alpha)_M = \beta(\alpha_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).


Name after Roger Godement.

Last revised on October 28, 2021 at 17:24:47. See the history of this page for a list of all contributions to it.