nLab
Godement product
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Category theory
category theory
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Universal constructions
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Extensions
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Higher category theory
higher category theory
Basic concepts
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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:
Definition
For categories A , B , C A,B,C , if α : F 1 → G 1 : A → B \alpha\colon F_1\to G_1\colon A\to B and β : F 2 → G 2 : B → C \beta\colon F_2\to G_2\colon B\to C are natural transformation s of functor s, the components ( β ∘ α ) M (\beta \circ \alpha)_M of the Godement product β ∘ α : F 2 ∘ F 1 → G 2 ∘ G 1 : A → C \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 2 → G 1 ; G 2 : A → C \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:
( β ∘ α ) M = β G 1 ( M ) ∘ F 2 ( α M )
(\beta\circ\alpha)_M = \beta_{G_1(M)}\circ F_2(\alpha_M)
( β ∘ α ) 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:
( β ∘ α ) 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 ) ↓ ↘ ( β ∘ α ) 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) 2 2 -categories (which in the case of Cat 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 2 natural transformations, but we can generalise from 2 2 to any natural number . The Godement product of 0 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 ).
References
Name after Roger Godement .
Last revised on October 28, 2021 at 17:24:47.
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