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fixed typo: "principle bundle" should be "principal bundle".
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John Baez
John Baez
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A finite group $G$ can be considered as a category with one object. Taking its nerve $NG$, and then geometrically realizing we get $BG$ the classifying space of $G$, which classifies principleprincipal $G$ bundles.

Instead starting with any category $C$, what does $NC$ classify? (Either before or after taking realization.) Does it classify something reasonable?

A finite group $G$ can be considered as a category with one object. Taking its nerve $NG$, and then geometrically realizing we get $BG$ the classifying space of $G$, which classifies principle $G$ bundles.

Instead starting with any category $C$, what does $NC$ classify? (Either before or after taking realization.) Does it classify something reasonable?

A finite group $G$ can be considered as a category with one object. Taking its nerve $NG$, and then geometrically realizing we get $BG$ the classifying space of $G$, which classifies principal $G$ bundles.

Instead starting with any category $C$, what does $NC$ classify? (Either before or after taking realization.) Does it classify something reasonable?

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Don Stanley
Don Stanley
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What does the classifying space of a category classify?

A finite group $G$ can be considered as a category with one object. Taking its nerve $NG$, and then geometrically realizing we get $BG$ the classifying space of $G$, which classifies principle $G$ bundles.

Instead starting with any category $C$, what does $NC$ classify? (Either before or after taking realization.) Does it classify something reasonable?