Franklin's Notes

Group action

Suppose $G$ is a group, with $\mathsf{B}G$ being regarded as a one-object category whose morphisms correspond the elements of $G$ such that the rules for composition align with the rules of multiplication in $G$. Then a functor $F:\mathsf{B}G\to\mathsf{C}$ defines an action of the group $G$ on an object $X\in\mathsf{C}$. In particular, the images of the morphisms of $\mathsf{B}G$ are automorphisms of $X$, endowed with a group structure under composition.




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