## 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.

• When $\mathsf{C}=\mathsf{Set}$, the object $X$ is called a $G$-set.

• When $\mathsf{C}=\mathsf{Vect}_\mathbb{k}$, the object $X$ is called a $G$-representation. (This definition forms the basis for representation theory, which studies representations of groups as sets of matrices.)

• When $\mathsf{C}=\mathsf{Top}$, the object $X$ is called a $G$-space.