[[Group action]]
# Category of G-spaces

The **category of (left) $G$-spaces** $\leftindex{G}\Set$ for a given [[group]] $G$ 
consists of sets (left-)[[group action|acted]] on by $G$ (called a **(left-)$G$-space**)
with **$G$-(left-)equivariant maps** as morphisms. #m/def/group 
A (left-)equivariant map $f \in {}_{G}\Set(X,Y)$ is a function $f : X \to Y$ satisfying
$$
\begin{align*}
f(g \cdot x) = g \cdot f(x)
\end{align*}
$$
for any $x \in X$ and $g \in G$. 
An isomorphism of $G$-spaces is sometimes called an **equivalence of actions**.

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#state/tidy | #lang/en | #SemBr