Category of G-spaces

The category of (left) $𝐺$ -spaces $𝐺 𝖲 𝖾 𝗍$ for a given group $𝐺$ consists of sets (left-)acted on by $𝐺$ (called a (left-) $𝐺$ -space) with $𝐺$ -(left-)equivariant maps as morphisms. #m/def/group A (left-)equivariant map $𝑓 \in 𝐺 𝖲 𝖾 𝗍 (𝑋, 𝑌)$ is a function $𝑓 : 𝑋 \to 𝑌$ satisfying

𝑓 (𝑔 \cdot 𝑥) = 𝑔 \cdot 𝑓 (𝑥)

for any $𝑥 \in 𝑋$ and $𝑔 \in 𝐺$ . An isomorphism of $𝐺$ -spaces is sometimes called an equivalence of actions.

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