Category of G-spaces

The category of (left) $𝐺$ -spaces \leftindex{G}\Set 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. group A (left-)equivariant map $𝑓 \in 𝐺 {(} 𝑋, 𝑌)$ 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.

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Category of G-spaces

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