Category of coverings

Given a topological space $𝑋$ , the category of coverings $𝖢 𝗈 𝗏 𝑋$ over $𝑋$ is a category where #m/def/homotopy

Each object $𝑝 \in 𝖢 𝗈 𝗏 𝑋$ is a covering $𝑝 : ˜ 𝑋 ↠ 𝑋$ where $˜ 𝑋$ is some covering space
Each morphism $𝑓 \in 𝖢 𝗈 𝗏 𝑋 (𝑝, 𝑞)$ is a map such that the following diagram commutes in $𝖳 𝗈 𝗉$ :

Such an $𝑓$ is sometimes referred to as a covering morphism. Two coverings $𝑝, 𝑞$ of $𝑋$ are called equivalent iff they are isomorphic in $𝖢 𝗈 𝗏 𝑋$

Category of coverings with basepoint

The category of coverings with basepoint $𝖢 𝗈 𝗏 (𝑋, 𝑥 0)$ is defined similarly

Since any $𝑓 \in 𝖢 𝗈 𝗏 (𝑋, 𝑥 0) (𝑝, 𝑞)$ is a lift of $𝑝$ over $𝑞$ there exists at most one.

Moreover for connected and locally path-connected coverings, there exists exactly one $𝑓 \in 𝖢 𝗈 𝗏 (𝑋, 𝑥 0) (𝑝, 𝑞)$ iff $i m (𝜋 1 𝑞) \subseteq i m (𝜋 1 𝑝)$ . Thus $𝖢 𝗈 𝗏 (𝑋, 𝑥 0)$ is a thin category or preorder.

Further terminology

An automorphism in $𝖢 𝗈 𝗏 𝑋$ is called a Deck transformation

Properties

Every covering morphism is a lift of a covering
Locally path-connected, connected covering morphism is a covering
Equivalence of coverings criterion

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