Covering

Category of coverings

Given a topological space , the category of coverings over is a category where homotopy

• Each object is a covering where is some covering space
• Each morphism is a map such that the following diagram commutes in :

Such an is sometimes referred to as a covering morphism. Two coverings of are called equivalent iff they are isomorphic in

Category of coverings with basepoint

The category of coverings with basepoint is defined similarly

Since any is a lift of over there exists at most one.

Moreover for connected and locally path-connected coverings, there exists exactly one iff . Thus is a thin category or preorder.

Further terminology

• An automorphism in 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

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