Continuous path

Category of paths

Let 𝑋 be a topological space, and 𝒫𝑋 be its path algebra. The following congruence relation on paths makes 𝒫𝑋 into a Small category 𝑃𝑋. Let 𝛼,𝛽 ∈𝒫𝑋(𝑥,𝑦). Then 𝛼 ∼𝛽 iff 𝛼 =𝛽 ∘𝜙 for some 𝜙 ∈𝖳𝗈𝗉(𝕀,𝕀). The category 𝑃𝑋 =𝒫𝑋/ ∼ is called the category of paths on 𝑋.

Properties

• By the Path traversal lemma we can factorise 𝜋1 through 𝑃 to get the Fundamental groupoid.

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