Mathematical Conversations

A covering space $C \\to M$ is classified by a subgroup of the fundamental group of $M$. If we refuse to choose a basepoint, then $C \\to M$ is equivalent to a functor from the fundamental groupoid of $M$ to $\\mathsf{Set}$. Suppose $(M,S)$ is a stratified space and $(C, T) \\to (M,S)$ a stratified covering. Then $(C, T) \\to (M,S)$ is equivalent to a functor from the entrance path category $\\mathsf{Ent}(M,S)$ of $(M,S)$ to $\\mathsf{Set}$. In general, any $S$-constructible cosheaf over $M$ is equivalent to a functor from $\\mathsf{Ent}(M,S)$. I will introduce the entrance path category and argue this equivalence.