Classification of Constructible Cosheaves

Informally speaking, a map is stratified if it is proper and there is a partition S of X into manifolds so that over each manifold S ∈ S the map f(S)→ S is a fiber bundle. One perspective on this question uses the open sets of X to define a functor F̂ : Open(X)→ Set that assigns to each open U the set π0(f (U)) of path components. This functor satisfies a gluing axiom reminiscent of the usual van Kampen theorem, which makes it the prototypical example of a cosheaf of sets. The assumption that f is stratified endows F̂ with a strong property called constructibility. Robert MacPherson observed that paths in X can be used to organize π0(f (x)). A path from x to x ′ lying entirely in a single stratum S ∈ S induces an isomorphism π0(f(x)) → π0(f (x ′)). Suppose x ′ lies in a stratum on the frontier of S. MacPherson noticed that a path from x to x ′ that leaves one stratum only by entering into a lower-dimensional stratum defines a map π0(f (x))→ π0(f(x ′)). Such paths are called entrance paths and this map between components is invariant under certain homotopic perturbations. One then thinks of this data as a functor F : Ent(X, S)→ Set from the entrance path category Ent(X, S) whose objects are points of X and whose morphisms are certain homotopy classes of entrance paths. One of the results of this paper is a proof that these two perspectives are equivalent. This equivalence is a consequence of a more general result, which we call the Classification Theorem for Constructible Cosheaves (Theorem 2). The purpose of this paper is to provide a concise and self-contained proof of this theorem. This requires two novel definitions: