The fundamental category of a stratified space

Understand

The fundamental groupoid of a locally 0 and 1-connected space classifies covering spaces, or equivalently local systems.

• When the space is topologically stratified Treumann, based on unpublished ideas of MacPherson, constructed an `exit category' (in the terminology of this paper, the `fundamental category') which classifies constructible sheaves, equivalently stratified etale covers.
• This paper generalises this construction to homotopically stratified sets, in addition showing that the fundamental category dually classifies constructible cosheaves, equivalently stratified branched covers.
• The more general setting has several advantages.