Explicit exit-path category from fan combinatorics

Determine an explicit, combinatorial presentation of the exit-path category associated to the FLTZ stratification induced by a fan Σ, describing its objects and morphisms directly in terms of cones, faces, and their incidences, so that exodromy yields a functor-category model of constructible sheaves.

Background

In zonotopal and unimodular cases, the FLTZ conic Lagrangian equals the conormal variety, and exodromy identifies constructible sheaves with functors on the exit-path category. Despite this, the authors do not have a direct combinatorial recipe that, given Σ, produces the exit-path category.

A concrete presentation of the exit-path category in terms of Σ would systematize exodromy-based descriptions of constructible sheaves and help connect toric combinatorics with categorical models across examples.