# Canonical stratifications along bisheaves

## TL;DR

This paper develops a method to explicitly determine the coarsest stratification of a space that preserves the constructibility of a bisheaf, which combines sheaf and cosheaf data, over triangulated spaces.

## Contribution

It introduces a way to compute the canonical stratification for bisheaves that are constructible with respect to a triangulation.

## Key findings

- Explicit construction of the coarsest stratification for bisheaves.
- Application to bisheaves that are locally constant on triangulations.
- Provides a foundation for analyzing homological stability via stratifications.

## Abstract

A theory of bisheaves has been recently introduced to measure the homological stability of fibers of maps to manifolds. A bisheaf over a topological space is a triple consisting of a sheaf, a cosheaf, and compatible maps from the stalks of the sheaf to the stalks of the cosheaf. In this note we describe how, given a bisheaf constructible (i.e., locally constant) with respect to a triangulation of its underlying space, one can explicitly determine the coarsest stratification of that space for which the bisheaf remains constructible.