# Graded topological spaces

## TL;DR

This paper introduces graded topological spaces, combining topology with sheaves of abelian groups to facilitate graded objects, and develops foundational sheaf theory and duality results for these spaces.

## Contribution

It presents the concept of graded topological spaces and establishes their fundamental sheaf-theoretic properties and duality theories, expanding the framework of classical topology.

## Key findings

- Defined graded topological spaces with sheaves of abelian groups
- Developed sheaf theory for graded spaces
- Established Poincaré-Verdier duality in this context

## Abstract

We introduce the notion of a "graded topological space": a topological space endowed with a sheaf of abelian groups which we think of as a sheaf of gradings. Any object living on a graded topological space will be graded by this sheaf of abelian groups. We work out the fundamentals of sheaf theory and Poincar\'e-Verdier duality for such spaces.