Introduction to graded geometry

TL;DR

This paper introduces the fundamentals of $ obreakbZ$-graded manifolds, covering their local models, properties, and algebraic correspondences, providing a foundational overview of graded geometry.

Contribution

It systematically develops the theory of $ obreakbZ$-graded manifolds, including their local models, function algebras, and vector fields, clarifying key constructions and properties.

Findings

Defined $ obreakbZ$-graded manifolds from local models

Explained the use of completed graded symmetric algebra for functions

Reviewed correspondences with algebraic structures

Abstract

This paper aims at setting out the basics of $\mathbb{Z}$-graded manifolds theory. We introduce $\mathbb{Z}$-graded manifolds from local models and give some of their properties. The requirement to work with a completed graded symmetric algebra to define functions is made clear. Moreover, we define vector fields and exhibit their graded local basis. The paper also reviews some correspondences between differential Z-graded manifolds and algebraic structures.