The category of $\mathbb{Z}$-graded manifolds: what happens if you do not stay positive

TL;DR

This paper explores the categorical structure of $Z$-graded manifolds, highlighting differences from the $N$-graded case, and establishes foundational theorems including an analogue of Batchelor's theorem.

Contribution

It provides a detailed analysis of the local and global structures of $Z$-graded manifolds, introducing intrinsic definitions and extending classical results.

Findings

Describes the local model differences between $Z$- and $N$-graded manifolds.

Introduces filtrations to define objects and morphisms intrinsically.

Formulates an analogue of Batchelor's theorem for $Z$-graded manifolds.

Abstract

In this paper we discuss the categorical properties of $\mathbb{Z}$-graded manifolds. We start by describing the local model paying special attention to the differences in comparison to the $\mathbb{N}$-graded case. In particular we explain the origin of formality for the functional space and spell-out the structure of the power series. Then we make this construction intrinsic using filtrations. This sums up to proper definitions of objects and morphisms in the category. We also formulate the analogue of Batchelor's theorem for the global structure of $\mathbb{Z}$-graded manifolds.