The functor between two categories of $\mathbb{Z}-$graded manifolds

TL;DR

This work treats ${oldsymbol{ m Z}}$-graded manifolds as semiformal homogeneity structures, contrasting two natural filtrations on the local polynomial model and showing their completions are isomorphic in finite dimensions, while providing a finer topology in infinite dimensions. Building on the Batchelor–Gawedzki framework, it interprets every smooth ${oldsymbol{ m Z}}$-graded manifold of finite graded dimension as the formal neighborhood of the zero section in a Batchelor bundle, with a canonical Euler vector field encoding the grading. A graded Borel–Whitney-type theorem is proved: morphisms between formal neighborhoods lift to smooth homogeneity maps between Batchelor–Gawedzki bundles, harnessing local graded Borel lemmas and jet-prolongation techniques. The paper also situates these results within a broader homogeneity-structure program, highlighting examples and open questions about the structure, diffeomorphisms, and calculus on ${oldsymbol{ m Z}}$-graded objects.

Abstract

This paper examines $\mathbb{Z}$-graded manifolds as semiformal homogeneity structures, comparing two polynomial filtrations from their local models. In finite dimensions, these are componentwise equivalent, yielding isomorphic graded completions; generally, one induces a finer topology. By the Batchelor-Gawedzki-type theorem (Kotov--Salnikov), every $\mathbb{Z}$-graded manifold over base $M$ is noncanonically isomorphic to one associated with its canonical $\mathbb{Z}$-graded bundle (Batchelor-Gawedzki bundle). In finite dimensions, this is the formal neighborhood of the zero section with the induced homogeneity structure. Kotov-Salnikov's graded Borel lemma extends weight-$k$ functions from the formal neighborhood to smooth ones of the same weight. Here, this generalizes to a Borel--Whitney theorem: homogeneity morphisms of formal neighborhoods lift to smooth homogeneity maps between Batchelor-Gawedzki bundles. Categorically, let $\mathsf{B}_{\mathbb{Z}}$ be the category of finite-dimensional $\mathbb{Z}$-graded vector bundles with homogeneity morphisms, and $\mathsf{Man}_{\mathbb{Z}}$ the category of finite-dimensional $\mathbb{Z}$-graded manifolds. The functor $\mathsf{F}\colon \mathsf{B}_{\mathbb{Z}} \to \mathsf{Man}_{\mathbb{Z}}$ sends bundles to formal neighborhoods of their zero sections. The graded Batchelor-Gawedzki and Borel-Whitney theorems imply $\mathsf{F}$ is full and surjective on objects.

Theorems & Definitions (31)

• Definition 1
• Theorem 1: Equivalence of completions
• proof
• Example 1: $\underline{F} \le F$ fails reversely in infinite dimensions
• Proposition 1
• Remark 1
• Theorem 2: Graded Borel-Whitney-type theorem
• Lemma 1
• proof
• Example 2: Homogeneity Lie groups KS-HCP