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. The Kotov–Salnikov 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 $B_{\mathbb{Z}}$ be the category of finite-dimensional $\mathbb{Z}$-graded vector bundles with homogeneity morphisms, and ${Man}_{\mathbb{Z}}$ the category of finite-dimensional $\mathbb{Z}$-graded manifolds. The functor $F:B_{\mathbb{Z}}\to {Man}_{\mathbb{Z}}$ sends bundles to formal neighborhoods of their zero sections. The graded Batchelor–Gawedzki and Borel–Whitney theorems imply $F$ is full and surjective on objects.

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