两类$\u2124$-分次流形范畴之间的函子
The functor between two categories of $\mathbb{Z}-$graded manifolds
AI总结：
本文研究$\u2124$-分次流形的齐次结构，对比两类多项式滤过的性质，推广得到Borel–Whitney定理，证明有限维情形下从分次向量丛范畴到分次流形范畴的函子是满且对象满的。
摘要：
本文将$\u2124$-分次流形作为半形式齐次结构展开研究，对比其局部模型导出的两种多项式滤过。有限维情形下，二者按分量等价，对应的分次完备化同构；一般情形下，其中一种诱导出更精细的拓扑。根据Batchelor-Gawedzki型定理（Kotov–Salnikov），任意以$M$为底空间的$\u2124$-分次流形，都与其典范$\u2124$-分次丛（Batchelor-Gawedzki丛）对应的分次流形存在非典范同构。有限维情形下，该分次流形就是零截面的形式邻域，带有诱导的齐次结构。Kotov-Salnikov的分次Borel引理可将形式邻域上的权$k$函数延拓为相同权的光滑函数。本文将这一结论推广为Borel–Whitney定理：形式邻域的齐次态射可提升为Batchelor-Gawedzki丛之间的光滑齐次映射。从范畴论角度，设$\u2a66_{\u2124}$为带齐次态射的有限维$\u2124$-分次向量丛范畴，$\u2a66an_{\u2124}$为有限维$\u2124$-分次流形范畴。函子$\u2a66:\u2a66_{\u2124}\to\u2a66an_{\u2124}$将丛映射到其零截面的形式邻域。分次Batchelor-Gawedzki定理与Borel-Whitney定理共同表明，$\u2a66$是满函子且在对象上是满射的。
英文摘要：
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.
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