样本光滑空间：可微概率编程的便捷范畴

中文摘要 AI 辅助

我们在混合站点上引入样本光滑空间范畴 $\mathbf{SSS}$。测试对象是笛卡尔空间与万有希尔伯特立方体 $\Omega$ 的乘积 $\Omega_n:= \mathbb{R}^n \times \Omega$，其中 $\Omega$ 携带所有万有可测集，而一个空间是带有在预复合下封闭的可容许图 $\Omega_n \to \mathcal{X}$ 族的集合。光滑性和可测性不再是沿公理粘合的两个结构，而是同一站点上的一个结构。该站点具有有限的非空乘积，因为 $\Omega$ 吸收其自身的平方；其 Karoubi 包络包含每个 $\mathbb{R}^n$；并且它具有混合态射 $\omega \mapsto (W(\omega),\Phi(\omega))$，这将光滑族的可测性从公理转变为推论。$\mathbf{SSS}$ 是一个具体的拟拓扑斯：完备、余完备、笛卡尔闭和局部笛卡尔闭，具有嵌入分类器。笛卡尔空间的态射恰好是 $C^\infty$ 映射，流形完全且忠实地嵌入，两者都无需 Boman 定理。每个对象都有切空间和余切空间，每个态射都有微分。模态位于伴随串 $\Pi \dashv \flat \dashv \natural \dashv \sharp \dashv \Lambda$ 中，使 $\mathbf{SSS}$ 在拟万有空间上具有内聚性。关键在于概率单子。将 $\mathsf{P}(\mathcal{X})$ 的图定义为在每个测试对象处 $\mathcal{X}$-图的推前，$\mathsf{P}$ 是 $\mathbf{SSS}$ 上所有对象上的无条件强交换仿射单子——函子、单位、核的乘积、乘法和单子律各是一行种子分裂——其 Kleisli 范畴（可微模拟器）是马尔可夫范畴。重参数化技巧通过构造成立：每个 Kleisli 态射在图上都是采样器，且在复合下稳定。一个反射定理将全部收益定位在单个图族中。

英文摘要

We introduce the category $\mathbf{SSS}$ of sample-smooth spaces over a mixed site. The test objects are the products $Ω_n := \mathbb{R}^n \times Ω$ of a Cartesian space with the universal Hilbert cube $Ω$ carrying all universally measurable sets, and a space is a set with a family of admissible plots $Ω_n \to \mathcal{X}$ closed under precomposition. Smoothness and measurability are then not two structures glued along an axiom, but one structure over one site. The site has finite non-empty products, because $Ω$ absorbs its own square; its Karoubi envelope contains every $\mathbb{R}^n$; and it has mixed morphisms $ω\mapsto (W(ω),Φ(ω))$, which turn measurability of a smooth family from an axiom into a consequence. $\mathbf{SSS}$ is a concrete quasitopos: complete, cocomplete, cartesian closed and locally cartesian closed, with a classifier for embeddings. Morphisms of Cartesian spaces are exactly the $C^\infty$ maps and manifolds embed full and faithfully, both without Boman's theorem. Every object has tangent and cotangent spaces, every morphism a differential. The modalities sit in an adjoint string $Π\dashv \flat \dashv \natural \dashv \sharp \dashv Λ$, making $\mathbf{SSS}$ cohesive over quasi-universal spaces. The point is the probability monad. Defining the plots of $\mathsf{P}(\mathcal{X})$ as push-forwards of $\mathcal{X}$-plots at every test object, $\mathsf{P}$ is an unconditional strong commutative affine monad on all of $\mathbf{SSS}$ -- functor, unit, product of kernels, multiplication and the monad laws are each one line of seed splitting -- and its Kleisli category, of differentiable simulators, is a Markov category. The reparametrisation trick holds by construction: every Kleisli morphism is plot-wise a sampler, stably under composition. A reflection theorem locates the whole gain in a single plot family.

发表机构

• Korteweg-de Vries Institute for Mathematics（科尔特韦格-德弗里斯数学研究所）
• University of Amsterdam（阿姆斯特丹大学）

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