具有量子模态的内聚 ∞-意象：来自有限维 C*-代数
A Cohesive $\infty$-Topos with a Quantum Modality from Finite-Dimensional $C^{*}$-Algebras
AI总结：
通过有限维 C*-代数的函子 ∞-意象构造了一个带有量子模态（幂等保积余单子）的内聚 ∞-意象，并证明了其满足贝克利瓦利相容性条件，从而为内聚线性同伦类型论提供了第一个具体模型。
摘要：
我们构造了一个内聚 ∞-意象 $\\\mathbf{H}_{\\\mathbb{Q}}$，配备了一个 \\emph{量子模态}——一个幂等的保积余单子 $Q^{\\\diamond}$，其右伴随 $Q_{\\bullet}$ 满足与内聚结构 $(\\\Pi,\\flat,\\\sharp)$ 的 Beck-Chevalley 相容性条件。该模型是函子 ∞-意象 $\\\operatorname{Fun}(\\\mathbf{C}^{*}\\\mathbf{Alg}_{\\\mathrm{fd}},\\\; \\\mathbf{H}_{\\\mathrm{sm}})$，其中 $\\\mathbf{H}_{\\\mathrm{sm}}$ 是光滑内聚 ∞-意象，$\\\mathbf{C}^{*}\\\mathbf{Alg}_{\\\mathrm{fd}}$ 是有限维 C*-代数及其保中心 *-同态构成的范畴。内聚性从 $\\\mathbf{H}_{\\\mathrm{sm}}$ 逐点提升；量子余单子是通过中心函子的预复合。我们赋予该意象由 C*-代数的张量积诱导的 Day 卷积幺半结构 $\\\otimes_{\\\mathrm{Day}}$，并证明 $Q^{\\\diamond}$ 是强幺半余单子。通过 Gelfand 对偶，$Q^{\\\diamond}$-余代数范畴等价于离散经典场论的意象 $\\\operatorname{Fun}(\\\mathbf{FinSet}^{\\\mathrm{op}},\\\mathbf{H}_{\\\mathrm{sm}})$。该余单子被解释为退相干。这产生了一个内聚线性 ∞-意象，其中笛卡尔线性逻辑结构退化，而 Day 卷积提供了乘法直觉主义线性逻辑的非退化仿射模型。我们还证明了一个合成不可克隆定理，并讨论了中心模态表示量子通道的局限性。这项工作提供了内聚线性框架的第一个严格实例，并解决了为内聚线性同伦类型论寻找具体模型的开放问题。
英文摘要：
We construct a cohesive $\infty$-topos $\mathbf{H}_{\mathbb{Q}}$ equipped with a \emph{quantum modality} -- an idempotent product-preserving comonad $Q^{\diamond}$ with right adjoint $Q_{\bullet}$ satisfying the Beck--Chevalley compatibility conditions with the cohesive structure $(Π,\flat,\sharp)$. The model is the functor $\infty$-topos $\operatorname{Fun}(\mathbf{C}^{*}\mathbf{Alg}_{\mathrm{fd}},\; \mathbf{H}_{\mathrm{sm}})$, where $\mathbf{H}_{\mathrm{sm}}$ is the smooth cohesive $\infty$-topos and $\mathbf{C}^{*}\mathbf{Alg}_{\mathrm{fd}}$ is the category of finite-dimensional $C^{*}$-algebras with centre-preserving $*$-homomorphisms. Cohesion is lifted pointwise from $\mathbf{H}_{\mathrm{sm}}$; the quantum comonad is precomposition with the centre functor. We endow the topos with the Day convolution monoidal structure $\otimes_{\mathrm{Day}}$ induced by the tensor product of $C^{*}$-algebras and prove that $Q^{\diamond}$ is a strong monoidal comonad. The category of $Q^{\diamond}$-coalgebras is equivalent, via Gelfand duality, to the topos $\operatorname{Fun}(\mathbf{FinSet}^{\mathrm{op}},\mathbf{H}_{\mathrm{sm}})$ of discrete classical field theories. The comonad is interpreted as decoherence. This yields a cohesive linear $\infty$-topos in which the cartesian linear-logic structure degenerates, while the Day convolution provides a non-degenerate affine model of multiplicative intuitionistic linear logic. We also prove a synthetic no-cloning theorem and discuss the limits of the centre modality for representing quantum channels. This work provides the first rigorous instance of the cohesive linear framework and settles the open problem of finding a concrete model for cohesive linear homotopy type theory.
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