A Cohesive $\infty$-Topos with a Quantum Modality from Finite-Dimensional $C^{*}$-Algebras

Abstract: 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.