Joey Woo | Searcharxiv

arXiv2026

The Degeneracy of the Centre Comonad Model and the Precomposition Obstruction for Quantum Modalities on Presheaf Topoi

The centre comonad model provided the first concrete cohesive linear $\infty$-topos, settling an open problem of Schreiber. However, the model is degenerate: the quantum modality annihilates all non-commutative algebras, and the associated linear logic collapses to classical cartesian logic. In this paper we give a complete mathematical diagnosis of this degeneracy. We prove that the centre comonad sends the representable sheaf of a simple non-commutative algebra to the empty presheaf, and that the state space of any such algebra is empty. We then prove that the Day convolution on the classical core is cartesian, forcing the Seely isomorphism to hold trivially and collapsing the linear logic. We isolate the structural reason behind this collapse: whenever the opposite of the classical core is monoidally equivalent to a cartesian monoidal category, any coreflective precomposition comonad will exhibit the same degeneracy. We conclude that a non-degenerate quantum modality must be constructed without precomposition, and we briefly discuss possible directions.

math.CT↗

arXiv2026

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

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 $(\Pi,\flat,\sharp)$. 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.

math.CT↗