This paper provides a compact method to lift the free exponential construction of Mellies-Tabareau-Tasson over the Hyland-Schalk double glueing for orthogonality categories. A condition ``reciprocity of orthogonality'' is presented simply enough to lift the free exponential over the double glueing in terms of the orthogonality. Our general method applies to the monoidal category TsK of the s-finite transition kernels with countable biproducts. We show (i) TsK^op has the free exponential, which is shown to be describable in terms of measure theory. (ii) The s-finite transition kernels have an orthogonality between measures and measurable functions in terms of Lebesgue integrals. The orthogonality is reciprocal, hence the free exponential of (i) lifts to the orthogonality category O_I(TsK^op), which subsumes Ehrhard et al's probabilistic coherent spaces as a full subcategory of countable measurable spaces. To lift the free exponential, the measure-theoretic uniform convergence theorem commuting Lebesgue integral and limit plays a crucial role as well as Fubini-Tonelli theorem for double integral in s-finiteness. Our measure-theoretic orthogonality is considered as a continuous version of the orthogonality of the probabilistic coherent spaces for linear logic, and in particular provides a two layered decomposition of Crubille et al's direct free exponential for these spaces.

翻译：本文提出了一种紧凑方法，将Mellies-Tabareau-Tasson的自由指数构造提升至Hyland-Schalk正交性范畴的双重粘合框架。我们提出了“正交互易性”条件，该条件表述简洁，足以基于正交性将自由指数提升至双重粘合结构。我们的通用方法适用于具有可数双积的s有限转移核幺半范畴TsK。我们证明：(i) TsK^op具有自由指数，且该指数可用测度论术语描述；(ii) s有限转移核通过勒贝格积分在测度与可测函数之间具有正交性。该正交性满足互易条件，因此(i)中的自由指数可提升至正交性范畴O_I(TsK^op)，该范畴将Ehrhard等人的概率相干空间包含为可测空间的可数子范畴。在提升自由指数的过程中，测度论中勒贝格积分与极限交换的一致收敛定理，以及s有限性条件下二重积分的Fubini-Tonelli定理，均起到关键作用。我们的测度论正交性被视为线性逻辑概率相干空间正交性的连续版本，特别地为Crubille等人针对这些空间的直接自由指数提供了双层分解结构。