Double Glueing over Free Exponential: with Measure Theoretic Applications
Abstract
This paper provides a compact method to lift the free exponential construction of MelliesTabareauTasson over the HylandSchalk 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 sfinite 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 sfinite 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 the full subcategory of countable measurable spaces. To lift the free exponential, the measuretheoretic uniform convergence theorem commuting Lebesgue integral and limit plays a crucial role. Our measuretheoretic 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.07726
 Bibcode:
 2021arXiv210707726H
 Keywords:

 Computer Science  Logic in Computer Science;
 Mathematics  Category Theory