Report
Weak Distributive Laws between Monads of Continuous Valuations and of Non-Deterministic Choice
| العنوان: | Weak Distributive Laws between Monads of Continuous Valuations and of Non-Deterministic Choice |
|---|---|
| المؤلفون: | Goubault-Larrecq, Jean |
| سنة النشر: | 2024 |
| المجموعة: | Computer Science Mathematics |
| مصطلحات موضوعية: | Mathematics - Category Theory, Computer Science - Logic in Computer Science, Mathematics - General Topology, Mathematics - Probability, 18C15 primary, secondary: 54B20, 28A33, 46E27 |
| الوصف: | We show that there is weak distributive law of the Smyth hyperspace monad $\mathcal Q_{\mathsf V}$ (resp., the Hoare hyperspace monad $\mathcal H_{\mathsf V}$, resp. the monad $\mathcal P\ell^{\mathrm q}_{\mathsf V}$ of quasi-lenses, resp. the monad $\mathcal P\ell_{\mathsf V}$ of lenses) over the continuous valuation monad $\mathbf V$, as well as over the subprobability valuation monad $\mathbf V_{\leq 1}$ and the probability valuation monad $\mathbf V_1$, on the whole category $\mathbf{Top}$ of topological spaces (resp., on certain full subcategories such as the category of locally compact spaces or of stably compact spaces). We show that the resulting weak composite monad is the author's monad of superlinear previsions (resp., sublinear previsions, resp. forks), possibly subnormalized or normalized depending on whether we consider $\mathbf V_{\leq 1}$ or $\mathbf V_1$ instead of $\mathbf V$. As a special case, we obtain a weak distributive law of the monad $\mathcal P\ell^{\mathrm q}_{\mathsf V} \cong \mathcal P\ell_{\mathsf V}$ over the monad of (sub)probability Radon measures $\mathbf R_\bullet$ on the category of stably compact spaces, which specializes further to a weak distributive laws of the Vietoris monad over $\mathbf R_\bullet$. The associated weak composite monad is the monad of (sub)normalized forks. Comment: 79 pages; in v2, credit now given to G. B\"ohm for weak distributive laws; in the Appendix, new example of an inner regular, non-locally finite measure with finite values on the compact sets; in v3, fixed Remarks 4.3, 7.4, 10.3, 10.5 and 11.7, which were faulty; mentioned Quentin Aristote's work, if very briefly; in v4, included modifications from errata cited as [21] |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2408.15977 |
| رقم الانضمام: | edsarx.2408.15977 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |