Corpus ID: 197481695

Probability, valuations, hyperspace: Three monads on Top and the support as a morphism

@article{Fritz2019ProbabilityVH,
  title={Probability, valuations, hyperspace: Three monads on Top and the support as a morphism},
  author={Tobias Fritz and Paolo Perrone and Sharwin Rezagholi},
  journal={ArXiv},
  year={2019},
  volume={abs/1910.03752}
}
We consider three monads on Top, the category of topological spaces, which formalize topological aspects of probability and possibility in categorical terms. The first one is the hyperspace monad H, which assigns to every space its space of closed subsets equipped with the lower Vietoris topology. The second is the monad V of continuous valuations, also known as the extended probabilistic powerdomain. Both monads are constructed in terms of double dualization. This not only reveals a strong… Expand
The zero-one laws of Kolmogorov and Hewitt--Savage in categorical probability
We state and prove the zero--one laws of Kolmogorov and Hewitt--Savage within the setting of Markov categories, a category-theoretic approach to the foundations of probability and statistics. ThisExpand
Homotopy Theoretic and Categorical Models of Neural Information Networks
TLDR
A novel mathematical formalism for the modeling of neural information networks endowed with additional structure in the form of assignments of resources, either computational or metabolic or informational, is developed. Expand
A synthetic approach to Markov kernels, conditional independence, and theorems on sufficient statistics
  • T. Fritz
  • Computer Science, Mathematics
  • ArXiv
  • 2019
TLDR
Markov categories are developed as a framework for synthetic probability and statistics, following work of Golubtsov as well as Cho and Jacobs, and provides a uniform treatment of various types of probability theory. Expand
Infinite products and zero-one laws in categorical probability
Markov categories are a recent category-theoretic approach to the foundations of probability and statistics. Here we develop this approach further by treating infinite products and the KolmogorovExpand

References

SHOWING 1-10 OF 71 REFERENCES
A Probability Monad as the Colimit of Spaces of Finite Samples
We define and study a probability monad on the category of complete metric spaces and short maps. It assigns to each space the space of Radon probability measures on it with finite first moment,Expand
On the categorical meaning of Hausdorff and Gromov distances, I
Abstract Hausdorff and Gromov distances are introduced and treated in the context of categories enriched over a commutative unital quantale V . The Hausdorff functor which, for every V -category X,Expand
The probabilistic powerdomain for stably compact spaces
TLDR
The Riesz Representation Theorem is used for a straightforward proof of the (known) fact that every valuation on a stably compact space extends uniquely to a Radon measure on the Borel algebra of the corresponding compact Hausdorff space. Expand
Algebras of the Extended Probabilistic Powerdomain Monad
TLDR
It is proved that every Eilenberg-Moore algebras of the extended probabilistic powerdomain monad $\mathcal V_w$-algebra in this setting is a weakly locally convex sober topological cone, and that a map is the structure map of a $\Mathcal V- algebra if and only if it is continuous and sends every continuous valuation to its unique barycentre. Expand
The monad of probability measures over compact ordered spaces and its Eilenberg–Moore algebras
Abstract The probability measures on compact Hausdorff spaces K form a compact convex subset P K of the space of measures with the vague topology. Every continuous map f : K → L of compact HausdorffExpand
Bimonoidal Structure of Probability Monads
TLDR
A conceptual treatment of the notion of joints, marginals, and independence in the setting of categorical probability by endowing the usual probability monads with a monoidal and an opmonoidal structure, mutually compatible. Expand
ON PROPERTY-LIKE STRUCTURES
A category may bear many monoidal structures, but (to within a unique isomorphism) only one structure of "category with finite products". To capture such distinctions, we consider on a 2-categoryExpand
Monads of sets
Publisher Summary This chapter focuses on monads of sets—monads in the category S of sets and (total) functions. Monads were first surfaced to codify resolutions for sheaf cohomology. They areExpand
The Extended Probabilistic Powerdomain Monad over Stably Compact Spaces
TLDR
It is the aim of this work to obtain similar results for the (extended) probabilistic power domain monad over stably compact spaces and to determine the algebras of this powerdomain monad and the algebra homomorphisms. Expand
A monad of valuation locales
If X is a locale then its valuation locale VX has for its points the valuations on X. V is the functor part of a strong monad on the category of locales, a localic analogue of the Giry monad. It isExpand
...
1
2
3
4
5
...