# A Probability Monad as the Colimit of Spaces of Finite Samples

@article{Fritz2017APM, title={A Probability Monad as the Colimit of Spaces of Finite Samples}, author={Tobias Fritz and Paolo Perrone}, journal={arXiv: Probability}, year={2017} }

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, equipped with the Kantorovich-Wasserstein distance. This monad is analogous to the Giry monad on the category of Polish spaces, and it extends a construction due to van Breugel for compact and for 1-bounded complete metric spaces.
We prove that this Kantorovich monad arises from a colimit construction…

## 15 Citations

Stochastic order on metric spaces and the ordered Kantorovich monad

- Computer Science, MathematicsArXiv
- 2018

The Kantorovich monad is extended further to a certain class of ordered metric spaces, by endowing the spaces of probability measures with the usual stochastic order, which can be considered a metric analogue of the probabilistic powerdomain.

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

- Mathematics, Computer ScienceArXiv
- 2019

It is obtained that taking the support of a $\tau$-smooth probability measure is also given by a morphism of monads, which implies that every H- algebra (topological complete semilattice) is also a V-algebra.

A Criterion for Kan Extensions of Lax Monoidal Functors

- Mathematics
- 2018

In this mainly expository note, we state a criterion for when a left Kan extension of a lax monoidal functor along a strong monoidal functor can itself be equipped with a lax monoidal structure, in a…

Monads, partial evaluations, and rewriting

- Computer Science, MathematicsMFPS
- 2020

It is proved that whenever the monad is weakly cartesian, partial evaluations can be composed via the usual Kan filler property of simplicial sets, of which the author gives an interpretation in terms of substitution of terms.

Entropy, Derivation Operators and Huffman Trees

- Computer Science, MathematicsArXiv
- 2021

A theory of binary trees on finite multisets that categorifies, or operationalizes, the entropy of a finite probability distribution, and shows how the derivation property of theEntropy of a joint distribution lifts to Huffman trees.

Dirichlet Polynomials and Entropy

- Computer Science, MathematicsEntropy
- 2021

The main result of this paper is the following: the rectangle-area formula A(d) = L (d)W( d) holds for any Dirichlet polynomial d, and the entropy of an empirical distribution can be calculated entirely in terms of the homomorphism h applied to its corresponding Dirichlets.

Operations on Metric Thickenings

- Mathematics
- 2021

Many simplicial complexes arising in practice have an associated metric space structure on the vertex set but not on the complex, e.g. the Vietoris–Rips complex in applied topology. We formalize a…

The Category Theory of Causal Models

- 2020

Building on work of Rubenstein et al, we consider a notion of structurepreserving transformations between structural causal models. We describe a category, in the sense of category theory, of such…

Are Banach spaces monadic?

- MathematicsCommunications in Algebra
- 2021

We will show that Banach spaces are monadic over complete metric spaces via the unit ball functor. For the forgetful functor, one should take complete pointed metric spaces.

Homotopy Theoretic and Categorical Models of Neural Information Networks

- Computer Science, MathematicsArXiv
- 2020

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.

## References

SHOWING 1-10 OF 50 REFERENCES

The monad of probability measures over compact ordered spaces and its Eilenberg–Moore algebras

- Mathematics
- 2008

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 Hausdorff…

From Kleisli Categories to Commutative C*-algebras: Probabilistic Gelfand Duality

- Mathematics, Computer ScienceLog. Methods Comput. Sci.
- 2015

This paper shows that there are functors from several Kleisli categories, of monads that are relevant to model probabilistic computations, to categories of C*-algebras, and shows that the state space functor from C-algeses to Eilenberg-Moore algebrs of the Radon monad is full and faithful.

A Categorical Approach to Probability Theory

- Mathematics, Computer ScienceStud Logica
- 2010

This work shows that the category ID of D-posets of fuzzy sets and sequentially continuous D-homomorphisms allows to characterize the passage from classical to fuzzy events as the minimal generalization having nontrivial quantum character.

Convex Spaces I: Definition and Examples

- Mathematics
- 2009

We propose an abstract definition of convex spaces as sets where one can take convex combinations in a consistent way. A priori, a convex space is an algebra over a finitary version of the Giry…

OPERADS AS POLYNOMIAL 2-MONADS

- Mathematics
- 2014

In this article we give a construction of a polynomial 2-monad from an operad and describe the algebras of the 2-monads which then arise. This construction is dierent from the standard construction…

A Bayesian Characterization of Relative Entropy

- Mathematics, Computer ScienceArXiv
- 2014

Any convex linear, lower semicontinuous functor from FinStat to the additive monoid $[0,\infty]$ which vanishes when s is optimal must be a scalar multiple of this relative entropy.

Algebraic Kan extensions along morphisms of internal algebra classifiers

- Mathematics
- 2015

Abstract An \algebraic left Kan extension" is a left Kan extension which interacts well with the alge- braic structure present in the given situation, and these appear in various subjects such as the…

ALGEBRAIC KAN EXTENSIONS IN DOUBLE CATEGORIES

- Mathematics
- 2015

We study Kan extensions in three weakenings of the Eilenberg-Moore dou- ble category associated to a double monad, that was introduced by Grandis and Par e. To be precise, given a normal oplax double…

Continuous valuations

- 1993

0 Introduction In this paper we study, for a certain type of topological rings A, the topological space Cont A of all equivalence classes of continuous valuations of A. The space ContA is defined as…

A Criterion for Kan Extensions of Lax Monoidal Functors

- Mathematics
- 2018

In this mainly expository note, we state a criterion for when a left Kan extension of a lax monoidal functor along a strong monoidal functor can itself be equipped with a lax monoidal structure, in a…