Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Commutativity and liftings of codensity monads of probability measures

Zev Shirazi

TL;DR

The paper develops a unified, codensity-based framework for five canonical probability monads ($ ext{D}$, $ ext{G}$, $ ext{E}$, $ ext{R}$, $ ext{K}$) by presenting them as pointwise codensity monads over small categories of stochastic maps. It clarifies how their measure-theoretic and metric structures relate to universal properties and liftings of the Giry monad, and it characterizes when these codensity presentations are (exactly) monoidal or commutative, yielding Day-convolution descriptions of tensor products of free algebras in the Radon case. A key finding is that the Radon monad is exactly pointwise monoidal, while the Giry monad is not in general Meas but is in the standard Borel setting, and that subprobability monads inherit affineness from their probability counterparts. Overall, the work provides a robust, reusable approach to deducing algebraic properties (commutativity, affineness, monoidal liftings) of probability monads from their codensity presentations, with concrete implications for universal constructions and tensorial behavior in categorical probability.

Abstract

Several well-studied probability monads have been expressed as codensity monads over small categories of stochastic maps, giving a limit description of spaces of probability measures. In this paper we show how properties of probability monads such as commutativity and affineness can arise from their codensity presentation. First we show that their codensity presentation is closely related to another universal property of probability monads as liftings of the Giry monad, which allows us to generalise a result by Van Breugel on the Kantorovich monad and provide a novel characterisation of the Radon monad. We then provide sufficient conditions for a codensity monad to lift to \textbf{MonCat}, and give a characterisation of exactly pointwise monoidal codensity monads; codensity monads that satisfy a strengthening of these conditions. We show that the Radon monad is exactly pointwise monoidal, and hence give a description of the tensor product of free algebras of the Radon monad in terms of Day convolution. Finally we show that the Giry monad is not exactly pointwise monoidal due to the existence of probability bimeasures that do not extend to measures, although its restriction to standard Borel spaces is.

Commutativity and liftings of codensity monads of probability measures

TL;DR

The paper develops a unified, codensity-based framework for five canonical probability monads (, , , , ) by presenting them as pointwise codensity monads over small categories of stochastic maps. It clarifies how their measure-theoretic and metric structures relate to universal properties and liftings of the Giry monad, and it characterizes when these codensity presentations are (exactly) monoidal or commutative, yielding Day-convolution descriptions of tensor products of free algebras in the Radon case. A key finding is that the Radon monad is exactly pointwise monoidal, while the Giry monad is not in general Meas but is in the standard Borel setting, and that subprobability monads inherit affineness from their probability counterparts. Overall, the work provides a robust, reusable approach to deducing algebraic properties (commutativity, affineness, monoidal liftings) of probability monads from their codensity presentations, with concrete implications for universal constructions and tensorial behavior in categorical probability.

Abstract

Several well-studied probability monads have been expressed as codensity monads over small categories of stochastic maps, giving a limit description of spaces of probability measures. In this paper we show how properties of probability monads such as commutativity and affineness can arise from their codensity presentation. First we show that their codensity presentation is closely related to another universal property of probability monads as liftings of the Giry monad, which allows us to generalise a result by Van Breugel on the Kantorovich monad and provide a novel characterisation of the Radon monad. We then provide sufficient conditions for a codensity monad to lift to \textbf{MonCat}, and give a characterisation of exactly pointwise monoidal codensity monads; codensity monads that satisfy a strengthening of these conditions. We show that the Radon monad is exactly pointwise monoidal, and hence give a description of the tensor product of free algebras of the Radon monad in terms of Day convolution. Finally we show that the Giry monad is not exactly pointwise monoidal due to the existence of probability bimeasures that do not extend to measures, although its restriction to standard Borel spaces is.
Paper Structure (24 sections, 22 theorems, 55 equations, 1 table)

This paper contains 24 sections, 22 theorems, 55 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Universal properties of probability monads
  4. Commutativity of probability monads
  5. Related work
  6. Notation
  7. Acknowledgements
  8. Five probability monads
  9. Countable distribution monad $D$ on Set
  10. Giry monad $\mathcal{G}$ on Meas
  11. Countable expectation monad $\mathbb{E}$ on Set
  12. Radon monad $\mathcal{R}$ on KHaus
  13. Kantorovich monad $\mathcal{K}$ on KMet
  14. Codensity presentations
  15. Codensity monads
  16. ...and 9 more sections

Key Result

Proposition 3.10

Suppose $\mathcal{C}$ and $\mathcal{D}$ have terminal objects. If $K \colon \mathcal{D} \to \mathcal{C}$ is such that $!_{K1} \colon K1 \to 1$ is an isomorphism, and for every $y \colon 1 \to KX$ there is a unique $x \colon 1 \to X$ with $y=Kx!_{K1}^{-1}$. Then if $T$ is the pointwise codensity mona

Theorems & Definitions (77)

  • Remark 2.1
  • Definition 3.1: Codensity monad
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Definition 3.5
  • Definition 3.6
  • Example 3.7
  • Example 3.8
  • Example 3.9
  • ...and 67 more