---
title: "Joinings in Markov categories"
authors:
  - "Suddhasattwa Das"
  - "Tomoharu Suda"
year: 2026
venue: "arXiv:math.CT"
arxiv: "2609.10912"
doi: null
source: "https://arxiv.org/abs/2609.10912"
topics:
  - "curiosity"
  - "maths-foundations"
  - "category-theory"
  - "ergodic-theory"
seed_rank: 1833
seed_batch: "curiosity-2026-09-15"
reviewed: "2026-09-15"
pool: "maths-foundations"
relevance_score: 9
lineage: markov-categories-ergodic
cites:
  - title: "Joinings in Markov categories"
    url: "https://arxiv.org/abs/2609.10912"
    year: 2026
    arxiv: "2609.10912"
    doi: null
see:
  - "944-representable-markov-categories-statistical-experiments"
  - "1146-a-synthetic-approach-to-markov-kernels"
---

# Joinings in Markov categories

## One-sentence takeaway

Joinings and ergodic decomposition are expressed inside Markov categories, so measure-theoretic irreducibility becomes a categorical diagram chase.

## Why it's lovely

Why you might love this: ergodic theory's joinings are the classical way to compare systems; Markov categories are Broadside's preferred synthetic home for kernels and stochastic maps. Putting joinings there turns 'two measures on a product agreeing on margins' into the same language as 944/1146 — synthetic probability meeting Furstenberg.

## Key ideas

- Frames invariance (sets, densities, measures, observables) in Markov-categorical language.
- Treats ergodic systems as measure-theoretically irreducible objects in that setting.
- Develops joinings as morphisms/diagrams comparing systems.
- Connects ergodic decomposition to the categorical viewpoint rather than raw measure algebras.

## Caveats

Still foundational; may not yet recover the full classical joining toolkit (e.g. relatively independent joinings over factors) in full generality. Read with Markov-category cards 944 and 1146.

## Links

- arXiv abs: https://arxiv.org/abs/2609.10912
- PDF: https://arxiv.org/pdf/2609.10912.pdf