nLab Stoch




The category whose objects are measurable spaces and whose morphisms are stochastic kernels (or Markov kernels) is often denotes Stoch, or similar.

It is the motivating example of a Markov category, and together with its subcategory BorelStoch, one of the most important categories of categorical probability.


Stoch is the category whose

  • Objects are measurable spaces, i.e. pairs (X,π’œ)(X,\mathcal{A}) where XX is a set and π’œ\mathcal{A} is a sigma-algebra on XX;
  • Morphisms(X,π’œ)β†’(Y,ℬ)(X,\mathcal{A})\to(Y,\mathcal{B}) are Markov kernels of entries k(B|x)k(B|x), for x∈Xx\in X and Bβˆˆβ„¬B\in\mathcal{B};
  • The identities (X,π’œ)β†’(X,π’œ)(X,\mathcal{A})\to(X,\mathcal{A}) are given by the Dirac delta kernels
    Ξ΄(A|x)=1 A(x)={1 x∈A; 0 xβˆ‰A; \delta(A|x) = 1_A(x) = \begin{cases} 1 & x\in A ; \\ 0 & x\notin A ; \end{cases}
  • The composition of kernels k:(X,π’œ)β†’(Y,ℬ)k:(X,\mathcal{A})\to(Y,\mathcal{B}) and h:(Y,ℬ)β†’(Z,π’ž)h:(Y,\mathcal{B})\to(Z,\mathcal{C}) is given by the Lebesgue integral
    (h∘k)(C|x)=∫ Yh(C|y)k(dy|x) (h\circ k) (C|x) = \int_Y h(C|y)\,k(dy|x)

    for all x∈Xx\in X and Cβˆˆπ’žC\in\mathcal{C}. This is sometimes called Chapman-Kolmogorov formula. (Compare with the composition formula for stochastic matrices.)

As a Kleisli category

Stoch can be equivalently described as the Kleisli category of the Giry monad on Meas.


Notable subcategories


BorelStoch is the full subcategory of Stoch whose objects are standard Borel measurable spaces. This is particularly important as a Markov category because it has conditionals and countable Kolmogorov products. It is the Kleisli category of the Giry monad restricted to standard Borel spaces.

As proven here, in BorelStoch all idempotents split, making it a Cauchy-complete category.


FinStoch is the full subcategory of Stoch whose objects are discrete finite sets. Equivalently, it is the category of finite stochastic matrices.

It is closely related to the Kleisli category of the distribution monad.


  • Bill Lawvere, The category of probabilistic mappings, ms. 12 pages, 1962

    (Lawvere Probability 1962)

  • N. N. Chentsov, The categories of mathematical statistics, Dokl. Akad. SSSR 164, 1965.

  • Prakash Panangaden, The category of Markov kernels, ENTCS, 1999. (full text)

  • Kenta Cho, Bart Jacobs, Disintegration and Bayesian Inversion via String Diagrams, Mathematical Structures of Computer Science 29, 2019. (arXiv:1709.00322)

  • Tobias Fritz, A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics, Advances of Mathematics 370, 2020. (arXiv:1908.07021)

  • Tobias Fritz, TomΓ‘Ε‘ Gonda, Antonio Lorenzin, Paolo Perrone, Dario Stein, Absolute continuity, supports and idempotent splitting in categorical probability, (arXiv:2308.00651)

Last revised on February 9, 2024 at 10:31:14. See the history of this page for a list of all contributions to it.