Markov category



Measure and probability theory

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



The formalism of Markov categories can be thought of as a way to express certain aspects of probability and statistics synthetically. In other words, it consists of structures and axioms which one can think of as “fundamental” in probability and statistics, which one can use to prove theorems, without having to use measure theory directly. One then proves that the usual measure-theoretic treatment is a model (or semantics) of this theory.

Intuitively, for the purposes of probability a Markov category can be seen as a category where morphisms behave like “random functions”, or “Markov kernels?” (hence the name). Canonical examples are Kleisli categories of probability monads. The formalism is however far more general.

Caveat: some of the content of this page reflects work in progress. Content may change.


A Markov category is a semicartesian symmetric monoidal category (C,,1)(C,\otimes,1) in which every object XX is equipped with the structure of a commutative internal comonoid. We denote the comultiplication and counit maps by copy:XXXcopy: X \to X \otimes X and delete:X1delete: X\to 1.

We require the following compatibility property between the copy map and the tensor product: for all objects XX and YY,

copy XY=(id Xb Y,Xid Y)(copy Xcopy Y), copy_{X\otimes Y} \; =\; (id_X\otimes b_{Y,X} \otimes id_Y) ( copy_X \otimes copy_Y ),

where bb denotes the braiding.

Note that the map delete:X1delete: X\to 1 is uniquely determined by the fact that 1 is terminal, hence it is also natural in XX (see semicartesian monoidal category for more). On the other hand, the copy map is not required to be natural.

In terms of string diagrams



See also the detailed list below.

Deterministic morphisms

A morphism f:XYf:X\to Y in a Markov category is called deterministic if it commutes with the copy map,

copyf=(ff)copy. copy \circ f \;=\; (f\otimes f) \circ copy .

A way to motivate the definition is the following. Suppose that ff is a “random” function between real numbers, which adds to the input the result of the roll of a die. Given a number xx, we can roll a die, add the resulting value (say, nn) to xx, and then copy the result, to get (x+n,x+n)(x+n,x+n). Or, we could copy the value xx, roll two dice (or roll the die twice), and add the two resulting values (say, mm and nn) to the two copies of xx, obtaining (x+m,x+n)(x+m,x+n). The two results are likely to differ. One can take this as a definition of randomness: it’s a process that may give a different result if you do it twice. Or equivalently, that copying the information before or after the process has taken place gives different results.

A deterministic morphism is instead one that does not exhibit this behavior, i.e. that commutes with the copy map.


Further structures and properties



Kolmogorov products


(to be expanded)

Representable Markov categories

For now, see probability monad.

Detailed list

(…to be expanded…)

Category Probability monad Conditionals Positivity Kolmogorov products Further references
Stoch? Giry monad on Meas No (Fritz'19, Example 11.3) (…) (…) Fritz'19
BorelStoch? Giry monad on Pol? Yes (Kallenberg '17, B-M'19) (…) (…) Fritz'19
FinStoch? Not representable Yes (Fritz'19, Example 11.6) (…) (…) Fritz'19

See also


Markov categories as defined here appear in:

See also the references therein.

The first idea of defining a “category of probabilistic mappings” seems to be due to Lawvere, in

W. Lawvere, The category of probabilistic mappings, ms. 12 pages, 1962 (Lawvere Probability 1962)

(…more to come…)

Further references:

  • Olaf Kallenberg, Random Measures, Theory and Applications, Springer, 2017.

  • Vladimir Bogachev and Il’ya Malofeev, Kantorovich problems and conditional measures depending on a parameter. (arXiv:1904.03642)

Last revised on October 14, 2021 at 04:48:18. See the history of this page for a list of all contributions to it.