markov

The free Markov category, i.e. a semicartesian category with a supply of commutative comonoid, see Fritz and Liang [FL23].

Summary

Diagram	A Markov diagram is a symmetric diagram with Copy boxes.
Box	A Markov box is a symmetric box in a Markov diagram.
Swap	Symmetric swap in a Markov diagram.
Copy	The copy of an atomic type x some n number of times.
Functor	A Markov functor is a symmetric functor that preserves copies.

Axioms

>>> x = Ty('x')

>>> copy, merge = Copy(x), Merge(x)
>>> unit, delete = Merge(x, n=0), Copy(x, n=0)

Commutative monoid

>>> unitality = Equation(unit @ x >> merge, Id(x), x @ unit >> merge)
>>> associativity = Equation(merge @ x >> merge, x @ merge >> merge)
>>> commutativity = Equation(Swap(x, x) >> merge, merge)
>>> assert unitality and associativity and commutativity
>>> Equation(unitality, associativity, commutativity, symbol='').draw(
...     path="docs/_static/frobenius/monoid.svg")

Cocommutative comonoid

>>> counitality = Equation(copy >> delete @ x, Id(x), copy >> x @ delete)
>>> coassociativity = Equation(copy >> copy @ x, copy >> x @ copy)
>>> cocommutativity = Equation(copy >> Swap(x, x), copy)
>>> assert counitality and coassociativity and cocommutativity
>>> Equation(counitality, coassociativity, cocommutativity, symbol='').draw(
...     path="docs/_static/frobenius/comonoid.svg")

Coherence

>>> assert Equation(Diagram.copy(x @ x, n=0), delete @ delete)
>>> assert Equation(Diagram.copy(x @ x),
...     copy @ copy >> x @ Swap(x, x) @ x)
>>> assert Equation(Diagram.merge(x @ x, n=0), unit @ unit)
>>> assert Equation(Diagram.merge(x @ x),
...     x @ Swap(x, x) @ x >> merge @ merge)

Note

Equality of Markov diagrams is computed by translation to hypergraph. Both copy and merge boxes are translated to spiders, thus when they appear in the same diagram they automatically satisfy the frobenius axioms.