braided

The free braided category, i.e. diagrams with braids.

Summary

Diagram	A braided diagram is a monoidal diagram with Braid boxes.
Box	A braided box is a monoidal box in a braided diagram.
Braid	The braiding of atomic types left and right.
Sum	A braided sum is a monoidal sum and a braided box.
Category	A braided category is a monoidal category with a method braid.
Functor	A braided functor is a monoidal functor that preserves braids.

Functions

hexagon

Take a factory for braids of atomic types and extend it recursively.

Axioms

Braids have their dagger as inverse, up to Diagram.simplify().

>>> x, y, z = map(Ty, "xyz")
>>> LHS = Braid(x, y) >> Braid(x, y)[::-1]
>>> RHS = Braid(y, x)[::-1] >> Braid(y, x)
>>> assert LHS.simplify() == Id(x @ y) == RHS.simplify()

>>> from discopy.drawing import Equation
>>> Equation(LHS, Id(x @ y), RHS).draw(
...     path='docs/_static/braided/inverse.png')

The hexagon equations hold on the nose.

>>> left_hexagon = Braid(x, y) @ z >> y @ Braid(x, z)
>>> assert left_hexagon == Diagram.braid(x, y @ z)
>>> right_hexagon = x @ Braid(y, z) >> Braid(x, z) @ y
>>> assert right_hexagon == Diagram.braid(x @ y, z)

>>> Equation(left_hexagon, right_hexagon, symbol='').draw(
...     space=2, path='docs/_static/braided/hexagons.png')