Source code for discopy.compact

# -*- coding: utf-8 -*-

"""
The free compact category, i.e. diagrams with swaps, cups and caps.

Summary
-------

.. autosummary::
    :template: class.rst
    :nosignatures:
    :toctree:

    Diagram
    Box
    Cup
    Cap
    Swap
    Category
    Functor

Axioms
------

>>> from discopy.drawing import Equation
>>> Diagram.use_hypergraph_equality = True
>>> x, y = Ty('x'), Ty('y')

* Snake equations:

>>> snake = Equation(Id(x.l).transpose(left=True), Id(x), Id(x.r).transpose())
>>> assert snake
>>> snake.draw(path="docs/_static/compact/snake.png")

.. image:: /_static/compact/snake.png
    :align: center

* Yanking (a.k.a. Reidemeister move 1):

>>> right_loop = x @ Cap(x, x.r) >> Swap(x, x) @ x.r >> x @ Cup(x, x.r)
>>> left_loop = Cap(x.r, x) @ x >> x.r @ Swap(x, x) >> Cup(x.r, x) @ x
>>> yanking = Equation(left_loop, Id(x), right_loop)
>>> assert yanking
>>> yanking.draw(path="docs/_static/compact/yanking.png")

.. image:: /_static/compact/yanking.png
    :align: center

>>> cap_yanking = Equation(Cap(x, x.r) >> Swap(x, x.r), Cap(x.r, x))
>>> cup_yanking = Equation(Swap(x, x.r) >> Cup(x.r, x), Cup(x, x.r))
>>> assert cap_yanking and cup_yanking
>>> Equation(cap_yanking, cup_yanking, symbol='', space=1).draw(
...     figsize=(6, 3), path="docs/_static/compact/yanking_cup_and_cap.png")

.. image:: /_static/compact/yanking_cup_and_cap.png
    :align: center

* Coherence:

>>> assert Diagram.caps(x @ y, y.r @ x.r)\\
...     == Cap(x, x.r) @ Cap(y, y.r) >> x @ Diagram.swap(x.r, y @ y.r)

>>> Diagram.use_hypergraph_equality = False
"""

from discopy import symmetric, ribbon
from discopy.cat import factory
from discopy.pivotal import Ob, Ty  # noqa: F401


[docs] @factory class Diagram(symmetric.Diagram, ribbon.Diagram): """ A compact diagram is a symmetric diagram and a ribbon diagram. Parameters: inside(Layer) : The layers of the diagram. dom (pivotal.Ty) : The domain of the diagram, i.e. its input. cod (pivotal.Ty) : The codomain of the diagram, i.e. its output. """ ty_factory = Ty trace_factory = ribbon.Diagram.trace_factory
[docs] class Box(symmetric.Box, ribbon.Box, Diagram): """ A compact box is a symmetric and ribbon box in a compact diagram. Parameters: name (str) : The name of the box. dom (pivotal.Ty) : The domain of the box, i.e. its input. cod (pivotal.Ty) : The codomain of the box, i.e. its output. """ __ambiguous_inheritance__ = (symmetric.Box, ribbon.Box, )
[docs] class Cup(ribbon.Cup, Box): """ A compact cup is a ribbon cup in a compact diagram. Parameters: left (pivotal.Ty) : The atomic type. right (pivotal.Ty) : Its adjoint. """ __ambiguous_inheritance__ = (ribbon.Cup, )
[docs] class Cap(ribbon.Cap, Box): """ A compact cap is a ribbon cap in a compact diagram. Parameters: left (pivotal.Ty) : The atomic type. right (pivotal.Ty) : Its adjoint. """ __ambiguous_inheritance__ = (ribbon.Cap, )
[docs] class Swap(symmetric.Swap, ribbon.Braid, Box): """ A compact swap is a symmetric swap and a ribbon braid. Parameters: left (pivotal.Ty) : The type on the top left and bottom right. right (pivotal.Ty) : The type on the top right and bottom left. """ __ambiguous_inheritance__ = (symmetric.Swap, ribbon.Braid, )
[docs] class Category(symmetric.Category, ribbon.Category): """ A compact category is both a symmetric category and a ribbon category. Parameters: ob : The objects of the category, default is :class:`pivotal.Ty`. ar : The arrows of the category, default is :class:`Diagram`. """ ob, ar = Ty, Diagram
[docs] class Functor(symmetric.Functor, ribbon.Functor): """ A compact functor is both a symmetric functor and a ribbon functor. Parameters: ob (Mapping[pivotal.Ty, pivotal.Ty]) : Map from atomic :class:`pivotal.Ty` to :code:`cod.ob`. ar (Mapping[Box, Diagram]) : Map from :class:`Box` to :code:`cod.ar`. cod (Category) : The codomain of the functor. """ dom = cod = Category() def __call__(self, other): if isinstance(other, Swap): return symmetric.Functor.__call__(self, other) return ribbon.Functor.__call__(self, other)
class Hypergraph(symmetric.Hypergraph): category, functor = Category, Functor Id = Diagram.id Diagram.braid_factory = Swap Diagram.hypergraph_factory = Hypergraph Diagram.cup_factory, Diagram.cap_factory = Cup, Cap