# -*- 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