Ty

class discopy.monoidal.Ty(*objects)

Bases: Ob

Implements a type as a list of discopy.cat.Ob, used as domain and codomain for monoidal.Diagram. Types are the free monoid on objects with product @ and unit Ty().

Parameters:

objects (list of discopy.cat.Ob) – List of objects or object names.

Important

Elements that are not instance of discopy.cat.Ob are implicitly taken to be the name of an object, i.e. Ty('x', 'y') == Ty(Ob('x'), Ob('y')).

Notes

We can check the axioms for a monoid.

>>> x, y, z, unit = Ty('x'), Ty('y'), Ty('z'), Ty()
>>> assert x @ unit == x == unit @ x
>>> assert (x @ y) @ z == x @ y @ z == x @ (y @ z)

property objects

List of objects forming a type.

Note

A type may be sliced into subtypes.

>>> t = Ty('x', 'y', 'z')
>>> assert t[0] == Ob('x')
>>> assert t[:1] == Ty('x')
>>> assert t[1:] == Ty('y', 'z')

tensor(*others)

Returns the tensor of types, i.e. the concatenation of their lists of objects. This is called with the binary operator @.

>>> Ty('x') @ Ty('y', 'z')
Ty('x', 'y', 'z')

Parameters:

other (monoidal.Ty) –

Returns:

t – such that t.objects == self.objects + other.objects.

Return type:

monoidal.Ty

Note

We can take the sum of a list of type, specifying the unit Ty().

>>> types = Ty('x'), Ty('y'), Ty('z')
>>> Ty().tensor(*types)
Ty('x', 'y', 'z')

We can take the exponent of a type by any natural number.

>>> Ty('x') ** 3
Ty('x', 'x', 'x')

count(obj)

Counts the occurrence of a given object.

Parameters:

obj (Ty or Ob) – either a type of length 1 or an object

Returns:

n – such that n == self.objects.count(ob).

Return type:

int

Examples

>>> x = Ty('x')
>>> xs = x ** 5
>>> assert xs.count(x) == xs.count(x[0]) == xs.objects.count(Ob('x'))

static upgrade(old)

Allows class inheritance for tensor and __getitem__.

downgrade()

Downgrades to discopy.monoidal.Ty.