shapes.Simplex

class rewalt.shapes.Simplex

Bases: Shape

Subclass of Shape for oriented simplices.

The methods of this class provide a full implementation of the category of simplices, which is generated by the face and degeneracy maps between simplices one dimension apart.

Use Shape.simplex() to construct.

Examples

We create a 1-simplex (arrow), a 2-simplex (triangle), and a 3-simplex (tetrahedron).

>>> arrow = Shape.simplex(1)
>>> triangle = Shape.simplex(2)
>>> tetra = Shape.simplex(3)

We can then check some of the simplicial relations between degeneracy and face maps.

>>> map1 = triangle.simplex_degeneracy(2).then(
...     arrow.simplex_degeneracy(1))
>>> map2 = triangle.simplex_degeneracy(1).then(
...     arrow.simplex_degeneracy(1))
>>> assert map1 == map2

>>> map3 = tetra.simplex_face(2).then(
...     triangle.simplex_degeneracy(2))
>>> assert map3 == triangle.id()

>>> map4 = tetra.simplex_face(0).then(
...     triangle.simplex_degeneracy(2))
>>> map5 = arrow.simplex_degeneracy(1).then(
...     triangle.simplex_face(0))
>>> assert map4 == map5

Methods

`simplex_degeneracy`(k)	Returns one of the collapse (degeneracy) maps of the simplex one dimension higher.
`simplex_face`(k)	Returns one of the face inclusion maps of the simplex.

simplex_face(k)

Returns one of the face inclusion maps of the simplex.

Parameters: k (int) – The index of the face map, ranging from 0 to self.dim.
Returns: simplex_face – The face map.
Return type: ShapeMap
Raises: ValueError – If the index is out of range.

simplex_degeneracy(k)

Returns one of the collapse (degeneracy) maps of the simplex one dimension higher.

Parameters: k (int) – The index of the degeneracy map, ranging from 0 to self.dim.
Returns: simplex_degeneracy – The degeneracy map.
Return type: ShapeMap
Raises: ValueError – If the index is out of range.