comch.simplicial: Simplicial tensor products

comch.simplicial.Simplex(iterable)	A simplex \((v_0, \dots, v_n)\).
comch.simplicial.SimplicialElement([data, ...])	Elements in an iterated tensor product of the chains on the infinite simplex.
comch.simplicial.Simplicial()	Produces simplicial elements of interest.

class comch.simplicial.Simplex(iterable)

A simplex \((v_0, \dots, v_n)\).

A simplex is a finite non-decreasing tuple of non-negative integers. We identity these with faces of the infinite simplex \(\Delta^\infty\).

__init__(iterable)

Initializes self.

Parameters:

interable (:class:'iterable') – Used to create a tuple of int.

Example

>>> print(Simplex((1,2,4)))
(1,2,4)

property dimension

The dimension of self.

Defined as the length of the tuple minus one.

Returns:

The dimension of self.

Return type:

int

Example

>>> Simplex((1,3,4,5)).dimension
3

face(i)

The i-th face of self.

Obtained by removing the i-th entry of self.

Returns:

The i-th face of self.

Return type:

comch.simplicial.Simplex

Example

>>> Simplex((1,3,4,5)).face(2)
(1, 3, 5)

degeneracy(i)

The i-th degeneracy of self.

Obtained by repeating the i-th entry of the tuple.

Returns:

The i-th face of self.

Return type:

comch.simplicial.Simplex

Example

>>> Simplex((1,3,4,5)).degeneracy(2)
(1, 3, 4, 4, 5)

coface(i)

The i-th coface of self.

Obtained by adding 1 to each j-th entries with j greater or equal to i.

Returns:

The i-th coface of self.

Return type:

comch.simplicial.Simplex

Example

>>> Simplex((1,3,4,5)).coface(2)
(1, 4, 5, 6)

codegeneracy(i)

The i-th codegeneracy of self.

Obtained by subtracting 1 from each j-th entries with j greater than i.

Returns:

The i-th codegeneracy of self.

Return type:

comch.simplicial.Simplex

Example

>>> Simplex((1,3,4,5)).codegeneracy(2)
(1, 2, 3, 4)

is_degenerate()

Returns True if self is degenerate and False if not.

A simplex is degenerate if it is empty or if contains equal consecutive values.

Returns:

True if self is degenerate and False if not.

Return type:

bool

Example

>>> Simplex(()).is_degenerate()
True
>>> Simplex((1,1,2)).is_degenerate()
True

is_nondegenerate()

Returns True if self is nondegenerate and False if not.

A simplex is nondegenerate if it is not empty and contains no equal consecutive values.

Returns:

True if self is nondegenerate and False if not.

Return type:

bool

Example

>>> Simplex((1,2,5)).is_nondegenerate()
True

class comch.simplicial.SimplicialElement(data=None, dimension=None, torsion=None)

Elements in an iterated tensor product of the chains on the infinite simplex.

The chains on the infinite simplex \(C = C_\bullet(\Delta^\infty; R)\) is the differential graded module \(C\) with degree-\(n\) part \(C_n\) freely generated as an \(R\)-module by simplices of dimension \(n\), and differential on these given by the sum of its faces with alternating signs. Explicitly,

\[\partial (v_0, \dots, v_n) = \sum_{i=0}^n (v_0, \dots, \widehat{v}_i, \dots, v_d).\]

The degree-\(n\) part of the tensor product \(C^{\otimes r}\) and its differential are recursively defined by

\[(C^{\otimes r})_n = \bigoplus_{i+j=n} C_i \otimes (C^{\otimes r})_n\]

and

\[\partial(c_1 \otimes c_2 \otimes \cdots \otimes c_r) = (\partial c_1) \otimes c_2 \otimes \cdots \otimes c_r + (-1)^{|c_1|} c_1 \otimes \partial (c_2 \otimes \cdots \otimes c_r).\]

torsion

The non-neg int \(n\) of the ring \(\mathbb Z/n \mathbb Z\).

Type:

int

dimension

NOT SURE IF NEEDED.

Type:

int non-negative

__init__(data=None, dimension=None, torsion=None)

Initializes self.

Parameters:

• data (int or None, default: None) – Dictionary representing a linear cobination of basis elements. Items in the dictionary correspond to basis_element: coefficient pairs. Each basis_element must create a tuple of comch.simplicial.Simplex and coefficient must be an int.

• dimension (int) – NOT SURE IF NEEDED.

• torsion (int) – The non-neg int \(n\) of the ring \(\mathbb Z/n \mathbb Z\).

Example

>>> x = SimplicialElement({((0,), (0, 1, 2)): 1,                                   ((0, 1), (1, 2)): -1,                                   ((0, 1, 2), (2,)): 1})
>>> print(x)
((0,),(0,1,2)) - ((0,1),(1,2)) + ((0,1,2),(2,))

property arity

Arity of self.

Defined as None if self is not homogeneous. The arity of a basis element is defined as the number of tensor factors making it.

Returns:

The length of the keys of self or None if not well defined.

Return type:

int positive or None.

Example

>>> x = SimplicialElement({((0,), (0, 1, 2)): 1})
>>> x.arity
2

property degree

Degree of self.

Defined as None if self is not homogeneous. The degree of a basis element agrees with the sum of the dimension of the simplices making it.

Returns:

The sum of the dimensions of the simplices of every key of self or None if not well defined.

Return type:

int positive or None.

Example

>>> x = SimplicialElement({((0,), (0, 1, 2)): 1})
>>> x.degree
2

boundary()

Boundary of self.

As defined in the class’s docstring.

Returns:

The boundary of self as an element in a tensor product of differential graded modules.

Return type:

comch.simplicical.SimplicialElement

Example

>>> x = SimplicialElement({((0, 1), (1, 2)): 1})
>>> print(x.boundary())
((1,),(1,2)) - ((0,),(1,2)) - ((0,1),(2,)) + ((0,1),(1,))

__rmul__(other)

Left action: other * self

Left multiplication by a symmetric group element or an integer. Defined up to signs on basis elements by permuting the tensor factor.

Parameters:

other (int or comch.symmetric.SymmetricElement.) – The symmetric ring element left acting on self.

Returns:

The product: other * self with Koszul’s sign convention.

Return type:

comch.simplicial.SimplicialElement

Example

>>> x = SimplicialElement({((0, 1), (1, 2)): 1})
>>> t = SymmetricRingElement({(2, 1): 1})
>>> print(t * x)
- ((1,2),(0,1))
>>> print(3 * x)
3((0,1),(1,2))

iterated_diagonal(times=1, coord=1)

Iterated Alexander-Whitney diagonal applied at a specific tensor factor.

The AW diagonal is the chain map \(\Delta \colon C \to C \otimes C\) defined on the chains of the infinite simplex by the formula

\[\Delta((v_0, dots, v_n)) = \sum_{i=0}^n (v_0, dots, v_i) \otimes (v_i, dots, v_n).\]

It is coassociative, \((\Delta \otimes \mathrm{id}) \Delta = (\mathrm{id} \otimes \Delta) \Delta\), so it has a well defined iteration \(\Delta^k\), and for every \(i \in \{1, \dots, r\}\), there is map \(C^{\otimes r} \to C^{\otimes k+r}\) sending \((x_1 \otimes \cdots \otimes x_n)\) to \((x_1 \otimes \cdots \otimes \Delta^k(k_i) \cdots \otimes x_n)\).

Parameters:

• times (int) – The number of times the AW diagonal is composed with itself.

• coord (int) – The tensor position on which the iterated diagonal acts.

Returns:

The action of the iterated AW diagonal on self.

Return type:

comch.simplicial.SimplicialElement

Example

>>> x = SimplicialElement({((0, 1, 2), ): 1})
>>> print(x.iterated_diagonal())
((0,),(0,1,2)) + ((0,1),(1,2)) + ((0,1,2),(2,))

one_reduced()

Returns the 1-reduction of self.

The 1-reduction map is the map induced by the collapse of the 1-skeleton of the infinite simplex.

Returns:

The preferred representative of self.

Return type:

comch.simplicial.SimplicialElement

Example

>>> x = SimplicialElement({((1,2), (2,3,4)): 1})
>>> print(x.one_reduced())
0

preferred_rep()

Preferred representative of self.

Removes pairs basis element: coefficient which satisfy either of: 1) The basis element has a degenerate tensor factor, or 2) the coefficient is 0.

Returns:

The preferred representative of self.

Return type:

comch.simplicial.SimplicialElement

Example

>>> print(SimplicialElement({((1,3), (1,1)): 1}))
0

class comch.simplicial.Simplicial

Produces simplicial elements of interest.

static standard_element(n, times=1, torsion=None)

The chain represented by the simplex \((0, \dots, n)\).

Parameters:

• n (int) – The dimension of the standard simplex considered.

• times (int) – The number of tensor copies.

• torsion (int) – The non-neg int \(n\) of the ring \(\mathbb Z/n \mathbb Z\).

Examples

>>> print(Simplicial.standard_element(3, 2))
((0,1,2,3),(0,1,2,3))

static basis(n, torsion=None)

Iterator of all basis elements in the chain complex of a n-simplex.

Parameters:

• n (int) – The dimension of the standard simplex considered.

• torsion (int) – The non-neg int \(n\) of the ring \(\mathbb Z/n \mathbb Z\).

Examples

>>> print([str(b) for b in Simplicial.basis(1)])
['((0,),)', '((1,),)', '((0,1),)']