Simplicial Sets#

class sage.categories.simplicial_sets.SimplicialSets#

Bases: Category_singleton

The category of simplicial sets.

A simplicial set \(X\) is a collection of sets \(X_i\), indexed by the non-negative integers, together with maps

\[\begin{split}d_i: X_n \to X_{n-1}, \quad 0 \leq i \leq n \quad \text{(face maps)} \\ s_j: X_n \to X_{n+1}, \quad 0 \leq j \leq n \quad \text{(degeneracy maps)}\end{split}\]

satisfying the simplicial identities:

\[\begin{split}d_i d_j &= d_{j-1} d_i \quad \text{if } i<j \\ d_i s_j &= s_{j-1} d_i \quad \text{if } i<j \\ d_j s_j &= 1 = d_{j+1} s_j \\ d_i s_j &= s_{j} d_{i-1} \quad \text{if } i>j+1 \\ s_i s_j &= s_{j+1} s_{i} \quad \text{if } i \leq j\end{split}\]

Morphisms are sequences of maps \(f_i : X_i \to Y_i\) which commute with the face and degeneracy maps.

EXAMPLES:

sage: from sage.categories.simplicial_sets import SimplicialSets
sage: C = SimplicialSets(); C
Category of simplicial sets
class Finite(base_category)#

Bases: CategoryWithAxiom

Category of finite simplicial sets.

The objects are simplicial sets with finitely many non-degenerate simplices.

class Homsets(category, *args)#

Bases: HomsetsCategory

class Endset(base_category)#

Bases: CategoryWithAxiom

class ParentMethods#

Bases: object

one()#

Return the identity morphism in \(\operatorname{Hom}(S, S)\).

EXAMPLES:

sage: T = simplicial_sets.Torus()                               # needs sage.graphs
sage: Hom(T, T).identity()                                      # needs sage.graphs
Simplicial set endomorphism of Torus
  Defn: Identity map
class ParentMethods#

Bases: object

is_finite()#

Return True if this simplicial set is finite, i.e., has a finite number of nondegenerate simplices.

EXAMPLES:

sage: simplicial_sets.Torus().is_finite()                               # needs sage.graphs
True
sage: C5 = groups.misc.MultiplicativeAbelian([5])                       # needs sage.graphs sage.groups
sage: simplicial_sets.ClassifyingSpace(C5).is_finite()                  # needs sage.graphs sage.groups
False
is_pointed()#

Return True if this simplicial set is pointed, i.e., has a base point.

EXAMPLES:

sage: # needs sage.graphs
sage: from sage.topology.simplicial_set import AbstractSimplex, SimplicialSet
sage: v = AbstractSimplex(0)
sage: w = AbstractSimplex(0)
sage: e = AbstractSimplex(1)
sage: X = SimplicialSet({e: (v, w)})
sage: Y = SimplicialSet({e: (v, w)}, base_point=w)
sage: X.is_pointed()
False
sage: Y.is_pointed()
True
set_base_point(point)#

Return a copy of this simplicial set in which the base point is set to point.

INPUT:

  • point – a 0-simplex in this simplicial set

EXAMPLES:

sage: # needs sage.graphs
sage: from sage.topology.simplicial_set import AbstractSimplex, SimplicialSet
sage: v = AbstractSimplex(0, name='v_0')
sage: w = AbstractSimplex(0, name='w_0')
sage: e = AbstractSimplex(1)
sage: X = SimplicialSet({e: (v, w)})
sage: Y = SimplicialSet({e: (v, w)}, base_point=w)
sage: Y.base_point()
w_0
sage: X_star = X.set_base_point(w)
sage: X_star.base_point()
w_0
sage: Y_star = Y.set_base_point(v)
sage: Y_star.base_point()
v_0
class Pointed(base_category)#

Bases: CategoryWithAxiom

class Finite(base_category)#

Bases: CategoryWithAxiom

class ParentMethods#

Bases: object

fat_wedge(n)#

Return the \(n\)-th fat wedge of this pointed simplicial set.

This is the subcomplex of the \(n\)-fold product \(X^n\) consisting of those points in which at least one factor is the base point. Thus when \(n=2\), this is the wedge of the simplicial set with itself, but when \(n\) is larger, the fat wedge is larger than the \(n\)-fold wedge.

EXAMPLES:

sage: # needs sage.graphs
sage: S1 = simplicial_sets.Sphere(1)
sage: S1.fat_wedge(0)
Point
sage: S1.fat_wedge(1)
S^1
sage: S1.fat_wedge(2).fundamental_group()                       # needs sage.groups
Finitely presented group < e0, e1 |  >
sage: S1.fat_wedge(4).homology()                                # needs sage.modules
{0: 0, 1: Z x Z x Z x Z, 2: Z^6, 3: Z x Z x Z x Z}
smash_product(*others)#

Return the smash product of this simplicial set with others.

INPUT:

  • others – one or several simplicial sets

EXAMPLES:

sage: # needs sage.graphs sage.groups
sage: S1 = simplicial_sets.Sphere(1)
sage: RP2 = simplicial_sets.RealProjectiveSpace(2)
sage: X = S1.smash_product(RP2)
sage: X.homology(base_ring=GF(2))                               # needs sage.modules
{0: Vector space of dimension 0 over Finite Field of size 2,
 1: Vector space of dimension 0 over Finite Field of size 2,
 2: Vector space of dimension 1 over Finite Field of size 2,
 3: Vector space of dimension 1 over Finite Field of size 2}

sage: T = S1.product(S1)                                        # needs sage.graphs sage.groups
sage: X = T.smash_product(S1)                                   # needs sage.graphs sage.groups
sage: X.homology(reduced=False)                                 # needs sage.graphs sage.groups sage.modules
{0: Z, 1: 0, 2: Z x Z, 3: Z}
unset_base_point()#

Return a copy of this simplicial set in which the base point has been forgotten.

EXAMPLES:

sage: # needs sage.graphs
sage: from sage.topology.simplicial_set import AbstractSimplex, SimplicialSet
sage: v = AbstractSimplex(0, name='v_0')
sage: w = AbstractSimplex(0, name='w_0')
sage: e = AbstractSimplex(1)
sage: Y = SimplicialSet({e: (v, w)}, base_point=w)
sage: Y.is_pointed()
True
sage: Y.base_point()
w_0
sage: Z = Y.unset_base_point()
sage: Z.is_pointed()
False
class ParentMethods#

Bases: object

base_point()#

Return this simplicial set’s base point

EXAMPLES:

sage: # needs sage.graphs
sage: from sage.topology.simplicial_set import AbstractSimplex, SimplicialSet
sage: v = AbstractSimplex(0, name='*')
sage: e = AbstractSimplex(1)
sage: S1 = SimplicialSet({e: (v, v)}, base_point=v)
sage: S1.is_pointed()
True
sage: S1.base_point()
*
base_point_map(domain=None)#

Return a map from a one-point space to this one, with image the base point.

This raises an error if this simplicial set does not have a base point.

INPUT:

  • domain – optional, default None. Use this to specify a particular one-point space as the domain. The default behavior is to use the sage.topology.simplicial_set.Point() function to use a standard one-point space.

EXAMPLES:

sage: # needs sage.graphs
sage: T = simplicial_sets.Torus()
sage: f = T.base_point_map(); f
Simplicial set morphism:
  From: Point
  To:   Torus
  Defn: Constant map at (v_0, v_0)
sage: S3 = simplicial_sets.Sphere(3)
sage: g = S3.base_point_map()
sage: f.domain() == g.domain()
True
sage: RP3 = simplicial_sets.RealProjectiveSpace(3)                  # needs sage.groups
sage: temp = simplicial_sets.Simplex(0)
sage: pt = temp.set_base_point(temp.n_cells(0)[0])
sage: h = RP3.base_point_map(domain=pt)                             # needs sage.groups
sage: f.domain() == h.domain()                                      # needs sage.groups
False

sage: C5 = groups.misc.MultiplicativeAbelian([5])                   # needs sage.graphs sage.groups
sage: BC5 = simplicial_sets.ClassifyingSpace(C5)                    # needs sage.graphs sage.groups
sage: BC5.base_point_map()                                          # needs sage.graphs sage.groups
Simplicial set morphism:
  From: Point
  To:   Classifying space of Multiplicative Abelian group isomorphic to C5
  Defn: Constant map at 1
connectivity(max_dim=None)#

Return the connectivity of this pointed simplicial set.

INPUT:

  • max_dim – specify a maximum dimension through which to check. This is required if this simplicial set is simply connected and not finite.

The dimension of the first nonzero homotopy group. If simply connected, this is the same as the dimension of the first nonzero homology group.

Warning

See the warning for the is_simply_connected() method.

The connectivity of a contractible space is +Infinity.

EXAMPLES:

sage: # needs sage.graphs sage.groups
sage: simplicial_sets.Sphere(3).connectivity()
2
sage: simplicial_sets.Sphere(0).connectivity()
-1
sage: K = simplicial_sets.Simplex(4)
sage: K = K.set_base_point(K.n_cells(0)[0])
sage: K.connectivity()
+Infinity
sage: X = simplicial_sets.Torus().suspension(2)
sage: X.connectivity()
2

sage: C2 = groups.misc.MultiplicativeAbelian([2])                   # needs sage.graphs sage.groups
sage: BC2 = simplicial_sets.ClassifyingSpace(C2)                    # needs sage.graphs sage.groups
sage: BC2.connectivity()                                            # needs sage.graphs sage.groups
0
cover(character)#

Return the cover of the simplicial set associated to a character of the fundamental group.

The character is represented by a dictionary, that assigns an element of a finite group to each nondegenerate 1-dimensional cell. It should correspond to an epimorphism from the fundamental group.

INPUT:

  • character – a dictionary

EXAMPLES:

sage: # needs sage.graphs sage.groups
sage: S1 = simplicial_sets.Sphere(1)
sage: W = S1.wedge(S1)
sage: G = CyclicPermutationGroup(3)
sage: (a, b) = W.n_cells(1)
sage: C = W.cover({a : G.gen(0), b : G.gen(0)^2})
sage: C.face_data()
{(*, ()): None,
 (*, (1,2,3)): None,
 (*, (1,3,2)): None,
 (sigma_1, ()): ((*, (1,2,3)), (*, ())),
 (sigma_1, ()): ((*, (1,3,2)), (*, ())),
 (sigma_1, (1,2,3)): ((*, (1,3,2)), (*, (1,2,3))),
 (sigma_1, (1,2,3)): ((*, ()), (*, (1,2,3))),
 (sigma_1, (1,3,2)): ((*, ()), (*, (1,3,2))),
 (sigma_1, (1,3,2)): ((*, (1,2,3)), (*, (1,3,2)))}
sage: C.homology(1)                                                 # needs sage.modules
Z x Z x Z x Z
sage: C.fundamental_group()
Finitely presented group < e0, e1, e2, e3 |  >
covering_map(character)#

Return the covering map associated to a character.

The character is represented by a dictionary that assigns an element of a finite group to each nondegenerate 1-dimensional cell. It should correspond to an epimorphism from the fundamental group.

INPUT:

  • character – a dictionary

EXAMPLES:

sage: # needs sage.graphs sage.groups
sage: S1 = simplicial_sets.Sphere(1)
sage: W = S1.wedge(S1)
sage: G = CyclicPermutationGroup(3)
sage: a, b = W.n_cells(1)
sage: C = W.covering_map({a : G.gen(0), b : G.one()}); C
Simplicial set morphism:
  From: Simplicial set with 9 non-degenerate simplices
  To:   Wedge: (S^1 v S^1)
  Defn: [(*, ()), (*, (1,2,3)), (*, (1,3,2)), (sigma_1, ()),
         (sigma_1, ()), (sigma_1, (1,2,3)), (sigma_1, (1,2,3)),
         (sigma_1, (1,3,2)), (sigma_1, (1,3,2))]
        --> [*, *, *, sigma_1, sigma_1, sigma_1, sigma_1, sigma_1, sigma_1]
sage: C.domain()
Simplicial set with 9 non-degenerate simplices
sage: C.domain().face_data()
{(*, ()): None,
 (*, (1,2,3)): None,
 (*, (1,3,2)): None,
 (sigma_1, ()): ((*, (1,2,3)), (*, ())),
 (sigma_1, ()): ((*, ()), (*, ())),
 (sigma_1, (1,2,3)): ((*, (1,3,2)), (*, (1,2,3))),
 (sigma_1, (1,2,3)): ((*, (1,2,3)), (*, (1,2,3))),
 (sigma_1, (1,3,2)): ((*, ()), (*, (1,3,2))),
 (sigma_1, (1,3,2)): ((*, (1,3,2)), (*, (1,3,2)))}
fundamental_group(simplify=True)#

Return the fundamental group of this pointed simplicial set.

INPUT:

  • simplify (bool, optional True) – if False, then return a presentation of the group in terms of generators and relations. If True, the default, simplify as much as GAP is able to.

Algorithm: we compute the edge-path group – see Section 19 of [Kan1958] and Wikipedia article Fundamental_group. Choose a spanning tree for the connected component of the 1-skeleton containing the base point, and then the group’s generators are given by the non-degenerate edges. There are two types of relations: \(e=1\) if \(e\) is in the spanning tree, and for every 2-simplex, if its faces are \(e_0\), \(e_1\), and \(e_2\), then we impose the relation \(e_0 e_1^{-1} e_2 = 1\), where we first set \(e_i=1\) if \(e_i\) is degenerate.

EXAMPLES:

sage: S1 = simplicial_sets.Sphere(1)                                # needs sage.graphs
sage: eight = S1.wedge(S1)                                          # needs sage.graphs
sage: eight.fundamental_group()  # free group on 2 generators       # needs sage.graphs sage.groups
Finitely presented group < e0, e1 |  >

The fundamental group of a disjoint union of course depends on the choice of base point:

sage: T = simplicial_sets.Torus()                                   # needs sage.graphs
sage: K = simplicial_sets.KleinBottle()                             # needs sage.graphs
sage: X = T.disjoint_union(K)                                       # needs sage.graphs

sage: # needs sage.graphs
sage: X_0 = X.set_base_point(X.n_cells(0)[0])
sage: X_0.fundamental_group().is_abelian()                          # needs sage.groups
True
sage: X_1 = X.set_base_point(X.n_cells(0)[1])
sage: X_1.fundamental_group().is_abelian()                          # needs sage.groups
False

sage: RP3 = simplicial_sets.RealProjectiveSpace(3)                  # needs sage.graphs sage.groups
sage: RP3.fundamental_group()                                       # needs sage.graphs sage.groups
Finitely presented group < e | e^2 >

Compute the fundamental group of some classifying spaces:

sage: C5 = groups.misc.MultiplicativeAbelian([5])                   # needs sage.graphs sage.groups
sage: BC5 = C5.nerve()                                              # needs sage.graphs sage.groups
sage: BC5.fundamental_group()                                       # needs sage.graphs sage.groups
Finitely presented group < e0 | e0^5 >

sage: # needs sage.graphs sage.groups
sage: Sigma3 = groups.permutation.Symmetric(3)
sage: BSigma3 = Sigma3.nerve()
sage: pi = BSigma3.fundamental_group(); pi
Finitely presented group < e1, e2 | e2^2, e1^3, (e2*e1)^2 >
sage: pi.order()
6
sage: pi.is_abelian()
False

The sphere has a trivial fundamental group:

sage: S2 = simplicial_sets.Sphere(2)                                # needs sage.graphs
sage: S2.fundamental_group()                                        # needs sage.graphs sage.groups
Finitely presented group <  |  >
is_simply_connected()#

Return True if this pointed simplicial set is simply connected.

Warning

Determining simple connectivity is not always possible, because it requires determining when a group, as given by generators and relations, is trivial. So this conceivably may give a false negative in some cases.

EXAMPLES:

sage: # needs sage.graphs sage.groups
sage: T = simplicial_sets.Torus()
sage: T.is_simply_connected()
False
sage: T.suspension().is_simply_connected()
True
sage: simplicial_sets.KleinBottle().is_simply_connected()
False

sage: # needs sage.graphs
sage: S2 = simplicial_sets.Sphere(2)
sage: S3 = simplicial_sets.Sphere(3)
sage: (S2.wedge(S3)).is_simply_connected()                          # needs sage.groups
True
sage: X = S2.disjoint_union(S3)
sage: X = X.set_base_point(X.n_cells(0)[0])
sage: X.is_simply_connected()
False

sage: C3 = groups.misc.MultiplicativeAbelian([3])                   # needs sage.graphs sage.groups
sage: BC3 = simplicial_sets.ClassifyingSpace(C3)                    # needs sage.graphs sage.groups
sage: BC3.is_simply_connected()                                     # needs sage.graphs sage.groups
False
twisted_chain_complex(twisting_operator=None, dimensions=None, augmented=False, cochain=False, verbose=False, subcomplex=None, check=False)#

Return the normalized chain complex twisted by some operator.

A twisting operator is a map from the set of simplices to some algebra. The differentials are then twisted by this operator.

INPUT:

  • twisting_operator – a dictionary, associating the twist of each simplex. If it is not given, the canonical one (associated to the laurent polynomial ring abelianization of the fundamental group, ignoring torsion) is used.

  • dimensions – if None, compute the chain complex in all dimensions. If a list or tuple of integers, compute the chain complex in those dimensions, setting the chain groups in all other dimensions to zero.

  • augmented (optional, default False) – if True, return the augmented chain complex (that is, include a class in dimension \(-1\) corresponding to the empty cell).

  • cochain (optional, default False) – if True, return the cochain complex (that is, the dual of the chain complex).

  • verbose (optional, default False) – ignored.

  • subcomplex (optional, default None) – if present, compute the chain complex relative to this subcomplex.

  • check (optional, default False) – If True, make sure that the chain complex is actually a chain complex: the differentials are composable and their product is zero.

The normalized chain complex of a simplicial set is isomorphic to the chain complex obtained by modding out by degenerate simplices, and the latter is what is actually constructed here.

EXAMPLES:

sage: W = simplicial_sets.Sphere(1).wedge(simplicial_sets.Sphere(2))
sage: W.nondegenerate_simplices()
[*, sigma_1, sigma_2]
sage: s1 = W.nondegenerate_simplices()[1]
sage: L.<t> = LaurentPolynomialRing(QQ)
sage: tw = {s1:t}
sage: ChC = W.twisted_chain_complex(tw)
sage: ChC.differential(1)
[-1 + t]
sage: ChC.differential(2)
[0]

sage: X = simplicial_sets.Torus()
sage: C = X.twisted_chain_complex()
sage: C.differential(1)
[      f3 - 1 f2*f3^-1 - 1       f2 - 1]
sage: C.differential(2)
[       1 f2*f3^-1]
[      f3        1]
[      -1       -1]
sage: C.differential(3)
[]

sage: Y = simplicial_sets.RealProjectiveSpace(2)
sage: C = Y.twisted_chain_complex()
sage: C.differential(1)
[-1 + F1]
sage: C.differential(2)
[1 + F1]
sage: C.differential(3)
[]
twisted_homology(n, reduced=False)#

The \(n\)-th twisted homology module of the simplicial set with respect to the abelianization of the fundamental_group.

It is a module over a polynomial ring, including relations to make some variables the multiplicative inverses of others.

INPUT:

  • n - a positive integer.

  • reduced - (default: False) if set to True, the presentation matrix will be reduced.

EXAMPLES:

sage: X = simplicial_sets.Sphere(1).wedge(simplicial_sets.Sphere(2))
sage: X.twisted_homology(1)
Quotient module by Submodule of Ambient free module of rank 0 over the integral domain Multivariate Polynomial Ring in f1, f1inv over Integer Ring
Generated by the rows of the matrix:
[]
sage: X.twisted_homology(2)
Quotient module by Submodule of Ambient free module of rank 1 over the integral domain Multivariate Polynomial Ring in f1, f1inv over Integer Ring
Generated by the rows of the matrix:
[f1*f1inv - 1]

sage: Y = simplicial_sets.Torus()
sage: Y.twisted_homology(1)
Quotient module by Submodule of Ambient free module of rank 5 over the integral domain Multivariate Polynomial Ring in f2, f2inv, f3, f3inv over Integer Ring
Generated by the rows of the matrix:
[           1            0            0            0            0]
[           0            1            0            0            0]
[           0            0            1            0            0]
[           0            0            0            1            0]
[           0            0            0            0            1]
[f2*f2inv - 1            0            0            0            0]
[           0 f2*f2inv - 1            0            0            0]
[           0            0 f2*f2inv - 1            0            0]
[           0            0            0 f2*f2inv - 1            0]
[           0            0            0            0 f2*f2inv - 1]
[f3*f3inv - 1            0            0            0            0]
[           0 f3*f3inv - 1            0            0            0]
[           0            0 f3*f3inv - 1            0            0]
[           0            0            0 f3*f3inv - 1            0]
[           0            0            0            0 f3*f3inv - 1]
sage: Y.twisted_homology(2)
Quotient module by Submodule of Ambient free module of rank 0 over the integral domain Multivariate Polynomial Ring in f2, f2inv, f3, f3inv over Integer Ring
Generated by the rows of the matrix:
[]
sage: Y.twisted_homology(1, reduced=True)
Quotient module by Submodule of Ambient free module of rank 5 over the integral domain Multivariate Polynomial Ring in f2, f2inv, f3, f3inv over Integer Ring
Generated by the rows of the matrix:
[1 0 0 0 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
universal_cover()#

Return the universal cover of the simplicial set. The fundamental group must be finite in order to ensure that the universal cover is a simplicial set of finite type.

EXAMPLES:

sage: # needs sage.groups
sage: RP3 = simplicial_sets.RealProjectiveSpace(3)
sage: C = RP3.universal_cover(); C
Simplicial set with 8 non-degenerate simplices
sage: C.face_data()
{(1, 1): None,
 (1, e): None,
 (f, 1): ((1, e), (1, 1)),
 (f, e): ((1, 1), (1, e)),
 (f * f, 1): ((f, e), s_0 (1, 1), (f, 1)),
 (f * f, e): ((f, 1), s_0 (1, e), (f, e)),
 (f * f * f, 1): ((f * f, e), s_0 (f, 1), s_1 (f, 1), (f * f, 1)),
 (f * f * f, e): ((f * f, 1), s_0 (f, e), s_1 (f, e), (f * f, e))}
sage: C.fundamental_group()
Finitely presented group <  |  >
universal_cover_map()#

Return the universal covering map of the simplicial set.

It requires the fundamental group to be finite.

EXAMPLES:

sage: RP2 = simplicial_sets.RealProjectiveSpace(2)                  # needs sage.groups
sage: phi = RP2.universal_cover_map(); phi                          # needs sage.groups
Simplicial set morphism:
  From: Simplicial set with 6 non-degenerate simplices
  To:   RP^2
  Defn: [(1, 1), (1, e), (f, 1), (f, e), (f * f, 1), (f * f, e)]
        --> [1, 1, f, f, f * f, f * f]
sage: phi.domain().face_data()                                      # needs sage.groups
    {(1, 1): None,
     (1, e): None,
     (f, 1): ((1, e), (1, 1)),
     (f, e): ((1, 1), (1, e)),
     (f * f, 1): ((f, e), s_0 (1, 1), (f, 1)),
     (f * f, e): ((f, 1), s_0 (1, e), (f, e))}
class SubcategoryMethods#

Bases: object

Pointed()#

A simplicial set is pointed if it has a distinguished base point.

EXAMPLES:

sage: from sage.categories.simplicial_sets import SimplicialSets
sage: SimplicialSets().Pointed().Finite()
Category of finite pointed simplicial sets
sage: SimplicialSets().Finite().Pointed()
Category of finite pointed simplicial sets
super_categories()#

EXAMPLES:

sage: from sage.categories.simplicial_sets import SimplicialSets
sage: SimplicialSets().super_categories()
[Category of sets]