Simplicial Sets

class sage.categories.simplicial_sets.SimplicialSets(s=None)

Bases: sage.categories.category_singleton.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: sage.categories.category_with_axiom.CategoryWithAxiom

Category of finite simplicial sets.

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

class Homsets(category, *args)

Bases: sage.categories.homsets.HomsetsCategory

class Endset(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom

class ParentMethods
one()

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

EXAMPLES:

sage: T = simplicial_sets.Torus()
sage: Hom(T, T).identity()
Simplicial set endomorphism of Torus
  Defn: Identity map
class ParentMethods
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()
True
sage: C5 = groups.misc.MultiplicativeAbelian([5])
sage: simplicial_sets.ClassifyingSpace(C5).is_finite()
False
is_pointed()

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

EXAMPLES:

sage: from sage.homology.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: from sage.homology.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: sage.categories.category_with_axiom.CategoryWithAxiom

class Finite(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom

class ParentMethods
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: 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()
Finitely presented group < e0, e1 |  >
sage: S1.fat_wedge(4).homology()
{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: S1 = simplicial_sets.Sphere(1)
sage: RP2 = simplicial_sets.RealProjectiveSpace(2)
sage: X = S1.smash_product(RP2)
sage: X.homology(base_ring=GF(2))
{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)
sage: X = T.smash_product(S1)
sage: X.homology(reduced=False)
{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: from sage.homology.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
base_point()

Return this simplicial set’s base point

EXAMPLES:

sage: from sage.homology.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.homology.simplicial_set.Point() function to use a standard one-point space.

EXAMPLES:

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)
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)
sage: f.domain() == h.domain()
False

sage: C5 = groups.misc.MultiplicativeAbelian([5])
sage: BC5 = simplicial_sets.ClassifyingSpace(C5)
sage: BC5.base_point_map()
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: 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])
sage: BC2 = simplicial_sets.ClassifyingSpace(C2)
sage: BC2.connectivity()
0
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)
sage: eight = S1.wedge(S1)
sage: eight.fundamental_group() # free group on 2 generators
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()
sage: K = simplicial_sets.KleinBottle()
sage: X = T.disjoint_union(K)

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

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

Compute the fundamental group of some classifying spaces:

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

sage: Sigma3 = groups.permutation.Symmetric(3)
sage: BSigma3 = Sigma3.nerve()
sage: pi = BSigma3.fundamental_group(); pi
Finitely presented group < e0, e1 | e0^2, e1^3, (e0*e1^-1)^2 >
sage: pi.order()
6
sage: pi.is_abelian()
False
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: 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: S2 = simplicial_sets.Sphere(2)
sage: S3 = simplicial_sets.Sphere(3)
sage: (S2.wedge(S3)).is_simply_connected()
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])
sage: BC3 = simplicial_sets.ClassifyingSpace(C3)
sage: BC3.is_simply_connected()
False
class SubcategoryMethods
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]