Simplicial Sets#
- class sage.categories.simplicial_sets.SimplicialSets(s=None)#
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
- 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() # optional - sage.graphs True sage: C5 = groups.misc.MultiplicativeAbelian([5]) # optional - sage.graphs sage.groups sage: simplicial_sets.ClassifyingSpace(C5).is_finite() # optional - sage.graphs sage.groups False
- is_pointed()#
Return
True
if this simplicial set is pointed, i.e., has a base point.EXAMPLES:
sage: from sage.topology.simplicial_set import AbstractSimplex, SimplicialSet # optional - sage.graphs sage: v = AbstractSimplex(0) # optional - sage.graphs sage: w = AbstractSimplex(0) # optional - sage.graphs sage: e = AbstractSimplex(1) # optional - sage.graphs sage: X = SimplicialSet({e: (v, w)}) # optional - sage.graphs sage: Y = SimplicialSet({e: (v, w)}, base_point=w) # optional - sage.graphs sage: X.is_pointed() # optional - sage.graphs False sage: Y.is_pointed() # optional - sage.graphs 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.topology.simplicial_set import AbstractSimplex, SimplicialSet # optional - sage.graphs sage: v = AbstractSimplex(0, name='v_0') # optional - sage.graphs sage: w = AbstractSimplex(0, name='w_0') # optional - sage.graphs sage: e = AbstractSimplex(1) # optional - sage.graphs sage: X = SimplicialSet({e: (v, w)}) # optional - sage.graphs sage: Y = SimplicialSet({e: (v, w)}, base_point=w) # optional - sage.graphs sage: Y.base_point() # optional - sage.graphs w_0 sage: X_star = X.set_base_point(w) # optional - sage.graphs sage: X_star.base_point() # optional - sage.graphs w_0 sage: Y_star = Y.set_base_point(v) # optional - sage.graphs sage: Y_star.base_point() # optional - sage.graphs 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: S1 = simplicial_sets.Sphere(1) # optional - sage.graphs sage: S1.fat_wedge(0) # optional - sage.graphs Point sage: S1.fat_wedge(1) # optional - sage.graphs S^1 sage: S1.fat_wedge(2).fundamental_group() # optional - sage.graphs sage.groups Finitely presented group < e0, e1 | > sage: S1.fat_wedge(4).homology() # optional - sage.graphs 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: S1 = simplicial_sets.Sphere(1) # optional - sage.graphs sage: RP2 = simplicial_sets.RealProjectiveSpace(2) # optional - sage.graphs sage.groups sage: X = S1.smash_product(RP2) # optional - sage.graphs sage.groups sage: X.homology(base_ring=GF(2)) # optional - sage.graphs sage.groups sage.modules sage.rings.finite_rings {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) # optional - sage.graphs sage: X = T.smash_product(S1) # optional - sage.graphs sage: X.homology(reduced=False) # optional - sage.graphs 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: from sage.topology.simplicial_set import AbstractSimplex, SimplicialSet # optional - sage.graphs sage: v = AbstractSimplex(0, name='v_0') # optional - sage.graphs sage: w = AbstractSimplex(0, name='w_0') # optional - sage.graphs sage: e = AbstractSimplex(1) # optional - sage.graphs sage: Y = SimplicialSet({e: (v, w)}, base_point=w) # optional - sage.graphs sage: Y.is_pointed() # optional - sage.graphs True sage: Y.base_point() # optional - sage.graphs w_0 sage: Z = Y.unset_base_point() # optional - sage.graphs sage: Z.is_pointed() # optional - sage.graphs False
- class ParentMethods#
Bases:
object
- base_point()#
Return this simplicial set’s base point
EXAMPLES:
sage: from sage.topology.simplicial_set import AbstractSimplex, SimplicialSet # optional - sage.graphs sage: v = AbstractSimplex(0, name='*') # optional - sage.graphs sage: e = AbstractSimplex(1) # optional - sage.graphs sage: S1 = SimplicialSet({e: (v, v)}, base_point=v) # optional - sage.graphs sage: S1.is_pointed() # optional - sage.graphs True sage: S1.base_point() # optional - sage.graphs *
- 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, defaultNone
. Use this to specify a particular one-point space as the domain. The default behavior is to use thesage.topology.simplicial_set.Point()
function to use a standard one-point space.
EXAMPLES:
sage: T = simplicial_sets.Torus() # optional - sage.graphs sage: f = T.base_point_map(); f # optional - sage.graphs Simplicial set morphism: From: Point To: Torus Defn: Constant map at (v_0, v_0) sage: S3 = simplicial_sets.Sphere(3) # optional - sage.graphs sage: g = S3.base_point_map() # optional - sage.graphs sage: f.domain() == g.domain() # optional - sage.graphs True sage: RP3 = simplicial_sets.RealProjectiveSpace(3) # optional - sage.graphs sage.groups sage: temp = simplicial_sets.Simplex(0) # optional - sage.graphs sage.groups sage: pt = temp.set_base_point(temp.n_cells(0)[0]) # optional - sage.graphs sage.groups sage: h = RP3.base_point_map(domain=pt) # optional - sage.graphs sage.groups sage: f.domain() == h.domain() # optional - sage.graphs sage.groups False sage: C5 = groups.misc.MultiplicativeAbelian([5]) # optional - sage.graphs sage.groups sage: BC5 = simplicial_sets.ClassifyingSpace(C5) # optional - sage.graphs sage.groups sage: BC5.base_point_map() # optional - 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: simplicial_sets.Sphere(3).connectivity() # optional - sage.graphs sage.groups 2 sage: simplicial_sets.Sphere(0).connectivity() # optional - sage.graphs sage.groups -1 sage: K = simplicial_sets.Simplex(4) # optional - sage.graphs sage: K = K.set_base_point(K.n_cells(0)[0]) # optional - sage.graphs sage: K.connectivity() # optional - sage.graphs sage.groups +Infinity sage: X = simplicial_sets.Torus().suspension(2) # optional - sage.graphs sage: X.connectivity() # optional - sage.graphs sage.groups 2 sage: C2 = groups.misc.MultiplicativeAbelian([2]) # optional - sage.graphs sage.groups sage: BC2 = simplicial_sets.ClassifyingSpace(C2) # optional - sage.graphs sage.groups sage: BC2.connectivity() # optional - 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: S1 = simplicial_sets.Sphere(1) # optional - sage.graphs sage: W = S1.wedge(S1) # optional - sage.graphs sage: G = CyclicPermutationGroup(3) # optional - sage.groups sage: (a, b) = W.n_cells(1) # optional - sage.graphs sage: C = W.cover({a : G.gen(0), b : G.gen(0)^2}) # optional - sage.graphs sage.groups sage: C.face_data() # optional - sage.graphs sage.groups {(*, ()): 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) # optional - sage.graphs sage.groups sage.modules Z x Z x Z x Z sage: C.fundamental_group() # optional - sage.graphs sage.groups 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: S1 = simplicial_sets.Sphere(1) # optional - sage.graphs sage: W = S1.wedge(S1) # optional - sage.graphs sage: G = CyclicPermutationGroup(3) # optional - sage.groups sage: a, b = W.n_cells(1) # optional - sage.graphs sage: C = W.covering_map({a : G.gen(0), b : G.one()}); C # optional - sage.graphs sage.groups 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() # optional - sage.graphs sage.groups Simplicial set with 9 non-degenerate simplices sage: C.domain().face_data() # optional - sage.graphs sage.groups {(*, ()): 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, optionalTrue
) – ifFalse
, then return a presentation of the group in terms of generators and relations. IfTrue
, 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) # optional - sage.graphs sage: eight = S1.wedge(S1) # optional - sage.graphs sage: eight.fundamental_group() # free group on 2 generators # optional - 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() # optional - sage.graphs sage: K = simplicial_sets.KleinBottle() # optional - sage.graphs sage: X = T.disjoint_union(K) # optional - sage.graphs sage: X_0 = X.set_base_point(X.n_cells(0)[0]) # optional - sage.graphs sage: X_0.fundamental_group().is_abelian() # optional - sage.graphs sage.groups True sage: X_1 = X.set_base_point(X.n_cells(0)[1]) # optional - sage.graphs sage: X_1.fundamental_group().is_abelian() # optional - sage.graphs sage.groups False sage: RP3 = simplicial_sets.RealProjectiveSpace(3) # optional - sage.graphs sage.groups sage: RP3.fundamental_group() # optional - sage.graphs sage.groups Finitely presented group < e | e^2 >
Compute the fundamental group of some classifying spaces:
sage: C5 = groups.misc.MultiplicativeAbelian([5]) # optional - sage.graphs sage.groups sage: BC5 = C5.nerve() # optional - sage.graphs sage.groups sage: BC5.fundamental_group() # optional - sage.graphs sage.groups Finitely presented group < e0 | e0^5 > sage: Sigma3 = groups.permutation.Symmetric(3) # optional - sage.graphs sage.groups sage: BSigma3 = Sigma3.nerve() # optional - sage.graphs sage.groups sage: pi = BSigma3.fundamental_group(); pi # optional - sage.graphs sage.groups Finitely presented group < e1, e2 | e2^2, e1^3, (e2*e1)^2 > sage: pi.order() # optional - sage.graphs sage.groups 6 sage: pi.is_abelian() # optional - sage.graphs sage.groups False
The sphere has a trivial fundamental group:
sage: S2 = simplicial_sets.Sphere(2) # optional - sage.graphs sage: S2.fundamental_group() # optional - 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: T = simplicial_sets.Torus() # optional - sage.graphs sage: T.is_simply_connected() # optional - sage.graphs sage.groups False sage: T.suspension().is_simply_connected() # optional - sage.graphs sage.groups True sage: simplicial_sets.KleinBottle().is_simply_connected() # optional - sage.graphs sage.groups False sage: S2 = simplicial_sets.Sphere(2) # optional - sage.graphs sage: S3 = simplicial_sets.Sphere(3) # optional - sage.graphs sage: (S2.wedge(S3)).is_simply_connected() # optional - sage.graphs sage.groups True sage: X = S2.disjoint_union(S3) # optional - sage.graphs sage: X = X.set_base_point(X.n_cells(0)[0]) # optional - sage.graphs sage: X.is_simply_connected() # optional - sage.graphs sage.groups False sage: C3 = groups.misc.MultiplicativeAbelian([3]) # optional - sage.graphs sage.groups sage: BC3 = simplicial_sets.ClassifyingSpace(C3) # optional - sage.graphs sage.groups sage: BC3.is_simply_connected() # optional - sage.graphs sage.groups False
- 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: RP3 = simplicial_sets.RealProjectiveSpace(3) # optional - sage.groups sage: C = RP3.universal_cover(); C # optional - sage.groups Simplicial set with 8 non-degenerate simplices sage: C.face_data() # optional - 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)), (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() # optional - sage.groups 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) # optional - sage.groups sage: phi = RP2.universal_cover_map(); phi # optional - 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() # optional - 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]