Morphisms and homsets for simplicial sets

Note

Morphisms with infinite domain are not implemented in general: only constant maps and identity maps are currently implemented.

AUTHORS:

  • John H. Palmieri (2016-07)

This module implements morphisms and homsets of simplicial sets.

class sage.homology.simplicial_set_morphism.SimplicialSetHomset(X, Y, category=None, base=None, check=True)

Bases: sage.categories.homset.Homset

A set of morphisms between simplicial sets.

Once a homset has been constructed in Sage, typically via Hom(X,Y) or X.Hom(Y), one can use it to construct a morphism \(f\) by specifying a dictionary, the keys of which are the nondegenerate simplices in the domain, and the value corresponding to \(\sigma\) is the simplex \(f(\sigma)\) in the codomain.

EXAMPLES:

sage: from sage.homology.simplicial_set import AbstractSimplex, SimplicialSet
sage: v = AbstractSimplex(0, name='v')
sage: w = AbstractSimplex(0, name='w')
sage: e = AbstractSimplex(1, name='e')
sage: f = AbstractSimplex(1, name='f')
sage: X = SimplicialSet({e: (v, w), f: (w, v)})
sage: Y = SimplicialSet({e: (v, v)})

Define the homset:

sage: H = Hom(X, Y)

Now define a morphism by specifying a dictionary:

sage: H({v: v, w: v, e: e, f: e})
Simplicial set morphism:
  From: Simplicial set with 4 non-degenerate simplices
  To:   Simplicial set with 2 non-degenerate simplices
  Defn: [v, w, e, f] --> [v, v, e, e]
an_element()

Return an element of this homset: a constant map.

EXAMPLES:

sage: S1 = simplicial_sets.Sphere(1)
sage: S2 = simplicial_sets.Sphere(2)
sage: Hom(S2, S1).an_element()
Simplicial set morphism:
  From: S^2
  To:   S^1
  Defn: Constant map at v_0

sage: K = simplicial_sets.Simplex(3)
sage: L = simplicial_sets.Simplex(4)
sage: d = {K.n_cells(3)[0]: L.n_cells(0)[0].apply_degeneracies(2, 1, 0)}
sage: Hom(K,L)(d) == Hom(K,L).an_element()
True
constant_map(point=None)

Return the constant map in this homset.

INPUT:

  • point – optional, default None. If specified, it must be a 0-simplex in the codomain, and it will be the target of the constant map.

If point is specified, it is the target of the constant map. Otherwise, if the codomain is pointed, the target is its base point. If the codomain is not pointed and point is not specified, raise an error.

EXAMPLES:

sage: S3 = simplicial_sets.Sphere(3)
sage: T = simplicial_sets.Torus()
sage: T.n_cells(0)[0].rename('w')
sage: Hom(S3,T).constant_map()
Simplicial set morphism:
  From: S^3
  To:   Torus
  Defn: Constant map at w

sage: S0 = simplicial_sets.Sphere(0)
sage: v, w = S0.n_cells(0)
sage: Hom(S3, S0).constant_map(v)
Simplicial set morphism:
  From: S^3
  To:   S^0
  Defn: Constant map at v_0
sage: Hom(S3, S0).constant_map(w)
Simplicial set morphism:
  From: S^3
  To:   S^0
  Defn: Constant map at w_0

This constant map is not pointed, since it doesn’t send the base point of \(S^3\) to the base point of \(S^0\):

sage: Hom(S3, S0).constant_map(w).is_pointed()
False
diagonal_morphism()

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

EXAMPLES:

sage: RP2 = simplicial_sets.RealProjectiveSpace(2)
sage: Hom(RP2, RP2.product(RP2)).diagonal_morphism()
Simplicial set morphism:
  From: RP^2
  To:   RP^2 x RP^2
  Defn: [1, f, f * f] --> [(1, 1), (f, f), (f * f, f * f)]
identity()

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

EXAMPLES:

sage: S1 = simplicial_sets.Sphere(1)
sage: Hom(S1, S1).identity()
Simplicial set endomorphism of S^1
  Defn: Identity map
sage: T = simplicial_sets.Torus()
sage: Hom(S1, T).identity()
Traceback (most recent call last):
...
TypeError: identity map is only defined for endomorphism sets
class sage.homology.simplicial_set_morphism.SimplicialSetMorphism(data=None, domain=None, codomain=None, constant=None, identity=False, check=True)

Bases: sage.categories.morphism.Morphism

Return a morphism of simplicial sets.

INPUT:

  • data – optional. Dictionary defining the map.
  • domain – simplicial set
  • codomain – simplicial set
  • constant – optional: if not None, then this should be a vertex in the codomain, in which case return the constant map with this vertex as the target.
  • identity – optional: if True, return the identity morphism.
  • check – optional, default True. If True, check that this is actually a morphism: it commutes with the face maps.

So to define a map, you must specify domain and codomain. If the map is constant, specify the target (a vertex in the codomain) as constant. If the map is the identity map, specify identity=True. Otherwise, pass a dictionary, data. The keys of the dictionary are the nondegenerate simplices of the domain, the corresponding values are simplices in the codomain.

In fact, the keys in data do not need to include all of the nondegenerate simplices, only those which are not faces of other nondegenerate simplices: if \(\sigma\) is a face of \(\tau\), then the image of \(\sigma\) need not be specified.

EXAMPLES:

sage: from sage.homology.simplicial_set_morphism import SimplicialSetMorphism
sage: K = simplicial_sets.Simplex(1)
sage: S1 = simplicial_sets.Sphere(1)
sage: v0 = K.n_cells(0)[0]
sage: v1 = K.n_cells(0)[1]
sage: e01 = K.n_cells(1)[0]
sage: w = S1.n_cells(0)[0]
sage: sigma = S1.n_cells(1)[0]

sage: f = {v0: w, v1: w, e01: sigma}
sage: SimplicialSetMorphism(f, K, S1)
Simplicial set morphism:
  From: 1-simplex
  To:   S^1
  Defn: [(0,), (1,), (0, 1)] --> [v_0, v_0, sigma_1]

The same map can be defined as follows:

sage: H = Hom(K, S1)
sage: H(f)
Simplicial set morphism:
  From: 1-simplex
  To:   S^1
  Defn: [(0,), (1,), (0, 1)] --> [v_0, v_0, sigma_1]

Also, this map can be defined by specifying where the 1-simplex goes; the vertices then go where they have to, to satisfy the condition \(d_i \circ f = f \circ d_i\):

sage: H = Hom(K, S1)
sage: H({e01: sigma})
Simplicial set morphism:
  From: 1-simplex
  To:   S^1
  Defn: [(0,), (1,), (0, 1)] --> [v_0, v_0, sigma_1]

A constant map:

sage: g = {e01: w.apply_degeneracies(0)}
sage: SimplicialSetMorphism(g, K, S1)
Simplicial set morphism:
  From: 1-simplex
  To:   S^1
  Defn: Constant map at v_0

The same constant map:

sage: SimplicialSetMorphism(domain=K, codomain=S1, constant=w)
Simplicial set morphism:
  From: 1-simplex
  To:   S^1
  Defn: Constant map at v_0

An identity map:

sage: SimplicialSetMorphism(domain=K, codomain=K, identity=True)
Simplicial set endomorphism of 1-simplex
  Defn: Identity map

Defining a map by specifying it on only some of the simplices in the domain:

sage: S5 = simplicial_sets.Sphere(5)
sage: s = S5.n_cells(5)[0]
sage: one = S5.Hom(S5)({s: s})
sage: one
Simplicial set endomorphism of S^5
  Defn: Identity map
associated_chain_complex_morphism(base_ring=Integer Ring, augmented=False, cochain=False)

Return the associated chain complex morphism of self.

INPUT:

  • base_ring – default ZZ
  • augmented – boolean, default False. If True, return the augmented complex.
  • cochain – boolean, default False. If True, return the cochain complex.

EXAMPLES:

sage: S1 = simplicial_sets.Sphere(1)
sage: v0 = S1.n_cells(0)[0]
sage: e = S1.n_cells(1)[0]
sage: f = {v0: v0, e: v0.apply_degeneracies(0)} # constant map
sage: g = Hom(S1, S1)(f)
sage: g.associated_chain_complex_morphism().to_matrix()
[1|0]
[-+-]
[0|0]
coequalizer(other)

Return the coequalizer of this map with other.

INPUT:

  • other – a morphism with the same domain and codomain as this map

If the two maps are \(f, g: X \to Y\), then the coequalizer \(P\) is constructed as the pushout

X v Y --> Y
  |       |
  V       V
  Y ----> P

where the upper left corner is the coproduct of \(X\) and \(Y\) (the wedge if they are pointed, the disjoint union otherwise), and the two maps \(X \amalg Y \to Y\) are \(f \amalg 1\) and \(g \amalg 1\).

EXAMPLES:

sage: L = simplicial_sets.Simplex(2)
sage: pt = L.n_cells(0)[0]
sage: e = L.n_cells(1)[0]
sage: K = L.subsimplicial_set([e])
sage: f = K.inclusion_map()
sage: v,w = K.n_cells(0)
sage: g = Hom(K,L)({v:pt, w:pt, e:pt.apply_degeneracies(0)})
sage: P = f.coequalizer(g)
sage: P
Pushout of maps:
  Simplicial set morphism:
    From: Disjoint union: (Simplicial set with 3 non-degenerate simplices u 2-simplex)
    To:   2-simplex
    Defn: ...
  Simplicial set morphism:
    From: Disjoint union: (Simplicial set with 3 non-degenerate simplices u 2-simplex)
    To:   2-simplex
    Defn: ...
coproduct(*others)

Return the coproduct of this map with others.

  • others – morphisms of simplicial sets.

If the relevant maps are \(f_i: X_i \to Y_i\), this returns the natural map \(\amalg X_i \to \amalg Y_i\).

EXAMPLES:

sage: S1 = simplicial_sets.Sphere(1)
sage: f = Hom(S1,S1).identity()
sage: f.coproduct(f).is_bijective()
True
sage: g = S1.constant_map(S1)
sage: g.coproduct(g).is_bijective()
False
equalizer(other)

Return the equalizer of this map with other.

INPUT:

  • other – a morphism with the same domain and codomain as this map

If the two maps are \(f, g: X \to Y\), then the equalizer \(P\) is constructed as the pullback

P ----> X
|       |
V       V
X --> X x Y

where the two maps \(X \to X \times Y\) are \((1,f)\) and \((1,g)\).

EXAMPLES:

sage: from sage.homology.simplicial_set import AbstractSimplex, SimplicialSet
sage: v = AbstractSimplex(0, name='v')
sage: w = AbstractSimplex(0, name='w')
sage: x = AbstractSimplex(0, name='x')
sage: evw = AbstractSimplex(1, name='vw')
sage: evx = AbstractSimplex(1, name='vx')
sage: ewx = AbstractSimplex(1, name='wx')
sage: X = SimplicialSet({evw: (w, v), evx: (x, v)})
sage: Y = SimplicialSet({evw: (w, v), evx: (x, v), ewx: (x, w)})

Here \(X\) is a wedge of two 1-simplices (a horn, that is), and \(Y\) is the boundary of a 2-simplex. The map \(f\) includes the two 1-simplices into \(Y\), while the map \(g\) maps both 1-simplices to the same edge in \(Y\).

sage: f = Hom(X, Y)({v:v, w:w, x:x, evw:evw, evx:evx})
sage: g = Hom(X, Y)({v:v, w:x, x:x, evw:evx, evx:evx})
sage: P = f.equalizer(g)
sage: P
Pullback of maps:
  Simplicial set morphism:
    From: Simplicial set with 5 non-degenerate simplices
    To:   Simplicial set with 5 non-degenerate simplices x Simplicial set with 6 non-degenerate simplices
    Defn: [v, w, x, vw, vx] --> [(v, v), (w, w), (x, x), (vw, vw), (vx, vx)]
  Simplicial set morphism:
    From: Simplicial set with 5 non-degenerate simplices
    To:   Simplicial set with 5 non-degenerate simplices x Simplicial set with 6 non-degenerate simplices
    Defn: [v, w, x, vw, vx] --> [(v, v), (w, x), (x, x), (vw, vx), (vx, vx)]
image()

Return the image of this morphism as a subsimplicial set of the codomain.

EXAMPLES:

sage: S1 = simplicial_sets.Sphere(1)
sage: T = S1.product(S1)
sage: K = T.factor(0, as_subset=True)
sage: f = S1.Hom(T)({S1.n_cells(0)[0]:K.n_cells(0)[0], S1.n_cells(1)[0]:K.n_cells(1)[0]})
sage: f
Simplicial set morphism:
  From: S^1
  To:   S^1 x S^1
  Defn: [v_0, sigma_1] --> [(v_0, v_0), (sigma_1, s_0 v_0)]
sage: f.image()
Simplicial set with 2 non-degenerate simplices
sage: f.image().homology()
{0: 0, 1: Z}

sage: B = simplicial_sets.ClassifyingSpace(groups.misc.MultiplicativeAbelian([2]))
sage: B.constant_map().image()
Point
sage: Hom(B,B).identity().image() == B
True
induced_homology_morphism(base_ring=None, cohomology=False)

Return the map in (co)homology induced by this map

INPUT:

  • base_ring – must be a field (optional, default QQ)
  • cohomology – boolean (optional, default False). If True, the map induced in cohomology rather than homology.

EXAMPLES:

sage: from sage.homology.simplicial_set import AbstractSimplex, SimplicialSet
sage: v = AbstractSimplex(0, name='v')
sage: w = AbstractSimplex(0, name='w')
sage: e = AbstractSimplex(1, name='e')
sage: f = AbstractSimplex(1, name='f')
sage: X = SimplicialSet({e: (v, w), f: (w, v)})
sage: Y = SimplicialSet({e: (v, v)})
sage: H = Hom(X, Y)
sage: f = H({v: v, w: v, e: e, f: e})
sage: g = f.induced_homology_morphism()
sage: g.to_matrix()
[1|0]
[-+-]
[0|2]
sage: g3 = f.induced_homology_morphism(base_ring=GF(3), cohomology=True)
sage: g3.to_matrix()
[1|0]
[-+-]
[0|2]
is_bijective()

Return True if this map is bijective.

EXAMPLES:

sage: RP5 = simplicial_sets.RealProjectiveSpace(5)
sage: RP2 = RP5.n_skeleton(2)
sage: RP2.inclusion_map().is_bijective()
False

sage: RP5_2 = RP5.quotient(RP2)
sage: RP5_2.quotient_map().is_bijective()
False

sage: K = RP5_2.pullback(RP5_2.quotient_map(), RP5_2.base_point_map())
sage: f = K.universal_property(RP2.inclusion_map(), RP2.constant_map())
sage: f.is_bijective()
True
is_constant()

Return True if this morphism is a constant map.

EXAMPLES:

sage: K = simplicial_sets.KleinBottle()
sage: S4 = simplicial_sets.Sphere(4)
sage: c = Hom(K, S4).constant_map()
sage: c.is_constant()
True
sage: X = S4.n_skeleton(3) # a point
sage: X.inclusion_map().is_constant()
True
sage: eta = simplicial_sets.HopfMap()
sage: eta.is_constant()
False
is_identity()

Return True if this morphism is an identity map.

EXAMPLES:

sage: K = simplicial_sets.Simplex(1)
sage: v0 = K.n_cells(0)[0]
sage: v1 = K.n_cells(0)[1]
sage: e01 = K.n_cells(1)[0]
sage: L = simplicial_sets.Simplex(2).n_skeleton(1)
sage: w0 = L.n_cells(0)[0]
sage: w1 = L.n_cells(0)[1]
sage: w2 = L.n_cells(0)[2]
sage: f01 = L.n_cells(1)[0]
sage: f02 = L.n_cells(1)[1]
sage: f12 = L.n_cells(1)[2]

sage: d = {v0:w0, v1:w1, e01:f01}
sage: f = K.Hom(L)(d)
sage: f.is_identity()
False
sage: d = {w0:v0, w1:v1, w2:v1, f01:e01, f02:e01, f12: v1.apply_degeneracies(0,)}
sage: g = L.Hom(K)(d)
sage: (g*f).is_identity()
True
sage: (f*g).is_identity()
False
sage: (f*g).induced_homology_morphism().to_matrix(1)
[0]

sage: RP5 = simplicial_sets.RealProjectiveSpace(5)
sage: RP5.n_skeleton(2).inclusion_map().is_identity()
False
sage: RP5.n_skeleton(5).inclusion_map().is_identity()
True

sage: B = simplicial_sets.ClassifyingSpace(groups.misc.MultiplicativeAbelian([2]))
sage: Hom(B,B).identity().is_identity()
True
sage: Hom(B,B).constant_map().is_identity()
False
is_injective()

Return True if this map is injective.

EXAMPLES:

sage: RP5 = simplicial_sets.RealProjectiveSpace(5)
sage: RP2 = RP5.n_skeleton(2)
sage: RP2.inclusion_map().is_injective()
True

sage: RP5_2 = RP5.quotient(RP2)
sage: RP5_2.quotient_map().is_injective()
False

sage: K = RP5_2.pullback(RP5_2.quotient_map(), RP5_2.base_point_map())
sage: f = K.universal_property(RP2.inclusion_map(), RP2.constant_map())
sage: f.is_injective()
True
is_pointed()

Return True if this is a pointed map.

That is, return True if the domain and codomain are pointed and this morphism preserves the base point.

EXAMPLES:

sage: S0 = simplicial_sets.Sphere(0)
sage: f = Hom(S0,S0).identity()
sage: f.is_pointed()
True
sage: v = S0.n_cells(0)[0]
sage: w = S0.n_cells(0)[1]
sage: g = Hom(S0,S0)({v:v, w:v})
sage: g.is_pointed()
True
sage: t = Hom(S0,S0)({v:w, w:v})
sage: t.is_pointed()
False
is_surjective()

Return True if this map is surjective.

EXAMPLES:

sage: RP5 = simplicial_sets.RealProjectiveSpace(5)
sage: RP2 = RP5.n_skeleton(2)
sage: RP2.inclusion_map().is_surjective()
False

sage: RP5_2 = RP5.quotient(RP2)
sage: RP5_2.quotient_map().is_surjective()
True

sage: K = RP5_2.pullback(RP5_2.quotient_map(), RP5_2.base_point_map())
sage: f = K.universal_property(RP2.inclusion_map(), RP2.constant_map())
sage: f.is_surjective()
True
mapping_cone()

Return the mapping cone defined by this map.

EXAMPLES:

sage: S1 = simplicial_sets.Sphere(1)
sage: v_0, sigma_1 = S1.nondegenerate_simplices()
sage: K = simplicial_sets.Simplex(2).n_skeleton(1)

The mapping cone will be a little smaller if we use only pointed simplicial sets. \(S^1\) is already pointed, but not \(K\).

sage: L = K.set_base_point(K.n_cells(0)[0])
sage: u,v,w = L.n_cells(0)
sage: e,f,g = L.n_cells(1)
sage: h = L.Hom(S1)({u:v_0, v:v_0, w:v_0, e:sigma_1, f:v_0.apply_degeneracies(0), g:sigma_1})
sage: h
Simplicial set morphism:
  From: Simplicial set with 6 non-degenerate simplices
  To:   S^1
  Defn: [(0,), (1,), (2,), (0, 1), (0, 2), (1, 2)] --> [v_0, v_0, v_0, sigma_1, s_0 v_0, sigma_1]
sage: h.induced_homology_morphism().to_matrix()
[1|0]
[-+-]
[0|2]
sage: X = h.mapping_cone()
sage: X.homology() == simplicial_sets.RealProjectiveSpace(2).homology()
True
n_skeleton(n, domain=None, codomain=None)

Return the restriction of this morphism to the n-skeleta of the domain and codomain

INPUT:

  • n – the dimension
  • domain – optional, the domain. Specify this to explicitly specify the domain; otherwise, Sage will attempt to compute it. Specifying this can be useful if the domain is built as a pushout or pullback, so trying to compute it may lead to computing the \(n\)-skeleton of a map, causing an infinite recursion. (Users should not have to specify this, but it may be useful for developers.)
  • codomain – optional, the codomain.

EXAMPLES:

sage: B = simplicial_sets.ClassifyingSpace(groups.misc.MultiplicativeAbelian([2]))
sage: one = Hom(B,B).identity()
sage: one.n_skeleton(3)
Simplicial set endomorphism of Simplicial set with 4 non-degenerate simplices
  Defn: Identity map
sage: c = Hom(B,B).constant_map()
sage: c.n_skeleton(3)
Simplicial set endomorphism of Simplicial set with 4 non-degenerate simplices
  Defn: Constant map at 1

sage: K = simplicial_sets.Simplex(2)
sage: L = K.subsimplicial_set(K.n_cells(0)[:2])
sage: L.nondegenerate_simplices()
[(0,), (1,)]
sage: L.inclusion_map()
Simplicial set morphism:
  From: Simplicial set with 2 non-degenerate simplices
  To:   2-simplex
  Defn: [(0,), (1,)] --> [(0,), (1,)]
sage: L.inclusion_map().n_skeleton(1)
Simplicial set morphism:
  From: Simplicial set with 2 non-degenerate simplices
  To:   Simplicial set with 6 non-degenerate simplices
  Defn: [(0,), (1,)] --> [(0,), (1,)]
product(*others)

Return the product of this map with others.

  • others – morphisms of simplicial sets.

If the relevant maps are \(f_i: X_i \to Y_i\), this returns the natural map \(\prod X_i \to \prod Y_i\).

EXAMPLES:

sage: S1 = simplicial_sets.Sphere(1)
sage: f = Hom(S1,S1).identity()
sage: f.product(f).is_bijective()
True
sage: g = S1.constant_map(S1)
sage: g.product(g).is_bijective()
False
pullback(*others)

Return the pullback of this morphism along with others.

INPUT:

  • others – morphisms of simplicial sets, the codomains of which must all equal that of self.

This returns the pullback as a simplicial set. See sage.homology.simplicial_set_constructions.PullbackOfSimplicialSets for more documentation and examples.

EXAMPLES:

sage: T = simplicial_sets.Torus()
sage: K = simplicial_sets.KleinBottle()
sage: term_T = T.constant_map()
sage: term_K = K.constant_map()
sage: P = term_T.pullback(term_K) # the product as a pullback
sage: P
Pullback of maps:
  Simplicial set morphism:
    From: Torus
    To:   Point
    Defn: Constant map at *
  Simplicial set morphism:
    From: Klein bottle
    To:   Point
    Defn: Constant map at *
pushout(*others)

Return the pushout of this morphism along with others.

INPUT:

  • others – morphisms of simplicial sets, the domains of which must all equal that of self.

This returns the pushout as a simplicial set. See sage.homology.simplicial_set_constructions.PushoutOfSimplicialSets for more documentation and examples.

EXAMPLES:

sage: T = simplicial_sets.Torus()
sage: K = simplicial_sets.KleinBottle()
sage: init_T = T._map_from_empty_set()
sage: init_K = K._map_from_empty_set()
sage: D = init_T.pushout(init_K) # the disjoint union as a pushout
sage: D
Pushout of maps:
  Simplicial set morphism:
    From: Empty simplicial set
    To:   Torus
    Defn: [] --> []
  Simplicial set morphism:
    From: Empty simplicial set
    To:   Klein bottle
    Defn: [] --> []
suspension(n=1)

Return the \(n\)-th suspension of this morphism of simplicial sets.

INPUT:

  • n (optional) – non-negative integer, default 1

EXAMPLES:

sage: eta = simplicial_sets.HopfMap()
sage: susp_eta = eta.suspension()
sage: susp_eta.mapping_cone().homology() == eta.mapping_cone().suspension().homology()
True

This uses reduced suspensions if the original morphism is pointed, unreduced otherwise. So for example, if a constant map is not pointed, its suspension is not a constant map:

sage: L = simplicial_sets.Simplex(1)
sage: L.constant_map().is_pointed()
False
sage: f = L.constant_map().suspension()
sage: f.is_constant()
False

sage: K = simplicial_sets.Sphere(3)
sage: K.constant_map().is_pointed()
True
sage: g = K.constant_map().suspension()
sage: g.is_constant()
True

sage: h = K.identity().suspension()
sage: h.is_identity()
True