def __call__(self, x, orientation=False):
"""
Input is a simplex of the domain. Output is the image simplex.
If optional argument ``orientation`` is True, return a pair
``(image simplex, oriented)`` where ``oriented`` is 1 or `-1`
depending on whether the map preserves or reverses the
orientation of the image simplex.
EXAMPLES::
sage: S = simplicial_complexes.Sphere(2)
sage: T = simplicial_complexes.Sphere(3)
sage: S
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 2, 3), (0, 1, 2), (1, 2, 3), (0, 1, 3)}
sage: T
Simplicial complex with vertex set (0, 1, 2, 3, 4) and 5 facets
sage: f = {0:0,1:1,2:2,3:3}
sage: H = Hom(S,T)
sage: x = H(f)
sage: from sage.homology.simplicial_complex import Simplex
sage: x(Simplex([0,2,3]))
(0, 2, 3)
An orientation-reversing example::
sage: X = SimplicialComplex(1, [[0,1]])
sage: g = Hom(X,X)({0:1, 1:0})
sage: g(Simplex([0,1]))
(0, 1)
sage: g(Simplex([0,1]), orientation=True)
((0, 1), -1)
"""
dim = self._domain.dimension()
if not isinstance(x, simplicial_complex.Simplex) or x.dimension(
) > dim or not x in self._domain.faces()[x.dimension()]:
raise ValueError, "x must be a simplex of the source of f"
tup = x.tuple()
fx = []
for j in tup:
fx.append(self._vertex_dictionary[j])
if orientation:
if len(set(fx)) == len(tup):
oriented = Permutation(convert_perm(fx)).signature()
else:
oriented = 1
return (simplicial_complex.Simplex(set(fx)), oriented)
else:
return simplicial_complex.Simplex(set(fx))
def fiber_product(self, other, rename_vertices=True):
"""
Fiber product of ``self`` and ``other``. Both morphisms should have
the same codomain. The method returns a morphism of simplicial
complexes, which is the morphism from the space of the fiber product
to the codomain.
EXAMPLES::
sage: S = SimplicialComplex([[0,1],[1,2]], is_mutable=False)
sage: T = SimplicialComplex([[0,2],[1]], is_mutable=False)
sage: U = SimplicialComplex([[0,1],[2]], is_mutable=False)
sage: H = Hom(S,U)
sage: G = Hom(T,U)
sage: f = {0:0,1:1,2:0}
sage: g = {0:0,1:1,2:1}
sage: x = H(f)
sage: y = G(g)
sage: z = x.fiber_product(y)
sage: z
Simplicial complex morphism {'L1R2': 1, 'L1R1': 1, 'L2R0': 0, 'L0R0': 0}
from Simplicial complex with 4 vertices and facets
{('L2R0',), ('L1R1',), ('L0R0', 'L1R2')} to Simplicial complex
with vertex set (0, 1, 2) and facets {(2,), (0, 1)}
"""
if self._codomain != other._codomain:
raise ValueError("self and other must have the same codomain.")
X = self._domain.product(other._domain,
rename_vertices=rename_vertices)
v = []
f = dict()
eff1 = self._domain._vertex_set
eff2 = other._domain._vertex_set
for i in eff1:
for j in eff2:
if self(simplicial_complex.Simplex([i])) == other(
simplicial_complex.Simplex([j])):
if rename_vertices:
v.append("L" + str(i) + "R" + str(j))
f["L" + str(i) + "R" +
str(j)] = self._vertex_dictionary[i]
else:
v.append((i, j))
f[(i, j)] = self._vertex_dictionary[i]
return SimplicialComplexMorphism(f, X.generated_subcomplex(v),
self._codomain)
def __init__(self, f, X, Y):
"""
Input is a dictionary f, the domain, and the codomain.
One can define the dictionary either on the vertices of X or on the effective vertices of X (X.effective_vertices()). Note that this
difference does matter. For instance, it changes the result of the image method, and hence it changes the result of the is_surjective method as well.
This is because two SimplicialComplexes with the same faces but different vertex sets are not equal.
EXAMPLES::
sage: S = SimplicialComplex(5,[[0,1],[3,4]])
sage: H = Hom(S,S)
sage: f = {0:0,1:1,2:2,3:3,4:4,5:5}
sage: g = {0:0,1:1,3:3,4:4}
sage: x = H(f)
sage: y = H(g)
sage: x==y
False
sage: x.image()
Simplicial complex with vertex set (0, 1, 2, 3, 4, 5) and facets {(3, 4), (0, 1)}
sage: y.image()
Simplicial complex with vertex set (0, 1, 3, 4) and facets {(3, 4), (0, 1)}
sage: x.image()==y.image()
False
"""
if not isinstance(
X, simplicial_complex.SimplicialComplex) or not isinstance(
Y, simplicial_complex.SimplicialComplex):
raise ValueError, "X and Y must be SimplicialComplexes."
if not set(f.keys()) == X._vertex_set.set() and not set(
f.keys()) == X.effective_vertices().set():
raise ValueError, "f must be a dictionary from the vertex set of X to single values in the vertex set of Y."
dim = X.dimension()
Y_faces = Y.faces()
for k in range(dim + 1):
for i in X.faces()[k]:
tup = i.tuple()
fi = []
for j in tup:
fi.append(f[j])
v = simplicial_complex.Simplex(set(fi))
if not v in Y_faces[v.dimension()]:
raise ValueError, "f must be a dictionary from the vertices of X to the vertices of Y."
self._vertex_dictionary = f
self._domain = X
self._codomain = Y
def __init__(self, f, X, Y):
"""
Input is a dictionary ``f``, the domain ``X``, and the codomain ``Y``.
One can define the dictionary on the vertices of `X`.
EXAMPLES::
sage: S = SimplicialComplex([[0,1],[2],[3,4],[5]], is_mutable=False)
sage: H = Hom(S,S)
sage: f = {0:0,1:1,2:2,3:3,4:4,5:5}
sage: g = {0:0,1:1,2:0,3:3,4:4,5:0}
sage: x = H(f)
sage: y = H(g)
sage: x == y
False
sage: x.image()
Simplicial complex with vertex set (0, 1, 2, 3, 4, 5) and facets {(3, 4), (5,), (2,), (0, 1)}
sage: y.image()
Simplicial complex with vertex set (0, 1, 3, 4) and facets {(3, 4), (0, 1)}
sage: x.image() == y.image()
False
"""
if not isinstance(
X, simplicial_complex.SimplicialComplex) or not isinstance(
Y, simplicial_complex.SimplicialComplex):
raise ValueError("X and Y must be SimplicialComplexes.")
if not set(f.keys()) == X._vertex_set.set():
raise ValueError(
"f must be a dictionary from the vertex set of X to single values in the vertex set of Y."
)
dim = X.dimension()
Y_faces = Y.faces()
for k in range(dim + 1):
for i in X.faces()[k]:
tup = i.tuple()
fi = []
for j in tup:
fi.append(f[j])
v = simplicial_complex.Simplex(set(fi))
if not v in Y_faces[v.dimension()]:
raise ValueError(
"f must be a dictionary from the vertices of X to the vertices of Y."
)
self._vertex_dictionary = f
self._domain = X
self._codomain = Y