def identity(self):
"""
Returns the identity morphism of `Hom(S,S)`.
EXAMPLES::
sage: S = simplicial_complexes.Sphere(2)
sage: H = Hom(S,S)
sage: i = H.identity()
sage: i.is_identity()
True
sage: T = SimplicialComplex([[0,1]], is_mutable=False)
sage: G = Hom(T,T)
sage: G.identity()
Simplicial complex morphism {0: 0, 1: 1} from
Simplicial complex with vertex set (0, 1) and facets {(0, 1)} to
Simplicial complex with vertex set (0, 1) and facets {(0, 1)}
"""
if self.is_endomorphism_set():
f = dict()
for i in self._domain.vertices().set():
f[i] = i
return simplicial_complex_morphism.SimplicialComplexMorphism(
f, self._domain, self._codomain)
else:
raise TypeError(
"Identity map is only defined for endomorphism sets.")
def an_element(self):
"""
Returns a (non-random) element of ``self``.
EXAMPLES::
sage: S = simplicial_complexes.KleinBottle()
sage: T = simplicial_complexes.Sphere(5)
sage: H = Hom(S,T)
sage: x = H.an_element()
sage: x
Simplicial complex morphism {0: 0, 1: 0, 2: 0, 3: 0, 4: 0, 5: 0, 6: 0, 7: 0} from Simplicial complex with vertex set (0, 1, 2, 3, 4, 5, 6, 7) and 16 facets to Simplicial complex with vertex set (0, 1, 2, 3, 4, 5, 6) and 7 facets
"""
X_vertices = self._domain.vertices().set()
try:
i = self._codomain.vertices().set().__iter__().next()
except StopIteration:
if len(X_vertices) == 0:
return dict()
else:
raise TypeError(
"There are no morphisms from a non-empty simplicial complex to an empty simplicial comples."
)
f = dict()
for x in X_vertices:
f[x] = i
return simplicial_complex_morphism.SimplicialComplexMorphism(
f, self._domain, self._codomain)
def __call__(self, f):
"""
INPUT:
- ``f`` -- a dictionary with keys exactly the vertices of the domain
and values vertices of the codomain
EXAMPLES::
sage: S = simplicial_complexes.Sphere(3)
sage: T = simplicial_complexes.Sphere(2)
sage: f = {0:0,1:1,2:2,3:2,4:2}
sage: H = Hom(S,T)
sage: x = H(f)
sage: x
Simplicial complex morphism {0: 0, 1: 1, 2: 2, 3: 2, 4: 2} from Simplicial complex with vertex set (0, 1, 2, 3, 4) and 5 facets to Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 2, 3), (0, 1, 2), (1, 2, 3), (0, 1, 3)}
"""
return simplicial_complex_morphism.SimplicialComplexMorphism(f,self.domain(),self.codomain())
def diagonal_morphism(self, rename_vertices=True):
r"""
Returns the diagonal morphism in `Hom(S, S \times S)`.
EXAMPLES::
sage: S = simplicial_complexes.Sphere(2)
sage: H = Hom(S,S.product(S, is_mutable=False))
sage: d = H.diagonal_morphism()
sage: d
Simplicial complex morphism {0: 'L0R0', 1: 'L1R1', 2: 'L2R2', 3: 'L3R3'} from
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 2, 3), (0, 1, 2), (1, 2, 3), (0, 1, 3)}
to Simplicial complex with 16 vertices and 96 facets
sage: T = SimplicialComplex([[0], [1]], is_mutable=False)
sage: U = T.product(T,rename_vertices = False, is_mutable=False)
sage: G = Hom(T,U)
sage: e = G.diagonal_morphism(rename_vertices = False)
sage: e
Simplicial complex morphism {0: (0, 0), 1: (1, 1)} from
Simplicial complex with vertex set (0, 1) and facets {(0,), (1,)}
to Simplicial complex with 4 vertices and facets {((1, 1),), ((1, 0),), ((0, 0),), ((0, 1),)}
"""
if self._codomain == self._domain.product(
self._domain, rename_vertices=rename_vertices):
X = self._domain.product(self._domain,
rename_vertices=rename_vertices)
f = dict()
if rename_vertices:
for i in self._domain.vertices().set():
f[i] = "L" + str(i) + "R" + str(i)
else:
for i in self._domain.vertices().set():
f[i] = (i, i)
return simplicial_complex_morphism.SimplicialComplexMorphism(
f, self._domain, X)
else:
raise TypeError(
"Diagonal morphism is only defined for Hom(X,XxX).")
