def an_element(self):
"""
Return 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:
From: Minimal triangulation of the Klein bottle
To: Minimal triangulation of the 5-sphere
Defn: [0, 1, 2, 3, 4, 5, 6, 7] --> [0, 0, 0, 0, 0, 0, 0, 0]
"""
X_vertices = self._domain.vertices().set()
try:
i = next(iter(self._codomain.vertices().set()))
except StopIteration:
if not X_vertices:
return {}
else:
raise TypeError(
"there are no morphisms from a non-empty simplicial complex to an empty simplicial complex"
)
f = {x: i for x in X_vertices}
return SimplicialComplexMorphism(f, self._domain, self._codomain)
def diagonal_morphism(self, rename_vertices=True):
r"""
Return 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:
From: Minimal triangulation of the 2-sphere
To: Simplicial complex with 16 vertices and 96 facets
Defn: 0 |--> L0R0
1 |--> L1R1
2 |--> L2R2
3 |--> L3R3
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:
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),)}
Defn: 0 |--> (0, 0)
1 |--> (1, 1)
"""
# Preserve whether the codomain is mutable when renaming the vertices.
mutable = self._codomain.is_mutable()
X = self._domain.product(self._domain,
rename_vertices=rename_vertices,
is_mutable=mutable)
if self._codomain != X:
raise TypeError("diagonal morphism is only defined for Hom(X,XxX)")
f = {}
if rename_vertices:
f = {
i: "L{0}R{0}".format(i)
for i in self._domain.vertices().set()
}
else:
f = {i: (i, i) for i in self._domain.vertices().set()}
return SimplicialComplexMorphism(f, self._domain, X)
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:
From: Minimal triangulation of the 3-sphere
To: Minimal triangulation of the 2-sphere
Defn: [0, 1, 2, 3, 4] --> [0, 1, 2, 2, 2]
"""
return SimplicialComplexMorphism(f, self.domain(), self.codomain())
def identity(self):
"""
Return 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 endomorphism of Simplicial complex with vertex set (0, 1) and facets {(0, 1)}
Defn: 0 |--> 0
1 |--> 1
"""
if not self.is_endomorphism_set():
raise TypeError("identity map is only defined for endomorphism sets")
f = {i: i for i in self._domain.vertices()}
return SimplicialComplexMorphism(f, self._domain, self._codomain)