def super_categories(self):
"""
EXAMPLES::
sage: from sage.categories.graphs import Graphs
sage: Graphs().super_categories()
[Category of simplicial complexes]
"""
return [SimplicialComplexes()]
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 {(2,), (5,), (0, 1), (3, 4)}
sage: y.image()
Simplicial complex with vertex set (0, 1, 3, 4) and facets {(0, 1), (3, 4)}
sage: x.image() == y.image()
False
"""
if not isinstance(X, SimplicialComplex) or not isinstance(
Y, SimplicialComplex):
raise ValueError("X and Y must be SimplicialComplexes")
if not set(f.keys()) == set(X.vertices()):
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 = Simplex(set(fi))
if v not 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
Morphism.__init__(self, Hom(X, Y, SimplicialComplexes()))