Coverage for local/lib/python2.7/site-packages/sage/homology/simplicial_complex_homset.py : 81%

r""" 

Homsets between simplicial complexes 

AUTHORS: 

- Travis Scrimshaw (2012-08-18): Made all simplicial complexes immutable to 

work with the homset cache. 

EXAMPLES:: 

sage: S = simplicial_complexes.Sphere(1) 

sage: T = simplicial_complexes.Sphere(2) 

sage: H = Hom(S,T) 

sage: f = {0:0,1:1,2:3} 

sage: x = H(f) 

sage: x 

Simplicial complex morphism: 

From: Minimal triangulation of the 1-sphere 

To: Minimal triangulation of the 2-sphere 

Defn: 0 |--> 0 

1 |--> 1 

2 |--> 3 

sage: x.is_injective() 

True 

sage: x.is_surjective() 

False 

sage: x.image() 

Simplicial complex with vertex set (0, 1, 3) and facets {(1, 3), (0, 3), (0, 1)} 

sage: from sage.homology.simplicial_complex import Simplex 

sage: s = Simplex([1,2]) 

sage: x(s) 

(1, 3) 

TESTS:: 

sage: S = simplicial_complexes.Sphere(1) 

sage: T = simplicial_complexes.Sphere(2) 

sage: H = Hom(S,T) 

sage: loads(dumps(H)) == H 

True 

""" 

#***************************************************************************** 

# Copyright (C) 2009 D. Benjamin Antieau <d.ben.antieau@gmail.com> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# This code is distributed in the hope that it will be useful, 

# but WITHOUT ANY WARRANTY; without even the implied warranty 

# of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. 

# 

# See the GNU General Public License for more details; the full text 

# is available at: 

# 

# http://www.gnu.org/licenses/ 

# 

#***************************************************************************** 

import sage.categories.homset 

from sage.homology.simplicial_complex_morphism import SimplicialComplexMorphism 

def is_SimplicialComplexHomset(x): 

""" 

Return ``True`` if and only if ``x`` is a simplicial complex homspace. 

EXAMPLES:: 

sage: S = SimplicialComplex(is_mutable=False) 

sage: T = SimplicialComplex(is_mutable=False) 

sage: H = Hom(S, T) 

sage: H 

Set of Morphisms from Simplicial complex with vertex set () and facets {()} 

to Simplicial complex with vertex set () and facets {()} 

in Category of finite simplicial complexes 

sage: from sage.homology.simplicial_complex_homset import is_SimplicialComplexHomset 

sage: is_SimplicialComplexHomset(H) 

True 

""" 

return isinstance(x, SimplicialComplexHomset) 

class SimplicialComplexHomset(sage.categories.homset.Homset): 

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 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()} 

else: 

f = {i: (i,i) for i in self._domain.vertices()} 

return SimplicialComplexMorphism(f, self._domain, X) 

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) 

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() 

try: 

i = next(iter(self._codomain.vertices())) 

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)