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Trac #19179: Chain homotopies
This ticket add chain homotopies, chain contractions, and duals of chain maps and chain homotopies. This is a dependency for #6101 and #6102. URL: http://trac.sagemath.org/19179 Reported by: jhpalmieri Ticket author(s): John Palmieri Reviewer(s): Frédéric Chapoton
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‎src/doc/en/reference/homology/index.rst

+1
Original file line numberDiff line numberDiff line change
@@ -13,6 +13,7 @@ cell complexes.
1313

1414
sage/homology/chain_complex
1515
sage/homology/chain_complex_morphism
16+
sage/homology/chain_homotopy
1617
sage/homology/chain_complex_homspace
1718
sage/homology/simplicial_complex
1819
sage/homology/simplicial_complex_morphism

‎src/sage/homology/chain_complex_homspace.py

+9-3
Original file line numberDiff line numberDiff line change
@@ -23,7 +23,9 @@
2323
sage: i = H.identity()
2424
sage: x = i.associated_chain_complex_morphism(augmented=True)
2525
sage: x
26-
Chain complex morphism from Chain complex with at most 4 nonzero terms over Integer Ring to Chain complex with at most 4 nonzero terms over Integer Ring
26+
Chain complex morphism
27+
From: Chain complex with at most 4 nonzero terms over Integer Ring
28+
To: Chain complex with at most 4 nonzero terms over Integer Ring
2729
sage: x._matrix_dictionary
2830
{-1: [1], 0: [1 0 0 0 0 0 0 0 0]
2931
[0 1 0 0 0 0 0 0 0]
@@ -62,11 +64,15 @@
6264
sage: i = A.identity()
6365
sage: x = i.associated_chain_complex_morphism()
6466
sage: x
65-
Chain complex morphism from Chain complex with at most 3 nonzero terms over Integer Ring to Chain complex with at most 3 nonzero terms over Integer Ring
67+
Chain complex morphism
68+
From: Chain complex with at most 3 nonzero terms over Integer Ring
69+
To: Chain complex with at most 3 nonzero terms over Integer Ring
6670
sage: y = x*4
6771
sage: z = y*y
6872
sage: (y+z)
69-
Chain complex morphism from Chain complex with at most 3 nonzero terms over Integer Ring to Chain complex with at most 3 nonzero terms over Integer Ring
73+
Chain complex morphism
74+
From: Chain complex with at most 3 nonzero terms over Integer Ring
75+
To: Chain complex with at most 3 nonzero terms over Integer Ring
7076
sage: f = x._matrix_dictionary
7177
sage: C = S.chain_complex()
7278
sage: G = Hom(C,C)

‎src/sage/homology/chain_complex_morphism.py

+124-34
Original file line numberDiff line numberDiff line change
@@ -27,7 +27,9 @@
2727
sage: G = Hom(C,C)
2828
sage: x = G(f)
2929
sage: x
30-
Chain complex morphism from Chain complex with at most 2 nonzero terms over Integer Ring to Chain complex with at most 2 nonzero terms over Integer Ring
30+
Chain complex morphism
31+
From: Chain complex with at most 2 nonzero terms over Integer Ring
32+
To: Chain complex with at most 2 nonzero terms over Integer Ring
3133
sage: x._matrix_dictionary
3234
{0: [0 0 0]
3335
[0 0 0]
@@ -71,8 +73,9 @@ def is_ChainComplexMorphism(x):
7173
sage: i = H.identity()
7274
sage: x = i.associated_chain_complex_morphism()
7375
sage: x # indirect doctest
74-
Chain complex morphism from Chain complex with at most 7 nonzero terms over
75-
Integer Ring to Chain complex with at most 7 nonzero terms over Integer Ring
76+
Chain complex morphism
77+
From: Chain complex with at most 7 nonzero terms over Integer Ring
78+
To: Chain complex with at most 7 nonzero terms over Integer Ring
7679
sage: is_ChainComplexMorphism(x)
7780
True
7881
"""
@@ -101,9 +104,9 @@ def __init__(self, matrices, C, D, check=True):
101104
sage: G = Hom(C,C)
102105
sage: x = G(f)
103106
sage: x
104-
Chain complex morphism from Chain complex with at most 2 nonzero terms
105-
over Integer Ring to Chain complex with at most 2 nonzero terms over
106-
Integer Ring
107+
Chain complex morphism
108+
From: Chain complex with at most 2 nonzero terms over Integer Ring
109+
To: Chain complex with at most 2 nonzero terms over Integer Ring
107110
sage: x._matrix_dictionary
108111
{0: [0 0 0]
109112
[0 0 0]
@@ -117,9 +120,9 @@ def __init__(self, matrices, C, D, check=True):
117120
sage: Y = simplicial_complexes.Simplex(0)
118121
sage: g = Hom(X,Y)({0:0, 1:0})
119122
sage: g.associated_chain_complex_morphism()
120-
Chain complex morphism from Chain complex with at most 2 nonzero
121-
terms over Integer Ring to Chain complex with at most 1 nonzero terms
122-
over Integer Ring
123+
Chain complex morphism
124+
From: Chain complex with at most 2 nonzero terms over Integer Ring
125+
To: Chain complex with at most 1 nonzero terms over Integer Ring
123126
124127
Check that an error is raised if the matrices are the wrong size::
125128
@@ -130,7 +133,9 @@ def __init__(self, matrices, C, D, check=True):
130133
...
131134
ValueError: matrix in degree 0 is not the right size
132135
sage: Hom(C,D)({0: matrix(2, 1, [1, 1])}) # 2x1 is right.
133-
Chain complex morphism from Chain complex with at most 1 nonzero terms over Integer Ring to Chain complex with at most 1 nonzero terms over Integer Ring
136+
Chain complex morphism
137+
From: Chain complex with at most 1 nonzero terms over Integer Ring
138+
To: Chain complex with at most 1 nonzero terms over Integer Ring
134139
"""
135140
if not C.base_ring() == D.base_ring():
136141
raise NotImplementedError('morphisms between chain complexes of different'
@@ -177,10 +182,43 @@ def __init__(self, matrices, C, D, check=True):
177182
mC = matrices[i+d] * C.differential(i)
178183
if mC != Dm:
179184
raise ValueError('matrices must define a chain complex morphism')
180-
self._matrix_dictionary = matrices
185+
self._matrix_dictionary = {}
186+
for i in matrices:
187+
m = matrices[i]
188+
# Use immutable matrices because they're hashable.
189+
m.set_immutable()
190+
self._matrix_dictionary[i] = m
181191
self._domain = C
182192
self._codomain = D
183193

194+
def domain(self):
195+
"""
196+
The domain of this chain map
197+
198+
EXAMPLES::
199+
200+
sage: C = ChainComplex({0: identity_matrix(ZZ, 1)})
201+
sage: D = ChainComplex({0: zero_matrix(ZZ, 1), 1: zero_matrix(ZZ, 1)})
202+
sage: f = Hom(C,D)({0: identity_matrix(ZZ, 1), 1: zero_matrix(ZZ, 1)})
203+
sage: f.domain()
204+
Chain complex with at most 2 nonzero terms over Integer Ring
205+
"""
206+
return self._domain
207+
208+
def codomain(self):
209+
"""
210+
The codomain of this chain map
211+
212+
EXAMPLES::
213+
214+
sage: C = ChainComplex({0: identity_matrix(ZZ, 1)})
215+
sage: D = ChainComplex({0: zero_matrix(ZZ, 1), 1: zero_matrix(ZZ, 1)})
216+
sage: f = Hom(C,D)({0: identity_matrix(ZZ, 1), 1: zero_matrix(ZZ, 1)})
217+
sage: f.codomain()
218+
Chain complex with at most 3 nonzero terms over Integer Ring
219+
"""
220+
return self._codomain
221+
184222
def in_degree(self, n):
185223
"""
186224
The matrix representing this morphism in degree n
@@ -213,9 +251,44 @@ def in_degree(self, n):
213251
return self._matrix_dictionary[n]
214252
except KeyError:
215253
from sage.matrix.constructor import zero_matrix
216-
rows = self._codomain.free_module_rank(n)
217-
cols = self._domain.free_module_rank(n)
218-
return zero_matrix(self._domain.base_ring(), rows, cols)
254+
rows = self.codomain().free_module_rank(n)
255+
cols = self.domain().free_module_rank(n)
256+
return zero_matrix(self.domain().base_ring(), rows, cols)
257+
258+
def dual(self):
259+
"""
260+
The dual chain map to this one.
261+
262+
That is, the map from the dual of the codomain of this one to
263+
the dual of its domain, represented in each degree by the
264+
transpose of the corresponding matrix.
265+
266+
EXAMPLES::
267+
268+
sage: X = simplicial_complexes.Simplex(1)
269+
sage: Y = simplicial_complexes.Simplex(0)
270+
sage: g = Hom(X,Y)({0:0, 1:0})
271+
sage: f = g.associated_chain_complex_morphism()
272+
sage: f.in_degree(0)
273+
[1 1]
274+
sage: f.dual()
275+
Chain complex morphism
276+
From: Chain complex with at most 1 nonzero terms over Integer Ring
277+
To: Chain complex with at most 2 nonzero terms over Integer Ring
278+
sage: f.dual().in_degree(0)
279+
[1]
280+
[1]
281+
sage: ascii_art(f.domain())
282+
[-1]
283+
[ 1]
284+
0 <-- C_0 <----- C_1 <-- 0
285+
sage: ascii_art(f.dual().codomain())
286+
[-1 1]
287+
0 <-- C_1 <-------- C_0 <-- 0
288+
"""
289+
matrix_dict = self._matrix_dictionary
290+
matrices = {i: matrix_dict[i].transpose() for i in matrix_dict}
291+
return ChainComplexMorphism(matrices, self.codomain().dual(), self.domain().dual())
219292

220293
def __neg__(self):
221294
"""
@@ -248,7 +321,7 @@ def __neg__(self):
248321
f = dict()
249322
for i in self._matrix_dictionary.keys():
250323
f[i] = -self._matrix_dictionary[i]
251-
return ChainComplexMorphism(f, self._domain, self._codomain)
324+
return ChainComplexMorphism(f, self.domain(), self.codomain())
252325

253326
def __add__(self,x):
254327
"""
@@ -278,12 +351,12 @@ def __add__(self,x):
278351
[0 0 0 2]}
279352
280353
"""
281-
if not isinstance(x,ChainComplexMorphism) or self._codomain != x._codomain or self._domain != x._domain or self._matrix_dictionary.keys() != x._matrix_dictionary.keys():
354+
if not isinstance(x,ChainComplexMorphism) or self.codomain() != x.codomain() or self.domain() != x.domain() or self._matrix_dictionary.keys() != x._matrix_dictionary.keys():
282355
raise TypeError("Unsupported operation.")
283356
f = dict()
284357
for i in self._matrix_dictionary.keys():
285358
f[i] = self._matrix_dictionary[i] + x._matrix_dictionary[i]
286-
return ChainComplexMorphism(f, self._domain, self._codomain)
359+
return ChainComplexMorphism(f, self.domain(), self.codomain())
287360

288361
def __mul__(self,x):
289362
"""
@@ -350,25 +423,27 @@ def __mul__(self,x):
350423
sage: f = ChainComplexMorphism({}, C0, C1)
351424
sage: g = ChainComplexMorphism({}, C1, C2)
352425
sage: g * f
353-
Chain complex morphism from Chain complex with at most 1 nonzero terms over Integer Ring to Chain complex with at most 1 nonzero terms over Integer Ring
426+
Chain complex morphism
427+
From: Chain complex with at most 1 nonzero terms over Integer Ring
428+
To: Chain complex with at most 1 nonzero terms over Integer Ring
354429
sage: f._matrix_dictionary
355430
{0: [], 1: []}
356431
sage: g._matrix_dictionary
357432
{1: [], 2: []}
358433
"""
359-
if not isinstance(x,ChainComplexMorphism) or self._domain != x._codomain:
434+
if not isinstance(x,ChainComplexMorphism) or self.domain() != x.codomain():
360435
try:
361-
y = self._domain.base_ring()(x)
436+
y = self.domain().base_ring()(x)
362437
except TypeError:
363438
raise TypeError("multiplication is not defined")
364439
f = dict()
365440
for i in self._matrix_dictionary:
366441
f[i] = self._matrix_dictionary[i] * y
367-
return ChainComplexMorphism(f,self._domain,self._codomain)
442+
return ChainComplexMorphism(f,self.domain(),self.codomain())
368443
f = dict()
369444
for i in self._matrix_dictionary:
370445
f[i] = self._matrix_dictionary[i]*x.in_degree(i)
371-
return ChainComplexMorphism(f,x._domain,self._codomain)
446+
return ChainComplexMorphism(f,x.domain(),self.codomain())
372447

373448
def __rmul__(self,x):
374449
"""
@@ -386,13 +461,13 @@ def __rmul__(self,x):
386461
False
387462
"""
388463
try:
389-
y = self._domain.base_ring()(x)
464+
y = self.domain().base_ring()(x)
390465
except TypeError:
391466
raise TypeError("multiplication is not defined")
392467
f = dict()
393468
for i in self._matrix_dictionary.keys():
394469
f[i] = y * self._matrix_dictionary[i]
395-
return ChainComplexMorphism(f,self._domain,self._codomain)
470+
return ChainComplexMorphism(f,self.domain(),self.codomain())
396471

397472
def __sub__(self,x):
398473
"""
@@ -435,8 +510,9 @@ def __eq__(self,x):
435510
sage: i = H.identity()
436511
sage: x = i.associated_chain_complex_morphism()
437512
sage: x
438-
Chain complex morphism from Trivial chain complex over Integer Ring
439-
to Trivial chain complex over Integer Ring
513+
Chain complex morphism
514+
From: Trivial chain complex over Integer Ring
515+
To: Trivial chain complex over Integer Ring
440516
sage: f = x._matrix_dictionary
441517
sage: C = S.chain_complex()
442518
sage: G = Hom(C,C)
@@ -445,10 +521,22 @@ def __eq__(self,x):
445521
True
446522
"""
447523
return isinstance(x,ChainComplexMorphism) \
448-
and self._codomain == x._codomain \
449-
and self._domain == x._domain \
524+
and self.codomain() == x.codomain() \
525+
and self.domain() == x.domain() \
450526
and self._matrix_dictionary == x._matrix_dictionary
451527

528+
def __hash__(self):
529+
"""
530+
TESTS::
531+
532+
sage: C = ChainComplex({0: identity_matrix(ZZ, 1)})
533+
sage: D = ChainComplex({0: zero_matrix(ZZ, 1)})
534+
sage: f = Hom(C,D)({0: identity_matrix(ZZ, 1), 1: zero_matrix(ZZ, 1)})
535+
sage: hash(f) # random
536+
17
537+
"""
538+
return hash(self.domain()) ^ hash(self.codomain()) ^ hash(tuple(self._matrix_dictionary.items()))
539+
452540
def _repr_(self):
453541
"""
454542
Return the string representation of ``self``.
@@ -460,11 +548,13 @@ def _repr_(self):
460548
sage: i = H.identity()
461549
sage: x = i.associated_chain_complex_morphism()
462550
sage: x
463-
Chain complex morphism from Trivial chain complex over Integer Ring
464-
to Trivial chain complex over Integer Ring
551+
Chain complex morphism
552+
From: Trivial chain complex over Integer Ring
553+
To: Trivial chain complex over Integer Ring
465554
sage: x._repr_()
466-
'Chain complex morphism from Trivial chain complex over Integer Ring
467-
to Trivial chain complex over Integer Ring'
555+
'Chain complex morphism\n From: Trivial chain complex over Integer Ring\n To: Trivial chain complex over Integer Ring'
468556
"""
469-
return "Chain complex morphism from {} to {}".format(self._domain, self._codomain)
470-
557+
s = 'Chain complex morphism'
558+
s += '\n From: {}'.format(self.domain())
559+
s += '\n To: {}'.format(self.codomain())
560+
return s

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