Chain homotopies

[jhpalmieri]

This ticket add chain homotopies, chain contractions, and duals of chain maps and chain homotopies. This is a dependency for #6101 and #6102.

CC: @tscrim @fchapoton

Component: algebraic topology

Keywords: homotopy

Author: John Palmieri

Branch/Commit: 4d53526

Reviewer: Frédéric Chapoton

Issue created by migration from https://trac.sagemath.org/ticket/19179

[jhpalmieri]

Branch: u/jhpalmieri/chains

[jhpalmieri]

New commits:

dece275

trac 19179: chain homotopies, chain contractions, and duals of chain maps

[jhpalmieri]

Commit: dece275

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

d7b9ed4

change AssertionError to other errors

[sagetrac-git]

Changed commit from dece275 to d7b9ed4

[jhpalmieri]

Dependencies: #6102

[jhpalmieri]

comment:4

It's too messy for me to disentangle this from #6102, so that ticket and this are now dependencies of each other. In particular, the most natural way to construct examples of chain contractions is to use the algebraic topological model of a complex (which is part of #6102). So some of the examples here use code that is part of #6102. The actual code here doesn't depend on #6102, just those few examples.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

d65ba8d

trac 19179: change `_repr_` for chain maps, chain homotopies

[sagetrac-git]

Changed commit from d7b9ed4 to d65ba8d

[sagetrac-git]

Changed commit from d65ba8d to f70e3a2

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

f70e3a2

trac 19179: doctest fixes for `_repr_` changes

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

4e50776

trac 19179: hashing of chain maps, chain homotopies

[sagetrac-git]

Changed commit from f70e3a2 to 4e50776

[jhpalmieri]

comment:8

#18246 broke the default hashing of chain homotopies, so I've added a __hash__ method, and also one for chain maps.

[fchapoton]

comment:9

This looks strange, in chain_complex_morphism.py:

+        self._matrix_dictionary = {}
+        for i in matrices:
+            m = matrices[i]
+            # Use immutable matrices because they're hashable.
+            m.set_immutable()
+            self._matrix_dictionary[i] = m
         self._matrix_dictionary = matrices

Should the last line be removed ?

[jhpalmieri]

comment:10

Yes, thank you.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

07c9c42

trac 19179: delete extra line

[sagetrac-git]

Changed commit from 4e50776 to 07c9c42

[cnassau]

comment:12

I've been playing around with this for the last week and I'm essentially ready to give the ticket a positive review. I can think of a couple of very minor improvements that might be better addressed in a follow-up ticket:

• add syntactic sugar to replace H._g with something like H[1], or H.endpoint(1) or whatever
• create a parent for chain homotopies so that linear combinations can be taken easily

I encountered one more serious surprise in my experiments, and I'm not quite sure how to address this: I was starting with a chain complex C over the integers, and I wanted to work with its reduction modulo 2. The code that I came up with actually messed up the original complex:

sage: d1 = matrix([[-1,1,0],[1,-1,0],[-1,0,1],[1,0,-1]]).transpose()
sage: d2 = matrix([[-1,-1,+1,+1],[1,1,1,1]]).transpose()
sage: C=ChainComplex({2:d2,1:d1},degree_of_differential=-1,base_ring=ZZ)
sage: ascii_art(C)
                                [-1  1]      
            [-1  1 -1  1]       [-1  1]      
            [ 1 -1  0  0]       [ 1  1]      
            [ 0  0  1 -1]       [ 1  1]      
 0 <-- C_0 <-------------- C_1 <-------- C_2 <-- 0 
sage: D=ChainComplex(C._diff,degree_of_differential=C.degree_of_differential(),base_ring=GF(2))
sage: ascii_art(C)
                            [1 1]      
            [1 1 1 1]       [1 1]      
            [1 1 0 0]       [1 1]      
            [0 0 1 1]       [1 1]      
 0 <-- C_0 <---------- C_1 <------ C_2 <-- 0 

I would have expected an error here, and possibly a more convenient way to carry out the base change.

As for the patchbot errors: these are a bit confusing; I think they're all due to the fact that the dependencies of the ticket have not been properly merged for the test. Given the dependencies with #6101, #6102 I wonder if it isn't the best approach to approve them all at once, eventually.... (?)

[jhpalmieri]

comment:13

With your example, using D = Chain Complex(C.differential(), ...) works better than with C._diff, because C.differential() returns a copy of the defining data, and so modifying it won't change C. It can be dangerous using hidden attributes, and the corresponding methods ought to be safer.

If you are willing to give this a positive review, then #6101 should come next, but neither will get merged without the other.

[fchapoton]

comment:14

a few remarks on the doc:

• please remove the dot at the end of the title line of the new file.

• Please remove the \circ in H \circ H at several places where it appears (or, if you really prefer, put such a composition symbol everywhere else), for coherence with other compositions.

• What happens in chain homotopy if the given g is None ? This should be explained in the doc.

• The def of "dual" of chain homotopy class should start with r"""

• In dual for chain contraction, skip one line after the first sentence

[fchapoton]

Changed keywords from none to homotopy

[sagetrac-git]

Changed commit from 07c9c42 to 4d53526

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

4d53526

trac 19179: minor doc fixes

[jhpalmieri]

comment:16

Replying to @fchapoton:

a few remarks on the doc:

• please remove the dot at the end of the title line of the new file.

Done

• Please remove the \circ in H \circ H at several places where it appears (or, if you really prefer, put such a composition symbol everywhere else), for coherence with other compositions.

Done (removed \circ)

• What happens in chain homotopy if the given g is None ? This should be explained in the doc.

This was already there, but I added a sentence.

• The def of "dual" of chain homotopy class should start with r"""

Right, fixed.

• In dual for chain contraction, skip one line after the first sentence

Done.

[fchapoton]

comment:17

ok, thanks. Then good to go. It will of course be pending until #6102 is reviewed.

[fchapoton]

Reviewer: Frédéric Chapoton

[vbraun]

comment:20

Circular dependency, please make up your mind what you want merged first

[fchapoton]

comment:21

This one first please, but it will not pass the doctests alone.

[fchapoton]

Changed dependencies from #6102 to none

[vbraun]

Changed branch from u/jhpalmieri/chains to 4d53526