Category of chain complexes: (co)homology functor

[mjungmath]

In this ticket, we equip the category of chain complexes with a method homology yielding the associated homology. The corresponding homology functor will use this method.

This happens in view of Čech cohomology, Morse homology and de Rham cohomology on manifolds.

Comments and opinions are as always welcome.

Depends on #31691

CC: @mkoeppe @tscrim @egourgoulhon @jhpalmieri

Component: categories

Keywords: homology

Author: Michael Jung

Branch/Commit: 4bceb9b

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/31669

[mjungmath]

Branch: u/gh-mjungmath/category_of_chain_complexes___co_homology_functor

[mjungmath]

comment:3

First draft. Comments are welcome.

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New commits:

bcdee3a	Trac #31669: add cochains and (co)homology method
0393c25	Trac #31669: (co)homology functor added

[mjungmath]

Commit: 0393c25

[sagetrac-git]

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

8bcdfdf

Trac #31669: minor typos

[sagetrac-git]

Changed commit from 0393c25 to 8bcdfdf

[mjungmath]

Description changed:

--- 
+++ 
@@ -1,4 +1,4 @@
-In this ticket, we equip the category of chain complexes with a method `homology` yielding the associated homology. The corresponding homology functor will use this method. We furthermore establish a new category of cochain complexes.
+In this ticket, we equip the category of chain complexes with a method `homology` yielding the associated homology. The corresponding homology functor will use this method.
 
 This happens in view of Čech cohomology, Morse homology and de Rham cohomology on manifolds.
 

[mjungmath]

Changed keywords from homology, cohomology to homology

[mjungmath]

Author: Michael Jung

[mjungmath]

comment:5

chain_complex doesn't make a distinction between chain and cochain complexes. Probably because it also allows multigrading.

Should we remove the category of cochain complexes then? I mean, in a broad sense, cochain complexes can be seen as generalized chain complexes, no?

Btw, is it customary to add a mathematical description in the category section?

[sagetrac-git]

Changed commit from 8bcdfdf to 718d354

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

6988be0	Trac #31669: homology functor
718d354	Trac #31669: add category of chain complexes to commutative_dga