1
$\begingroup$

Most of the homological algebra books that I've checked develop chain complexes in the context of modules over a ring. The two exceptions I know would be Schapira notes "Categories and homological algebra" and the section 12.13 on stacks project.

I wanted to ask if you can point me to other references (es. notes) that develop them in the context of abelian categories, possibly giving more details than the aforementioned. My knowledge of abelian categories comes from Borceux Categorical Algebra II, so for example I would expect the definition of homology to be something similar to the following (ie. making use of exactness and additivity):

enter image description here

I understand that there is no particular merit of writing down those results without using elements, I ask this as personal linguistic preference.

$\endgroup$
1
$\begingroup$

Perhaps this paper of Heller is what you're looking for (it is called, very aptly, Homological Algebra in Abelian Categories.)

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Is the theory ("regular" additive categories) related to "Quillen exactness"? Is the term "proper maps" related to proper morphisms or used elsewhere? Thanks for the link, I'll try to read it. $\endgroup$ – piombino Oct 22 at 23:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.