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• CommentRowNumber1.
• CommentAuthorzskoda
• CommentTimeMay 23rd 2011
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMay 23rd 2011

Probably in the first line you meant to write “homological” instead of “regular”.

• CommentRowNumber3.
• CommentAuthorjim_stasheff
• CommentTimeMay 24th 2011
It would help to have a brief summary of what makes a cat homological!
I assume it is NOT just a dg cat?
• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeMay 24th 2011

Of course it is not. It is not an enriched category, but an ordinary one category satisfying some conditions. Of course, the homological algebra is then done with an appropriate notion of complexes in it and the five lemma is a lemma in the original category. Typical examples are categories of algebras over certain subclass of algebraic theories in Set. Every semi-abelian category is an example. There is also a generalization, relative homological categories.

Jim, I and mislead and do not like the terminology in this case either. It is misleading. Bourn and Janelidze thank Johnstone for the help with terminology, and my experience in past is that Johnstone often has strange inventions in terminology.

• CommentRowNumber5.
• CommentAuthorTim_Porter
• CommentTimeDec 9th 2018

Fixed some formatting

1. Added sidebar, TOC, section headings, related entries, and more details to a reference.

Rongmin Lu

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeSep 22nd 2021

I have added (here) the example of $Grp$, with pointers to why this is homological (a fact that the references listed here forget to make explicit)