Timeline for Abelian category equivalent to a non-abelian category
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Apr 1, 2020 at 3:08 | comment | added | Peter LeFanu Lumsdaine | @Maxime: That’s the main definition I think of too; but even without realising that the enrichment is determined by the other properties (which I agree is not at all obvious), I think it’s pretty straightforward to see that Ab-enrichment transfers along équivalences of categories. The only subtle aspect of this is having the confidence to read “invariant under equivalence” as something you can assert about a structure, not just a property. | |
| Mar 29, 2020 at 11:23 | comment | added | Maxime Ramzi | @PeterLeFanuLumsdaine :the first time I read the definition of an abelian category, it was as an $\mathbf{Ab}$-enriched category with such-and-such properties, so the $\mathbf{Ab}$-enrichment was taken as additional structure in that definition (that was essentially because the documents I was reading wanted to prove some general stuff about $\mathbf{Ab}$-enriched categories before going on to abelian ones). With this in mind, if you've never seen how the $\mathbf{Ab}$-enrichment actually follows from the properties of your abelian category, it's not obviously clear that it should be invariant | |
| Mar 28, 2020 at 18:32 | history | edited | Todd Trimble | CC BY-SA 4.0 |
added the English language reference
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| Mar 28, 2020 at 16:54 | comment | added | Peter LeFanu Lumsdaine | Just to make very clear: the proof is not difficult — it could be a slightly boring homework question in any course that introduces Abelian categories. All the components of the definition of Abelian category are pretty straightforwardly invariant under equivalence of categories for all of the definitions of Abelian category I know. If the OP has read some definition of Abelian category for which this isn’t straightforward, I’d be interested to hear what that definition is. | |
| Mar 26, 2020 at 16:01 | vote | accept | CommunityBot | ||
| Mar 26, 2020 at 14:19 | history | answered | Fred Rohrer | CC BY-SA 4.0 |