Timeline for Stable infinite category counterpart of pathological behaviours around the AB3,AB4 and AB5 axioms of abelian categories
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| 1 hour ago | comment | added | Piotr Achinger | +1 for "weirdness of the heart" | |
| 1 hour ago | history | edited | David Roberts♦ |
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| 5 hours ago | answer | added | user509184 | timeline score: 1 | |
| 6 hours ago | comment | added | Denis T | To sum up: try to formulate exactly what you mean by "pathological behaviour". Most likely there's an example of a stable derivator with the properties you require. | |
| 7 hours ago | comment | added | Denis T | The point is that you do not see "highly counter-intuitive behavior" of usual unbounded derived categories of module categories/Grothendieck categories just because you do not know where to look. Those abelian categories themselves are very complicated and weird already, as soon as you look above countably-presented objects. | |
| 7 hours ago | comment | added | Denis T | For example, you can think a bit harder about homotopy category of complexes of abelian groups. This category is already pretty wild; it does not satisfy any form of Brown presentability whatsoever, it has a non-localising subcategory closed under direct sums, etc, etc. | |
| 7 hours ago | comment | added | Denis T | I think that this example is only seems strange to you (and to most people who think that the category of abelian groups is "easy and familiar"), because it pushes the discrepancy between Mittag-Leffler property and surjectiveness of transition maps all the way down to the cardinal $\omega$. There's basically no way to control exactness of cofiltered limits even in the nicest abelian categories (for example, abelian groups), if you consider uncountable diagrams. | |
| 7 hours ago | history | asked | dh35jvn | CC BY-SA 4.0 |