Timeline for Axioms of derivators
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Oct 29, 2022 at 19:29 | vote | accept | user234212323 | ||
| Oct 24, 2022 at 9:42 | comment | added | David Roberts♦ | There is a bit more discussion nearer the end of section 71, around desiderata for derivators taking more general values. There's a little more general discussion in section 100. I think it worth looking through these comments, even if they are a bit mysterious, since it shows exactly the kinds of problems Grothendieck was thinking about as he invented the concept (here's PS on the arXiv). A far cry from the much more dry and traditional document Les Derivateurs | |
| Oct 24, 2022 at 9:27 | comment | added | David Roberts♦ | Aha, I remembered correctly. Section 69 of Pursuing Stacks discusses derivators, and the example given is that triangulated categories were not enough, and one should consider, associated to an abelian category $\mathcal{A}$, for each 'diagram shape' $I$, the derived category $D(Hom(I,\mathcal{A}))$ of the abelian category of $I$-shaped diagrams in $\mathcal{A}$, and this data was to encode the missing limits and colimits in the ordinary derived category $D(\mathcal{A})$. This was the inspiration to consider the analogue for the homotopy category of an arbitrary localiser... | |
| Oct 24, 2022 at 8:08 | comment | added | David Roberts♦ | I'm wondering if there was anything in Pursuing Stacks that prefigured Grothendieck's Les Derivateurs. I have a vague memory there might be, but it was a long time ago I read PS through. Grothendieck would certainly discuss his thought process, because PS was more like a research diary than a finished article, which LD is more in the style of. | |
| Oct 24, 2022 at 4:15 | answer | added | Kevin Carlson | timeline score: 5 | |
| Oct 19, 2022 at 18:28 | comment | added | user234212323 | @The reason nLab was not enough was that the notion is well motivated, but I would like to know about the particular choice of axioms. You can motivate a notion, for example, a group (resp. topological space) as a theory of symmetry (with localization) but the choice of axioms seems to be part of an experiment (Hausdorff axioms, open subsets,...). I am interested in a motivation for the choice of axioms. | |
| Oct 18, 2022 at 20:09 | comment | added | Kevin Carlson | All of the axioms are designed to hone in on those prederivators which look something like the 2-functors of homotopy categories of diagrams in some abstract homotopy theory, that is, $J\mapsto \mathrm{Ho}(C^J),$ where $C$ is a nice model category or $\infty$-category. If that much is already clear then please be more specific about what you’re looking for. | |
| Oct 16, 2022 at 22:18 | comment | added | Zhen Lin | The nLab article you link to has some explanatory remarks about the axioms. Do you want us to explain those remarks, or...? | |
| Oct 16, 2022 at 15:37 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals
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| Oct 16, 2022 at 15:08 | history | asked | user234212323 | CC BY-SA 4.0 |