Timeline for Functors with Mayer-Vietoris Sequences
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 18, 2014 at 8:53 | history | edited | Matthias Ludewig | CC BY-SA 3.0 |
added 132 characters in body
|
| May 15, 2014 at 22:06 | comment | added | Matthias Ludewig | Isn't the right arrow always zero in K-Theory? | |
| May 15, 2014 at 21:28 | comment | added | Tom Goodwillie | Oh, the left!? I take it all back. But what kind of $K$-theory gives you a zero on the right like that? | |
| May 15, 2014 at 21:02 | history | edited | Matthias Ludewig | CC BY-SA 3.0 |
added 52 characters in body
|
| May 15, 2014 at 20:43 | comment | added | Matthias Ludewig | I am confused. Sheaf Cohomology does give me a Mayer-Vietoris in positive direction, but here I would like to continue my sequence to the left. The example to have in mind is probably K theory, which does not arise from sheaf cohomology. | |
| May 15, 2014 at 17:21 | comment | added | Qiaochu Yuan | Derived functors make sense in a great deal more generality than abelian categories! See, for example, math.harvard.edu/~eriehl/cathtpy.pdf. Anyway, I think a keyword you might want to look up is "homotopy sheafification." | |
| May 15, 2014 at 15:03 | comment | added | Tom Goodwillie | Or maybe you do not need to bother with direct limits (or with requiring $F(\emptyset)=0$). It appears that $F$ is a sheaf with respect to the topology generated by ordinary two-set open covers, so that sheaf cohomology for that topology should do the job. | |
| May 15, 2014 at 14:16 | comment | added | Johannes Hahn | $F$ is a presheaf on any manifold and the mayer-vietoris axiom is a special case of the gluing axiom. If one has an additional continuity assumption (I'm thinking of something along the lines of $F(U) = \lim F(U_i)$ for any directed system of open subsets), it should be possible to derive the full gluing axiom. Then you have a sheaf of abelian groups and can do the usual construction. | |
| May 15, 2014 at 14:05 | comment | added | Paul Siegel | Presumably $F$ should satisfy some sort of excision axiom? | |
| May 15, 2014 at 13:33 | history | asked | Matthias Ludewig | CC BY-SA 3.0 |