Timeline for K-theory of finite diagram categories
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Nov 2, 2018 at 15:41 | comment | added | Marc Hoyois | @TylerLawson Ah yes, my mistake. Thanks for the correction! | |
| Nov 2, 2018 at 9:45 | comment | added | Tyler Lawson | @MarcHoyois $B\Bbb Z$ turns out to be finite. There's a presentation of $B\Bbb Z$ with one object, three morphisms $l, f, r$, and two homotopies $lf = id$, $fr = id$. (If you demand that the left inverse and right inverse are equal then you get the wrong answer and have to add higher cells; this is related to $B\Bbb Z$ not having a finite $C_2$-equivariant description) | |
| Nov 2, 2018 at 5:25 | comment | added | Marc Hoyois | I don't think $B\mathbb Z$ is a finite ∞-category either, but $B\mathbb N$ is. | |
| Nov 2, 2018 at 2:43 | comment | added | John Berman | Yes, you're right... I was being too naive. But I am still curious if there is some form of splitting like in the paper I referenced. The terms would not be Z, but K_0 of something coming from the automorphism groups in I. | |
| Oct 31, 2018 at 0:06 | comment | added | Phil Tosteson | Great, I think this right. I also think answer would change if we took the K theory of finitely generated (i.e. compact) $I$ spaces. | |
| Oct 30, 2018 at 23:49 | history | edited | Phil Tosteson | CC BY-SA 4.0 |
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| Oct 30, 2018 at 23:41 | comment | added | Tom Goodwillie | Even for $I=\mathbb Z$ the answer is no. A diagram is a finite complex $X$ with automorphism $a:X\to X$, right? And there are at least two independent maps from the $K$-group to $\mathbb Z$, the Euler characterisitc of $X$ and the Lefschetz number of $a$. | |
| Oct 30, 2018 at 21:15 | history | edited | Phil Tosteson | CC BY-SA 4.0 |
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| Oct 30, 2018 at 21:14 | comment | added | Phil Tosteson | OK thanks, I thought finite might mean postnikov finite instead of CW finite. | |
| Oct 30, 2018 at 20:29 | comment | added | John Berman | It's a finite category, but not a finite $\infty$-category, because the nerve $\mathbb{R}P^\infty=B\mathbb{Z}/2\mathbb{Z}$ is not a finite cell complex. If you want to think about ordinary categories, I am interested in categories like finite posets, or groups like $\mathbb{Z}$, $\mathbb{Z}\times\mathbb{Z}$, or $\mathbb{Z}\ast\mathbb{Z}$. (Finiteness is rather different for categories and $\infty$-categories.) | |
| Oct 30, 2018 at 18:38 | history | edited | Phil Tosteson | CC BY-SA 4.0 |
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| Oct 30, 2018 at 17:48 | history | answered | Phil Tosteson | CC BY-SA 4.0 |