Timeline for Is there a Hopf algebra-style description of chain complexes?
Current License: CC BY-SA 4.0
14 events
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| Oct 3, 2023 at 13:32 | comment | added | Bbb | Ah perhaps you were not asking about stable $\infty$-categories in the first place... | |
| Oct 3, 2023 at 13:26 | comment | added | Bbb | @TimCampion am I missing something that all the answers so far give geometric objects over $\mathbb Z$, so they'd work for something like $\mathbb Z$-linear stable $\infty$ categories, rather than arbitrary infinity categories? For example, in spectra, filtered objects are sheaves over $\mathbb A^1/ \mathbb G_m$, where $\mathbb A^1 = Spec(\mathbb S[\mathbb N]), \mathbb G_, = Spec(\mathbb S[\mathbb Z])$. Over the sphere it's not so clear to me what Achim's $\Lambda$ should be, even less clear what Sanath's $\mathbb G_a^\#$ should be.... | |
| Apr 28, 2023 at 14:02 | comment | added | Tim Campion | For my own reference, I want to record Aaron Mazel-Gee's observation on Discord that this stuff is related to Khovanov's Hopfological Algebra | |
| Apr 28, 2023 at 12:04 | answer | added | skd | timeline score: 5 | |
| Apr 28, 2023 at 11:42 | history | became hot network question | |||
| Apr 28, 2023 at 11:13 | vote | accept | Tim Campion | ||
| Apr 28, 2023 at 6:33 | answer | added | Achim Krause | timeline score: 8 | |
| Apr 28, 2023 at 4:19 | comment | added | Tim Campion | @skd I'd love to hear more about this! I don't understand the role played by Cartier duality here, for one thing. | |
| Apr 28, 2023 at 3:54 | comment | added | skd | Sorry, I meant the classifying stack of the semidirect product G_a^# x| G_m, where G_m is acting with weight 1. | |
| Apr 28, 2023 at 3:47 | comment | added | skd | Unless I’m severely misunderstanding your question, it seems that one candidate for your S is BG_a^#, where G_a^# is the divided power hull of the origin in G_a. (This’ll recover cochain complexes, corresponding to complete filtered objects.) This is coming from the fact that G_a^# is Cartier dual to the formal completion of G_a at the origin. | |
| Apr 28, 2023 at 3:20 | comment | added | Kapil | Perhaps you are interested in the work of Balmer et al on Tensor Triangulated Categories. Briefly, a space is associated with such a category. Note that filtered objects carry a natural tensor structure. Without a tensor structure, it is not obvious where the product/co-product for your Hopf-like structure will come from. | |
| Apr 28, 2023 at 2:51 | history | edited | Tim Campion |
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| Apr 28, 2023 at 2:42 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Apr 28, 2023 at 2:36 | history | asked | Tim Campion | CC BY-SA 4.0 |