Timeline for A theory of higher limits of (1-)functors, after higher hochschild homology
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| Nov 14, 2022 at 15:46 | comment | added | David Ben-Zvi | In your setup for C a small R-linear category we can take $Ind(C)$ which is a dualizable presentable R-linear category with dual $Ind(C^{op})$. Your bimodule D defines an object of $End(Ind(C))=Ind(C)\otimes Ind(C^{op})$, and you are I believe calculating its Hochschild cohomology in this sense. It also has a trace ($HH_*$) and such traces are the subject of a very active field of research (categorical traces -- see eg papers of Xinwen Zhu applying them to number theory). | |
| Nov 14, 2022 at 15:43 | comment | added | David Ben-Zvi | I might be getting confused but it seems to me you're writing something like the following notion: given a dualizable object of any symmetric monoidal higher category and an endomorphism, we can consider its trace (Hochschild homology) and I believe there's also a notion of centralizer (Hochschild cohomology)- I haven't thought about that in that generality, but it certainly makes sense in the present context | |
| Nov 14, 2022 at 14:58 | comment | added | Emily | (I have edited the question to clarify this) | |
| Nov 14, 2022 at 14:58 | comment | added | Emily | I also mention these being orthogonal in a sense to higher category theory because e.g. derivations between two functors form a notion of "2-natural transformation" that is different from the usual one (and of course that usual one doesn't make sense here, as $F$ and $G$ are 1-functors) | |
| Nov 14, 2022 at 14:58 | comment | added | Emily | @DylanWilson I understand that some of these may be expressed as homotopy ends/limits (I don't immediately see how all of these are though, e.g. do you know how to express the module of derivations $\mathrm{Ker}(d^2)$ between two functors $F$ and $G$ defined above as a homotopy limit/end?), but what I'm insisting on is that this doesn't make any of these questions trivial ("this is basically just homotopy coends/colimits"), nor does it make any of these invariants any easier to compute. | |
| Nov 14, 2022 at 14:52 | history | edited | Emily | CC BY-SA 4.0 |
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| Nov 14, 2022 at 13:15 | comment | added | Dylan Wilson | @Emily: Zhen is correct- these really are homotopy ends or homotopy limits. In your "motivation" case they are homotopy ends taken in the derived category of R. The fact that the various invariants you're considering have negative homotopy groups does not take them outside the realm of higher category theory. Invariants of the sort you are considering (including higher centers, and other homotopy limits or mapping objects) are well-studied and basically always have homotopy groups in negative degrees (often they are the homotopy groups of some spectrum). | |
| Nov 14, 2022 at 1:41 | comment | added | Emily | Second, some of the above notions aren't really reducible to homotopy co/ends; e.g. the notion of derivation between two functors above (and perhaps the de Rham cohomology theory of categories we might develop from it), the higher limits defined as the representing objects of $\pi_n(\mathrm{Trans}^\bullet(\Delta_{(-)},D))$ (where it isn't even clear what the correct notion of $\pi_n$ would be), or e.g. the whole accompanying cyclic co/homology version of all the above... How would one even define any of this in terms of homotopy coends? | |
| Nov 14, 2022 at 1:40 | comment | added | Emily | Or has anyone tried taking a group $G$, computing the $\pi_n$'s of the Hochschild simplicial set associated to its hom as a one-object groupoid, and then relating those (which might be viewed as higher characteristic classes of $G$) to group-theoretic information about $G$? | |
| Nov 14, 2022 at 1:40 | comment | added | Emily | First, I'm wondering if the $\pi_n$ themselves have been studied before as the main object of interest, being thought of as "higher co/limits" of $D$. For example, has anyone tried taking the diagram $D$ computing the ring of $p$-adic integers as its limit, and then studying the homotopy (co)groups of $\mathrm{HH}^\bullet(\mathrm{Ring};D)$, relating them to number-theoretic information about $\mathbb{Z}_p$? | |
| Nov 14, 2022 at 1:39 | comment | added | Emily | @ZhenLin I know about the connection to homotopy co/ends, Zhen (i.e. that $\pi_n(\mathrm{HH}_\bullet(\mathcal{C};D))$ is the $\pi_n$ of the homotopy coend of $D$), but the questions I'm asking here go in a different direction than this, and most of them can't be subsumed as just being part of the theory of homotopy co/ends. | |
| Nov 13, 2022 at 12:29 | comment | added | Zhen Lin | Well, as you say, this is basically just homotopy coends… | |
| Nov 13, 2022 at 7:00 | history | edited | Emily | CC BY-SA 4.0 |
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| Nov 13, 2022 at 6:51 | history | edited | Emily | CC BY-SA 4.0 |
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| Nov 13, 2022 at 5:17 | comment | added | Emily | (By the way, there's an analogue of all this for cyclic homology, too =) | |
| Nov 13, 2022 at 5:16 | history | asked | Emily | CC BY-SA 4.0 |