5

This question is about the relation between the category of spectra and the category of chain complexes of abelian groups. Specifically, I am trying to understand the examples from Deligne cohomology.

The Deligne complex $\mathbb{Z}(n)$ is defined as the homotopy pullback of the following diagram of complexes sheaves

$$\Omega^n_{\mathrm{cl}}[-n]\simeq \Omega^{\bullet\geq n}\rightarrow \Omega^{n}\simeq \mathbb{R}[0] \leftarrow \mathbb{Z}[0]$$

where we denote $A[n]$ to be the complex concentrated in degree $n$ for an abelian group $A$, and $\mathbb{R}$ stands for locally constant real functions. We have the following hypercohomologies $$H^n(M; \Omega^n_{\mathrm{cl}}[-n])=\Omega^n_{\mathrm{cl}}(M)$$

$$H^n(M; \mathbb{R}[0])=H^n_{\mathrm{ord}}(M; \mathbb{R})=[M, H\mathbb{R}]_{-n}$$

$$H^n(M; \mathbb{Z}[0])=H^n_{\mathrm{ord}}(M; \mathbb{Z})=[M, H\mathbb{Z}]_{-n}$$

where $H^n_{\mathrm{ord}}$ is the ordinary cohomology functor, and $H\mathbb{R}$, $H\mathbb{Z}$ are Eilenberg-MacLane spectra.

It seems to me there is a theorem on Higher Algebra that implies that the Eilenberg-MacLane functor $H$ sends a complex of abelian groups to a spectrum. However, I am not quite familiar with the language of higher categories and couldn't figure out how this works precisely.

My questions are as follows:

  1. For a complex of abelian groups $A^{\bullet}=(A_0\rightarrow A_1\rightarrow \cdots)$, how to construct the spectrum $HA^{\bullet}$?
  2. For a complex of sheaves (with values in abelian groups) $F^{\bullet}=(F_0\rightarrow F_1\rightarrow \cdots)$, how to construct $HF^{\bullet}$, probably a sheaf of spectra?
  3. What's the relation between $HF^{\bullet}$ and the hypercohomology $H^n(M; F^{\bullet})$?

Edit:

  • One motivation for this question is on page 26 of this book, 3.2.2, where $\infty$-categories of derived chain complexes and spectra are considered.

  • As for $HF^{\bullet}$ being a sheaf of spectra (instead of a spectrum), this is based on page 78, 7.3.4 of the same book, where they apply the Eilenberg-MacLane functor degreewise to a sheaf of chain complexes $F^{\bullet}$, and $HF^{\bullet}$ is identified with a sheaf of spectra $\widehat{H\mathbb{Z}}(k)$ defined on page 77.

  • In 7.3.6, an example is presented to show how to calculate differential cohomology via sheaf cohomology. I think this might be a good example to show how a sheaf of chain complexes and a sheaf of spectra are related by the Eilenberg-MacLane functor. For the sheaf of chain complexes $\mathbb{Z}(1)$, or the Deligne complex when $n=1$ we have the following: $$\mathbb{Z}(1)\mapsto H^1(M;\mathbb{Z}(1)),$$ this is taking the sheaf cohomology group of a sheaf of chain complexes. After applying the Eilenberg-MacLane functor degreewise, we get a sheaf of spectra $H\mathbb{Z}(1)$, and its first cohomology group can be calculated as follows: $$H\mathbb{Z}(1) \mapsto H\mathbb{Z}(1)(M) \mapsto \pi_{-1}H\mathbb{Z}(1).$$ I think this is basically evaluating the sheaf on $M$, getting a spectrum then taking the $-1$th homotopy group. I could understand why $H^1(M;\mathbb{Z}(1))$ and $\pi_{-1}H\mathbb{Z}(1)$ are both called "the first cohomology group" but somehow failed to see why they agree. It seems to me I have to understand the Eilenberg-MacLane functor in the first place.

timaeus
  • 331
  • 2
    In question 2., when you talk about spectra, do you have a $1$-category of spectra in mind (for instance the $1$-category of $\Omega$-spectra)? Because you want to talk about the homotopy theory of spectra in 3. most likely, and while different model categories for spectra (of which the category of $\Omega$-spectra is not one) model the same homotopy theory, you do get genuinely different $1$-categories, and therefore genuinely different $1$-categories of sheaves valued in a $1$-category of spectra. I doubt it is even possible to define the $H$-functor in such a way that (cont'd) – Daniël Apol Jan 13 '24 at 23:01
  • 2
    (cont'd) it preserves limits and therefore sends sheaves to sheaves. You would want this in order to have $HF^\bullet$ actually be a sheaf of spectra. However, we can still argue how natural a $1$-sheaf of spectra is, as we really only care about the homotopy theory of spectra, in particular in point 3. But if, on the other hand, you are working with the (unique, up to equivalence) $\infty$-category of spectra, then while you are working purely in their homotopy theory (so that things like limits actually mean what you want them to mean), now you have the problem that (cont'd) – Daniël Apol Jan 13 '24 at 23:05
  • 2
    Limits in the $1$-category $\mathrm{Ch}(\mathbb{Z})$ are not limits in the derived $\infty$-category $\mathcal{D}(\mathbb{Z})$, and therefore the Eilenberg-MacLane functor $\mathrm{Ch}(\mathbb{Z})\to\mathcal{D}(\mathbb{Z})\to\mathrm{Sp}$ does not preserve limits. Therefore it does not induce a functor on sheaves, so that again $HF^\bullet$ is not a(n $\infty$-)sheaf of spectra. (cont'd) – Daniël Apol Jan 13 '24 at 23:10
  • 2
    Since you really want $F^\bullet$ to be a complex of sheaves of abelian groups, I see regardless of whether you consider spectra to form some $1$-category or their natural $\infty$-category no way to actually make $HF^\bullet$ a sheaf of spectra (maybe it is possible if we use the $1$-category of $\Omega$-spectra, as you get quite close at least with Dold-Kan, but it is not trivial at all). Given all this, did you actually read somewhere that $HF^\bullet$ would be a sheaf of spectra? Because if it isn't, then it also becomes less likely that there is an interesting connection in point 3. – Daniël Apol Jan 13 '24 at 23:13
  • @DaniëlApol Thank you for the great comments! I have edited the question accordingly. – timaeus Jan 14 '24 at 09:57
  • The first question is answered by Higher Algebra 1.3.3.5. Basically, the heart of the $\infty$-category of spectra is $\mathrm{Ab}$ and the inclusion $\mathrm{Ab}\subset \mathrm{Sp}$ extends to a functor $\mathcal{D}^-(\mathrm{Ab})\to \mathrm{Sp}$ (HA 1.3.3.2). – Vincent Boelens Jan 16 '24 at 16:52

0 Answers0