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Let's stipulate that

  1. Connective -- i.e. nonnegatively-(homologically)-graded -- chain complexes have a very natural geometric interpretation: by the Dold-Kan theorem, they are a way of thinking about simplicial abelian groups.

  2. Dually, coconnective -- i.e. nonpositively-graded -- chain complexes have a very natural geometric interpretation as complexes of functions on spaces.

The motivations for considering the category of all (unbounded) chain complexes, then, are quite good -- this category provides a home for both the connective and coconnective chain complexes, and has excellent formal properties like stability and a good duality theory.

However, these motivations are quite formal in nature -- they don't provide a geometric interpretation in line with (1) or (2) above. For the most part, these motivations operate "one category level higher", discussing properties of the category of chain complexes. I'm specifically looking for something which gives a geometric, natural way to think about an individual chain complex.

Question 1: What is a good geometric interpretation of nonconnective, noncoconnective chain complexes?

Notes:

  • A similar discussion would more generally stipulate that grouplike $E_\infty$-spaces have a natural geometric interpretation, and ask for a similarly "geometric" interpretation of more general spectra. I'd be equally happy with a discussion in this setting.

  • Similarly, I'd be happy with a discussion in the context of complexes of sheaves of various flavors.

Guess: Here's a guess of a picture which might be appropriate, based on my understanding of Tyler and Adeel's answers to this question, and inspired by Sanath's comment below.

  • A general $E_\infty$ ring spectrum $A$ can be thought of as the global sections $A = \Gamma(X, \mathcal O_X)$ of a spectral scheme $X$ (note that the structure sheaf $\mathcal O_X$ takes values $\Gamma(U,\mathcal O_X)$ in connective $E_\infty$ ring spectra when $U$ is affine open, but its global sections can be nonconnective).

    • Thus the negative-dimensional homotopy groups of $A$ can be thought of as measuring the cohomology of $X$, i.e. the global structure of how $A$ is glued together from affines. The positive-dimensional homotopy groups of $A$ can be thought of as some sort of nilpotent thickening / infinitesimal structure of $X$ -- or perhaps it's better to think of them as encoding the "stacky" part of the structure of $X$.

  • A spectrum $M$ is a module over an $E_\infty$-ring spectrum $A = \Gamma(X, \mathcal O_X)$, and so we should think of $M = \Gamma(X,\mathcal F)$ as the global sections of a quasicoherent sheaf on $X$ (where the values $\Gamma(U,\mathcal F)$ of $\mathcal F$ on affine opens $U$ are connective, but its global sections need not be).

    • Thus the negative-dimensional homotopy groups of $M$ can be thought of as measuring the cohomology of $\mathcal F$, and the positive-dimensional homotopy groups of $M$ can be thought of as the infinitesimal (or rather "stacky") structure of $\mathcal F$.

Question 2: Is this a good picture to have in mind when trying to think about nonconnective, noncoconnective chain complexes / spectra geometrically?

I think I feel a bit more confident in thinking about $E_\infty$ ring spectra this way than I do about thinking about general spectra (or module spectra) in this way.

Question 3: For example, do we have $KU = \Gamma(X, \mathcal O_X)$ for a natural (or even canonical) spectral scheme $X$?

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    $\begingroup$ Are "reduced excisive functors" an acceptable answer? $\endgroup$
    – Denis Nardin
    Commented Apr 2, 2020 at 18:20
  • $\begingroup$ @DenisNardin Hmmm... Maybe? I guess my issue is that a functor is typically an object that lives a category level up. I suppose we've learned to think of sheaves as first-class geometric objects -- so maybe if you could convince me that reduced excisive functors are as "geometric" as sheaves are? $\endgroup$
    – Tim Campion
    Commented Apr 2, 2020 at 18:43
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    $\begingroup$ Are you happy with the answer that the unbounded chain complexes are combinatorial spectrum objects in Abelian groups? Also, there's this paper of Paul Lessard where he identifies spectra with locally finite (∞,Z)-categories: arxiv.org/abs/1812.00122 . Somehow the identification between simplicial abelian groups and bounded below complexes is kind of too strict to be a homotopy-invariant statement. It's an equivalence of presentations of homotopy theories rather than an equivalence of homotopy theories. $\endgroup$
    – Harry Gindi
    Commented Apr 3, 2020 at 12:51
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    $\begingroup$ @skd You know what -- if I had a good "geometric" way to think about the fact that a sheaf of connective $E_\infty$ ring spectra can have nonconnective global sections, I think that would satisfy me. Actually, what's a good example of this? Eg is $KU$ the global sections of a natural sheaf of connective ring spectra on something? The sense I got here is that nonconnective $E_\infty$ rings inevitably take one away from geometric intuition. Should I resign myself to this? $\endgroup$
    – Tim Campion
    Commented Apr 3, 2020 at 13:16
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    $\begingroup$ As for question 2: I agree with the interpretation of positive homotopy/homology groups as nilpotent fuzz, and negative homotopy groups as cohomology (concentrated in positive cohomological degree). One comment: while one will get examples of spectra/chain complexes with homotopy concentrated in positive and negative dimensions, most natural algebro-geometric objects have finite cohomological dimension, so the resulting spectra will be bounded below. $\endgroup$
    – skd
    Commented Apr 3, 2020 at 16:01

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