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.

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**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](https://mathoverflow.net/questions/268614/what-is-the-relationship-between-connective-and-nonconnective-derived-algebraic), 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 $X$.

 - 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 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 $X$?