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:** 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.