Let's stipulate that

1. 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, nonpositively-graded chain complexes have a very natural geometric 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 nonnegatively-graded and nonpositively-graded 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 unbounded chain complexes?

**Notes:**

 - I say "unbounded", but really I'm mostly interested more broadly in an interpretation of chain complexes which are neither nonnegatively nor nonpositively graded. For example, I'd be very happy with an interpretation which applies only to perfect complexes.

 - 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.