Chain complexes indexed over the reals

Question

One way to think of chain complexes is as a certain subcategory of a functor category: each chain complex is a functor from $\mathbb{Z}$ regarded as a poset category to, say, abelian groups, where consecutive maps compose to 0. Is there a corresponding notion for other partially ordered categories which are not discrete, for example the real numbers, producing a sort of "continuous" complex?