We study direct systems of chain complexes which commute up to (coherent) homotopies. We propose a model for the homotopy colimit in the ∞-categorical sense. We define an algebraic structure on such systems, which induces an associative product on (the honest direct limit of) the homology direct system. We lift this structure on the homotopy colimit by considering the bar-cobar resolution of a certain nonsymmetric operad. The lifted structure is shown to be an A∞-algebra.