What is the Seiberg-Witten map exactly?

We give a conceptual treatment of the Seiberg-Witten map as a quasi-isomorphism of differential graded algebras. The corresponding algebras have a very simple form, leading to explicit recurrence formulas for the quasi-isomorphism. Unlike most previous papers, our recurrence relations are nonperturbative in the parameter of non-commutativity. Using the language of quasi-isomorphisms, we give a homotopy classification of ambiguities in Seiberg-Witten maps. Possible generalizations to the Wess-Zumino complexes and some other algebras are briefly discussed.