structure theorems for -algebras is the existence and uniqueness of minimal models – which are defined as -algebras where the differential map is zero. Taking the cohomology algebra of an -algebra from the differential , so as a graded algebra, , with multiplication map . It turns out this graded algebra can then canonically be equipped with an -structure, , which is unique up-to quasi-isomorphisms of -algebras. In fact, the statement is even stronger: there is a canonical -morphism , which lifts the identity map of . Note these higher products are given by the Massey product. Motivation This theorem is very important for the study of differential graded algebras because they were originally introduced to study the homotopy theory of rings. Since the cohomology operation kills the homotopy information, and not every differential graded algebra is quasi-isomorphic to its cohomology algebra, information is lost by taking this operation. But, the minimal models let you recover the quasi-isomorphism class while still forgetting the differential. There is an analogous result for A∞-categories by Maxim Kontsevich and Yan Soibelman, giving an A∞-category structure on the cohomology category of the dg-category consisting of cochain complexes of coherent sheaves on a non-singular variety over a field of characteristic and morphisms given by the total complex of the Cech bi-complex of the differential graded sheaf pg 586-593. In this was, the degree morphisms in the category are given by . Applications There are several applications of this theorem. In particular, given a dg-algebra, such as the de Rham algebra , or the Hochschild cohomology algebra, they can be equipped with an -structure. Massey structure from DGA's Given a differential graded algebra its minimal model as an -algebra is constructed using the Massey products. That is, It turns out that any -algebra structure on is closely related to this construction. Given another -structure on with maps , there is the relation , where . Hence all such -enrichments on the cohomology algebra are related to one another. Graded algebras from its ext algebra Another structure theorem is the reconstruction of an algebra from its ext algebra. Given a connected graded algebra , it is canonically an associative algebra. There is an associated algebra, called its Ext algebra, defined as , where multiplication is given by the Yoneda product. Then, there is an -quasi-isomorphism between and . This identification is important because it gives a way to