Transporting the A-infinity structure of the Dolbeault differential graded algebra onto its cohomology, using the conjugate Dolbeault operator composed with the Green operator, so that Ext groups of hermitian holomorphic bundles carry higher products m_k alongside ordinary composition. The resulting refined derived category has vanishing differential and the usual Ext composition as m_2, so forgetting the higher products recovers the ordinary derived category, while the higher products measure the failure of a product of harmonic forms to be harmonic and are metric-independent up to homotopy. Students learn how this puts A-infinity categories on both sides of mirror symmetry, how the products are cyclically symmetric with respect to Serre duality, and how triple products of line bundles on an elliptic curve match Fukaya-category products up to a canonical homotopy.
An A-infinity category weakens ordinary composition: instead of one associative product it carries maps m_k on k-fold tensors of morphism spaces, associative only up to a coherent hierarchy of higher…