ABSTRACT
Many natural examples of operads are operads whose components are chain complexes, or at least homologically graded vector spaces. In fact, there are two conceptually important sources of examples of that sort. Some algebraic operads, like the celebrated operad of Gerstenhaber algebras [100], are the homology operads of some operad whose components are topological spaces (and composition maps are continuous). Some other algebraic operads, like the operad of L∞-algebras [227], or the operad of A∞-algebras [241], are operads where classical identities, like the Jacobi identity, or associativity, are relaxed up to a system of coherent homotopies. This is crucial for the questions on homotopy categories of algebras over operads, a nonabelian analogue of the questions on derived categories of modules over algebras that we discussed in Theorem 2.1.2.3.
