In mathematics, an algebra such as has multiplication whose associativity is well-defined on the nose. This means for any real numbers we have . But, there are algebras which are not necessarily associative, meaning if then in general. There is a notion of algebras, called -algebras, which still have a property on the multiplication which still acts like the first relation, meaning associativity holds, but only holds up to a homotopy, which is a way to say after an operation "compressing" the information in the algebra, the multiplication is associative. This means although we get something which looks like the second equation, the one of inequality, we actually get equality after "compressing" the information in the algebra. The study of -algebras is a subset of homotopical algebra, where there is a homotopical notion of associative algebras through a differential graded algebra with a multiplication operation and a series of higher homotopies giving the failure for the multiplication to be associative. Loosely, an -algebra is a -graded vector space over a field with a series of operations on the -th tensor powers of . The corresponds to a chain complex differential, is the multiplication map, and the higher are a measure of the failure of associativity of the . When looking at the underlying cohomology algebra , the map should be an associative map. Then, these higher maps should be interpreted as higher homotopies, where is the failure of to be associative, is the failure for to be higher associative, and so forth. Their structure was originally discovered by Jim Stasheff while studying A∞-spaces, but this was interpreted as a purely algebraic structure later on. These are spaces equipped with maps that are associative only up to homotopy, and the A∞ structure keeps track of these homotopies, homotopies of homotopies, and so forth. They are ubiquitous in homological mirror symmetry because of their necessity in defining the structure of the Fukaya category of D-branes on a Calabi–Yau manifold who have only a homotopy associative structure. Definition Definition For a fixed field an -algebra is a -graded vector space such that for there exist degree , -linear maps which satisfy a coherence condition: , where . Understanding the coherence conditions The coherence conditions are easy to write down for low degreespgs 583–584. d=1 For this is the condition that , since giving and . These two inequalities force in the coherence condition,