Structure of algebras

One of the principal activities in the description of mathematical objects is the characterization by comparing the algebraic structures embedded in these objects. The best situation is when these structures are natural, because usually they come with new information about the object, and sometimes it needs new tools to be handled. Coded in the algebraic structure, the information about the object, in this case a topological space, will be necessary in order to reconstruct the space only from the algebraic part. The spaces obtained will be approximations to the original space, that is, they will be the same but isomorphisms in homology. This type of relation is called quasiisomorphism. In this sens, the A∞-algebras, are algebraic structures associated to topological spaces in order to characterize their type of homotopy. To better understand this affirmation we will see an example with loop spaces. Take a pointed topological space (X, ∗). Now consider the loops over X which begin and end in ∗, or in more technical words, the continuous applications from the unit interval I = [0, 1] to X, such that each application f satisfies f(0) = f(1) = ∗. The space of loops comes with a natural loop product: the concatenation of loops. Imagine a loop as the interval I = [0, 1] together with a finite collection of open intervals I1, . . . , In embedded in I. The idea is that the complement of the open intervals represents the parts of the loop with value the base point ∗. Note that different loops can have the same representation. Now, take the interval I with (0, 1 2 ) and ( 1 2 , 1), and the product of two loops is the result of embed in each open interval (0, 12 ) and ( 1 2 , 1), the loops we want to multiply. One of the principal properties of loop product is that the axiom of associativity is lost: only in homology the product is associative. So, with loop spaces, we are dealing with an operation which is not associative but homotopy associative. In his PhD thesis, the American mathematician Jon Stasheff, describe in detail this new structure of homotopy associativity.