Roughly, by an algebraic structure I mean a space equipped with a bunch of
structure maps which are subject to certain relations. In general, algebraic
structures are rigid and do not behave well with respect to homotopy
operations on their underlying space. However, some algebraic structures are
sufficiently flexible and much better behaved in this regard. We call these
algebraic structure up to homotopy. Sounds vague and confusing? In this
post, we’ll consider a concrete and easy example: associative algebras.