A Infinity

Algebraic Structures up to Homotopy

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.