"Infinity-groupoids encode all the paths in a space, including paths of paths, and paths of paths of paths. They crop up in other frontiers of mathematical research as ways of encoding similar higher-order relationships, but they are unwieldy objects from the point of view of set theory. Because of this, they were thought to be useless for Voevodsky’s goal of formalizing mathematics. Yet Voevodsky was able to create an interpretation of type theory in the language of infinity-groupoids, an advance that allows mathematicians to reason efficiently about infinity-groupoids without ever having to think of them in terms of sets. This advance ultimately led to the development of univalent foundations."