One of the realizations that led to the development of Homotopy Type Theory (HoTT) is that the ideas of homotopy theory have very broad applicability in mathematics. Indeed, Quillen model categories comprise very general ideas that arise in a variety of places. To quote Mike Shulman:
Personally, I don’t find it especially surprising that homotopy theory has more than one application to some other subject, any more than I would find it surprising that category theory does. I think it’s becoming increasingly clear that both of them are general organizing principles of mathematics.
I would like to build a list of interesting examples of this, especially unexpected applications of the ideas of homotopy theory in otherwise far removed areas of mathematics.
Best Answer
Algebraic K-theory would be a huge example. "Unexpected" is in the eye of the beholder, but I would imagine that every intrusion of homotopy theoretic ideas into the subject was unexpected by many.
An interesting classical example is the theory of "scissors-congruence" (http://ncatlab.org/nlab/show/scissors+congruence), which can be interpreted in terms of K-theory (Zakharevich, http://arxiv.org/abs/1101.3833).