What is the intuition behind abstract homotopy theory (or homotopical algebra)? What was the motivation behind its historical development?
For homology and cohomology theory, the way I understand it, we have the notion of exact sequences and the homology/cohomology groups kind of measure how far a sequence is from being exact. This is used with the intuitive notion of cycles and boundaries in order to study objects like topological spaces. But exact sequences are ubiquitous in mathematics, not only in the context of topology, and so we have homological algebra, etc. Are there similar ideas behind homotopy theory?
I am aware, for example, of the existence of algebraic k-theory as an application of ideas of homotopy theory beyond classical algebraic topology, but also wonder as to the intuition behind how such "geometric" ideas can be applied to algebraic objects, which strikes me as one of the most beautiful but mysterious aspects of mathematics.