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.
Best Answer
Here is an element of answer, which is by no means complete, but it provides at least a first motivation for abstract homotopy theory. I assume that by "abstract homotopy theory" you mean the study of Quillen model categories.
Daniel Quillen realized that the basic machinery of homotopy theory of topological spaces can be set up in the more general context of categories equipped with three classes of morphisms satisfying a few axioms reminiscent of properties of topological spaces, the so-called Quillen model categories.
This situation has some advantages. First, it gives you an abstract approach and a unique language to talk about homotopy theory in a large number of different settings. Some of them are geometric in nature, like spaces, diagrams of spaces, spectra, while others are not, like chain complexes and simplicial sets. To the delight of a topologist, now a question like "What is the suspension of an augmented commutative algebra" makes sense. Second, all these settings have their computational peculiarities, and one of them, at least in some context, can reveal itself more convenient.
Moreover, the category of spaces is not particularly nice and it lacks some good categorical properties, indeed it is not cartesian closed or locally cartesian closed. To get a cartesian closed category you need to restrict to the subcategory of compactly generated spaces, but this category is not locally cartesian closed. In order to get this last property you can use a more combinatorial/algebraic setting instead, like the topos of simplicial sets.
Finally, it is possible to capture in a specific sense, through the notion of Quillen equivalence, what it means for two homotopy theories to be the same. In particular, there exist a model category on topological spaces and a model category on simplicial sets that are Quillen equivalent.