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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.

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  • $\begingroup$ It's a pretty broad subject. Can you ask about something more specific than "homotopy theory?" $\endgroup$ – Kevin Carlson Sep 27 '16 at 5:57
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The notion of homotopy, or deformation, occurs in many contexts, and so it is useful to give an abstract setting which allows analogies and comparisons. Sometimes the abstract setting allows for the simplest proof, because the proof is based on extracting the essential details. It also means that the theory is available for new examples.

See this expository paper for a discussion of analogy in relation to category theory. I also like the examples and friendly exposition in the book Abstract Homotopy and Simple Homotopy Theory.

Generalisation is a basic tool in research. I found this when writing the 1968 edition of what is now Topology and Groupoids (T&G) . I was curious about the fact that a not necessarily based homotopy equivalence $f:Y \to Z$ of spaces induced an isomorphism $f_*:\pi_n(Y,y) \to \pi_n(Z,f(y))$ of based homotopy groups. What happened if you replaced $(S^n,1)$ by $(X,A)$ and translated the proof to that situation? It worked well and implied a then new gluing theorem for homotopy equivalences (now 7.5.7 of T&G)! That theorem is now part of Abstract Homotopy Theory (though the proofs there usually do not give the same control of the homotopies).

Grothendieck's Pursuing Stacks stimulated much work on abstract homotopy. Pierre Cartier in his fascinating article comments on Grothendieck's "immoderate taste for extreme generality" but I believe this taste was a tool in Grothendieck's search for understanding, for the essence of a situation. Here is a quote from a letter (12/04/83):

"Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out would be suitable for this or that, but just tried to understand -- and it always turned out that understanding was all that mattered."

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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.

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