I have a good grasp of ordinary pullbacks and pushouts; in particular, there are many categorical constructions that can be seen as special cases: e.g., equalizers/coequalizers, kernerls/cokernels, binary products/coproducts, preimages,...


I know the (a?) definition of homotopy pullbacks/pushouts, but I am lacking two things: examples and intuition. So here are my questions:


1. What are the canonical examples of homotopy pullbacks/pushouts? E.g., in the category of pointed topological spaces the loop space $\Omega X$ is a homotopy pullback of the map $\ast \to X$ along itself.
2. How should I think about homotopy pullbacks/pushouts? What is the intuition behind the concept?



Homotopy pullbacks and homotopy pushouts
You can think of the pushout of two maps f : A → B, g : A → C in Set as computing the disjoint union of B and C with an identification f(a) = g(a) for each element a of A. We could imagine forming this as either the quotient by an equivalence relation, or by gluing in a segment joining f(a) to g(a) for each a, and taking π0 of the resulting space. If two elements a, a' of A satisfy f(a) = f(a') and g(a) = g(a'), the pushout is unaffected by removing a' from A. The homotopy pushout is formed by gluing in a segment joining f(a) to g(a) for each a and *not* forgetting the number of ways in which two elements of B ∐ C are identified; instead we take the entire space as the result. It is the "derived" version of the pushout.


In general you can think of the homotopy pushout of A → B, A → C as the "free" thing generated by B and C with "relations" coming from A. But it's important that the "relations" are imposed exactly once, since in the homotopical/derived setting we keep track of such things (and have "relations between relations" etc.)


Another, possibly more familiar example: In a derived category, the mapping cone of a morphism f : A → B is the homotopy pushout of f and the zero map A → 0. This certainly depends on A, even when B is the zero object: it is the suspension of A.


