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:
- 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.
- How should I think about homotopy pullbacks/pushouts? What is the intuition behind the concept?