I've read about model categories from an Appendix to one of Lurie's papers.


What are the examples of **model categories**? What should be my intuition about them?


E.g. I understand the typical examples come from taking homotopy of something — but are all model categories homotopy categories?



How to think about model categories?
Model categories are 1-categorical presentations of (∞,1)-categories, which you can just think of as categories enriched in topological spaces, such as the category of spaces itself. (Actually, there are conditions on (∞,1)-categories that come from model categories--most importantly they must have all homotopy limits and colimits.) They're particularly convenient for computations, as with any kind of presentation.


