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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 are homotopy categories of something — but are all model categories homotopy categories?

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?