When two topological spaces are linked by a homotopy equivalence, an algebraic topologist would usually think of them as being “the same” in some b sense.  In other words, we think of homtopy equivalences are sort of “isomorphisms”.  To make this intuition concrete, to turn into correct mathematics that applies to any model category, I introduce in this lecture the notion of the homotopy category of a model category M, as a category built from M, in which all weak equivalences in M admit inverses.  

To prove the existence of the homotopy category of any model category in the next lecture, I define and analyze the notions of fibrant and cofibrant replacement of an object in a model category.