The homotopy category is a fundamental concept in algebraic topology and homotopy theory that captures the idea of "homotopy equivalence" between topological spaces (or more generally, between objects in a category) in a categorical framework. To understand the homotopy category, we begin with the following components: 1. **Topological Spaces and Continuous Maps**: In topology, we often deal with spaces that can be continuously deformed into each other.
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The homotopy category of based CW complexes has spaces as objects and based homotopy classes as morphisms. Composition is well-defined because composition with continuous maps preserves homotopies. A based homotopy equivalence becomes an isomorphism. Rational localization instead inverts rational homotopy equivalences, producing a category represented by rational spaces. More general model categories have analogous homotopy categories obtained by inverting their weak equivalences.