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Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.
9
votes
What are surprising examples of Model Categories?
(more detail on the answer by mmm: Gavrilovich in http://arxiv.org/abs/1006.4647 and then further works of Gavrilovich and Hasson http://arxiv.org/abs/1102.5562 and then Gavrilovich, Hasson and Kaplan …