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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.
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Homotopy pushout independent of factorization and symmetric in cofibration category
(The following was intended as a comment to Karol's answer, that after using the "Brown Type Factorization" trick, we can prove the result by applying a result in the book, which does not assume cofib …