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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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$\ell$-adic schematic homotopy type

Étale homotopy type of scheme is rather classical object, see original Artin–Mazur paper or/and Quillen works on Adams conjecture with brilliant lectures of Sullivan "geometric topology" on the same t …
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