homotopy theory, (∞,1)-category theory, homotopy type theory flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed… models: topological, simplicial, localic, … see also algebraic topology Introductions Introduction to Basic Homotopy Theory Introduction to Abstract Homotopy Theory geometry of physics – homotopy types Definitions Paths and cylinders Homotopy groups Basic facts Theorems A a homotopy class is an equivalence class under homotopy: For a continuous function between topological spaces which admit the structure of CW-complexes, its homotopy class is the morphism in the classical homotopy category that is represented by . The sets of such homotopy classes if often denoted or similar. Similarly, if is a base-point preserving function between pointed topological spaces admitting the structure of CW-complexes, then its homotopy class (“pointed homomotopy class”) represents a morphism in the homotopy category of pointed homotopy types. The sets of such pointed homotopy classes are often denoted or similar. If the domain is an n-sphere, then the homotopy classes of maps form the th homotopy group of . If the codomain is an n-sphere , then the homotopy classes of maps form the Cohomotopy set of .