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 Background Basic concepts Universal constructions Local presentation (∞,1)-Yoneda lemma (∞,1)-Grothendieck construction adjoint (∞,1)-functor theorem (∞,1)-monadicity theorem Extra stuff, structure, properties Models homotopy hypothesis-theorem delooping hypothesis-theorem periodic table stabilization hypothesis-theorem exactness hypothesis holographic principle The notion of an -groupoid is a generalization of that of a group and groupoids to higher category theory. An -groupoid – equivalently an (∞,0)-category – is an ∞-category in which, for all natural numbers , all k-morphisms are equivalences. The collection of all -groupoids forms the (∞,1)-category ∞Grpd. Special cases of -groupoids include groupoids, 2-groupoids, 3-groupoids, n-groupoids, deloopings of groups, 2-groups, and ∞-groups. There are many ways to model the (∞,1)-category ∞Grpd of all -groupoids, or at least obtain its homotopy category. A simple and very useful incarnation of -groupoids is available using a geometric definition of higher categories in the form of simplicial sets that are Kan complexes. The -cells of the underlying simplicial set are the k-morphisms of the -groupoid, and the Kan horn-filler conditions encode the fact that adjacent -morphisms have a (non-unique) composite -morphism and that every -morphism is invertible with respect to this composition. See Kan complex for a detailed discussion of how these incarnate -groupoids. The (∞,1)-category of all -groupoids is modelled along these lines by the Quillen model structure on simplicial sets, whose fibrant-cofibrant...