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 One of the central theorems in (∞,1)-category theory is the fully faithfulness of the Rezk nerve where is the (∞,1)-category of simplicial animae. The mapping anima is simply the subcategory of the functor (∞,1)-category consisting of equivalences. The relative Rezk nerve replaces equivalences by maps in a prescribed wide subcategory and gives an alternative presentation of the localization of an (∞,1)-category of . (See below) Let be an (∞,1)-category and be a wide sub (∞,1)-category. The relative Rezk nerve of is the simplicial anima given by where is the left adjoint to the inclusion. Relative Rezk nerve was introduced by Rezk in Rezk01, Section 3 under the name classification diagram. One of the fundamental property of relative Rezk nerve is that its associated (∞,1)-category computes localization of an (∞,1)-category: For every (∞,1)-category and every wide subcategory , we have where denotes the left adjoint to the Rezk nerve (computing associated (,1)-category). Another fundamental property of the relative Rezk nerve is that its Segalification? admits an explicit presentation: The Segalification? of can be described in terms of zig-zags in in which one is allowed to go backward only via maps in . The mapping animae of the Segalification agrees (or generalizes) with the one appearing in hammock localization. (AC26, Theorems A and B) Classical references: Charles Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001), no. 3, 973–1007 (arXiv:math/9811037, doi:10.1090/S0002-9947-00-02653-2). Julia E. Bergner,...