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

homotopy, higher homotopy

homotopy type

Pi-algebra, spherical object and Pi(A)-algebra

homotopy coherent category theory

homotopical category

model category

category of fibrant objects, cofibration category

Waldhausen category

homotopy category

(∞,1)-category

Paths and cylinders

left homotopy

cylinder object

mapping cone

right homotopy

path object

mapping cocone

universal bundle

interval object

homotopy localization

infinitesimal interval object

Homotopy groups

homotopy group

fundamental group

Brown-Grossman homotopy group

categorical homotopy groups in an (∞,1)-topos

geometric homotopy groups in an (∞,1)-topos

fundamental ∞-groupoid

fundamental groupoid

fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos

fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos

fundamental (∞,1)-category

Basic facts

Theorems

fundamental theorem of covering spaces

Freudenthal suspension theorem

Blakers-Massey theorem

higher homotopy van Kampen theorem

nerve theorem

Whitehead's theorem

Hurewicz theorem

Galois theory

homotopy hypothesis-theorem

A homotopical functor $F : C \to D$ between two homotopical categories $C$ and $D$ is a functor which preserves the weak equivalences of $C$ (i.e., which sends all weak equivalences of $C$ to weak equivalences of $D$).

category with weak equivalences, homotopical category

derived functor

simplicial localization

homological functor

Ken Brown's lemma