Homotopy Theory
Introduction
Homotopy theory is a branch of topology, a field of mathematics, that deals with the study of continuous transformations between topological spaces. This theory is a fundamental concept in algebraic topology and differential geometry. It provides a method to distinguish between different topological spaces, but it also allows us to treat spaces as equivalent if they can be continuously deformed into each other.
Basic Concepts
The central concept in homotopy theory is that of a homotopy. A homotopy is a continuous transformation from one function to another. More formally, given two continuous functions f and g from a topological space X to a topological space Y, a homotopy from f to g is a continuous function H : X × [0,1] → Y such that H(x,0) = f(x) and H(x,1) = g(x) for all x in X.
The concept of homotopy leads to the definition of homotopy equivalence. Two topological spaces X and Y are said to be homotopy equivalent if there exist continuous maps f : X → Y and g : Y → X such that g ◦ f is homotopic to the identity map on X and f ◦ g is homotopic to the identity map on Y.
Homotopy Groups
Homotopy groups are algebraic invariants that are used to classify topological spaces up to homotopy equivalence. The most fundamental of these is the fundamental group, which captures information about the one-dimensional hole structure of a space.
The fundamental group of a topological space X at a base point x is the set of all homotopy classes of loops in X based at x, with the group operation given by concatenation of loops.
Higher homotopy groups are defined similarly, but with spheres of higher dimensions instead of loops. The n-th homotopy group of a space X is the set of homotopy classes of maps from the n-dimensional sphere to X.
Homotopy Type Theory
Homotopy type theory is a new branch of mathematics that combines aspects of several different fields, including topology, type theory, and category theory. It is a form of intuitionistic type theory that interprets types as topological spaces and terms as continuous maps.
In homotopy type theory, the identity type is interpreted as a path in a space, and equality is replaced by the concept of homotopy. This leads to a new understanding of the foundations of mathematics, where traditional logical principles are replaced by homotopical ones.
Applications
Homotopy theory has wide-ranging applications in various fields of mathematics and beyond. In algebraic topology, it is used to classify topological spaces and to study their properties. In differential geometry, it is used to study the topology of manifolds and their continuous deformations.
In theoretical physics, homotopy theory plays a crucial role in the study of quantum field theories, particularly in the classification of topological quantum field theories.
In computer science, homotopy type theory is being used as a foundation for a new kind of formal proof system, which has potential applications in automated theorem proving and software verification.