Homotopy Theory
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Homotopy Theory

In mathematics, homotopy theory is a systematic study of situations in which maps come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry (e.g., A¹ homotopy theory) and category theory (specifically the study of higher categories).

## Concepts

### Spaces and maps

In homotopy theory and algebraic topology, the word "space" denotes a topological space. In order to avoid pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being compactly generated, or Hausdorff, or a CW complex.

In the same vein as above, a "map" is a continuous function, possibly with some extra constraints.

Often, one works with a pointed space -- that is, a space with a "distinguished point", called a basepoint. A pointed map is then a map which preserves basepoints; that is, it sends the basepoint of the domain to that of the codomain. In contrast, a free map is one which needn't preserve basepoints.

### Homotopy

Let I denote the unit interval. A family of maps indexed by I, ${\displaystyle h_{t}:X\to Y}$ is called a homotopy from ${\displaystyle h_{0}}$ to ${\displaystyle h_{1}}$ if ${\displaystyle h:I\times X\to Y,(t,x)\mapsto h_{t}(x)}$ is a map (e.g., it must be a continuous function). When X, Y are pointed spaces, the ${\displaystyle h_{t}}$ are required to preserve the basepoints. A homotopy can be shown to be an equivalence relation. Given a pointed space X and an integer ${\displaystyle n\geq 1}$, let ${\displaystyle \pi _{n}(X)=[S^{n},X]_{*}}$ be the homotopy classes of based maps ${\displaystyle S^{n}\to X}$ from a (pointed) n-sphere ${\displaystyle S^{n}}$ to X. As it turns out, ${\displaystyle \pi _{n}(X)}$ are groups; in particular, ${\displaystyle \pi _{1}(X)}$ is called the fundamental group of X.

If one prefers to work with a space instead of a pointed space, there is the notion of a fundamental groupoid (and higher variants): by definition, the fundamental groupoid of a space X is the category where the objects are the points of X and the morphisms are paths.

### Cofibration and fibration

A map ${\displaystyle f:A\to X}$ is called a cofibration if given (1) a map ${\displaystyle h_{0}:X\to Z}$ and (2) a homotopy ${\displaystyle g_{t}:A\to Z}$, there exists a homotopy ${\displaystyle h_{t}:X\to Z}$ that extends ${\displaystyle h_{0}}$ and such that ${\displaystyle h_{t}\circ f=g_{t}}$. To some loose sense, it is an analog of the defining diagram of an injective module in abstract algebra. The most basic example is a CW pair ${\displaystyle (X,A)}$; since many work only with CW complexes, the notion of a cofibration is often implicit.

A fibration in the sense of Serre is the dual notion of a cofibration: that is, a map ${\displaystyle p:X\to B}$ is a fibration if given (1) a map ${\displaystyle Z\to X}$ and (2) a homotopy ${\displaystyle g_{t}:Z\to B}$, there exists a homotopy ${\displaystyle h_{t}:Z\to X}$ such that ${\displaystyle h_{0}}$ is the given one and ${\displaystyle p\circ h_{t}=g_{t}}$. A basic example is a covering map (in fact, a fibration is a generalization of a covering map). If ${\displaystyle E}$ is a principal G-bundle, that is, a space with a free and transitive (topological) group action of a (topological) group, then the projection map ${\displaystyle p:E\to X}$ is an example of a fibration.

### Classifying spaces and homotopy operations

Given a topological group G, the classifying space for principal G-bundles ("the" up to equivalence) is a space ${\displaystyle BG}$ such that, for each space X,

${\displaystyle [X,BG]=}$ { principal G-bundle on X } / ~ ${\displaystyle ,\,\,[f]\mapsto f^{*}EG}$

where

• the left-hand side is the set of homotopy classes of maps ${\displaystyle X\to BG}$,
• ~ refers isomorphism of bundles, and
• = is given by pulling-back the distinguished bundle ${\displaystyle EG}$ on ${\displaystyle BG}$ (called universal bundle) along a map ${\displaystyle X\to BG}$.

Brown's representability theorem guarantees the existence of classifying spaces.

### Spectrum and generalized cohomology

The idea that a classifying space classifies principal bundles can be pushed further. For example, one might try to classify cohomology classes: given an abelian group A (such as ${\displaystyle \mathbb {Z} }$),

${\displaystyle [X,K(A,n)]=\operatorname {H} ^{n}(X;A)}$

where ${\displaystyle K(A,n)}$ is the Eilenberg-MacLane space. The above equation leads to the notion of a generalized cohomology theory; i.e., a contravariant functor from the category of spaces to the category of abelian groups that satisfies the axioms generalizing ordinary cohomology theory. As it turns out, such a functor may not be representable by a space but it can always be represented by a sequence of (pointed) spaces with structure maps called a spectrum. In other words, to give a generalized cohomology theory is to give a spectrum.

A basic example of a spectrum is a sphere spectrum: ${\displaystyle S^{0}\to S^{1}\to S^{2}\to \cdots }$

## Specific theories

There are several specific theories

## Homotopy hypothesis

One of the basic questions in the foundations of homotopy theory is the nature of a space. The homotopy hypothesis asks whether a space is something fundamentally algebraic.