Inmathematics,homotopy theory is a systematic study of situations in whichmaps can come withhomotopies between them. It originated as a topic inalgebraic topology, but nowadays is learned 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)
• Category theory (specifically the study ofhigher categories)
• Noncommutative geometry (e.g.,KK-theory andNoncommutative topology)

In homotopy theory and algebraic topology, the word "space" denotes atopological space. In order to avoidpathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as beingcompactly generated weak Hausdorff or aCW complex.

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

Often, one works with apointed 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.

The Cartesian product of two pointed spaces${\displaystyle X,Y}$ are not naturally pointed. A substitute is thesmash product${\displaystyle X\wedge Y}$ which is characterized by theadjoint relation

${\displaystyle \mathrm{Map}(X\wedge Y,Z)=\mathrm{Map}(X,\mathrm{Map}(Y,Z))}$,

that is, a smash product is an analog of atensor product in abstract algebra (seetensor-hom adjunction). Explicitly,${\displaystyle X\wedge Y}$ is the quotient of${\displaystyle X\times Y}$ by thewedge sum${\displaystyle X\vee Y}$.

LetI denote the unit interval${\displaystyle [0,1]}$. A map

${\displaystyle h:X\times I\to Y}$

is called ahomotopy from the map${\displaystyle h_0}$ to the map${\displaystyle h_1}$, where${\displaystyle h_t(x)=h(x,t)}$. Intuitively, we may think of${\displaystyle h}$ as a path from the map${\displaystyle h_0}$ to the map${\displaystyle h_1}$. Indeed, a homotopy can be shown to be anequivalence relation. WhenX,Y are pointed spaces, the maps${\displaystyle h_t}$ are required to preserve the basepoint and the homotopy${\displaystyle h}$ is called abased homotopy. A based homotopy is the same as a (based) map${\displaystyle X\wedge I_{+}\to Y}$ where${\displaystyle I_{+}}$ is${\displaystyle I}$ together with a disjoint basepoint.^[1]

Given a pointed spaceX and aninteger${\displaystyle n\ge 0}$, 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${\displaystyle S^n}$ toX. As it turns out,

• for${\displaystyle n\ge 1}$,${\displaystyle {\pi }_{n}X}$ aregroups calledhomotopy groups; in particular,${\displaystyle {\pi }_{1}X}$ is called thefundamental group ofX,
• for${\displaystyle n\ge 2}$,${\displaystyle {\pi }_{n}X}$ areabelian groups by theEckmann–Hilton argument,
• ${\displaystyle {\pi }_{0}X}$ can be identified with the set of path-connected components in${\displaystyle X}$.

Every group is the fundamental group of some space.^[2]

A map${\displaystyle f}$ is called ahomotopy equivalence if there is another map${\displaystyle g}$ such that${\displaystyle f\circ g}$ and${\displaystyle g\circ f}$ are both homotopic to the identities. Two spaces are said to be homotopy equivalent if there is a homotopy equivalence between them. A homotopy equivalence class of spaces is then called ahomotopy type. There is a weaker notion: a map${\displaystyle f:X\to Y}$ is said to be aweak homotopy equivalence if${\displaystyle f_{\ast }:{\pi }_n(X)\to {\pi }_n(Y)}$ is an isomorphism for each${\displaystyle n\ge 0}$ and each choice of a base point. A homotopy equivalence is a weak homotopy equivalence but the converse need not be true.

Through the adjunction

${\displaystyle \mathrm{Map}(X\times I,Y)=\mathrm{Map}(X,\mathrm{Map}(I,Y)),h\mapsto (x\mapsto h(x,\cdot ))}$,

a homotopy${\displaystyle h:X\times I\to Y}$ is sometimes viewed as a map${\displaystyle X\to Y^I=\mathrm{Map}(I,Y)}$.

ACW complex is a space that has a filtration${\displaystyle X\supset \cdots \supset X^n\supset X^{n-1}\supset \cdots \supset X^0}$ whose union is${\displaystyle X}$ and such that

1. ${\displaystyle X^0}$ is a discrete space, called the set of 0-cells (vertices) in${\displaystyle X}$.
2. Each${\displaystyle X^n}$ is obtained by attaching severaln-disks,n-cells, to${\displaystyle X^{n-1}}$ via maps${\displaystyle S^{n-1}\to X^{n-1}}$; i.e., the boundary of an n-disk is identified with the image of${\displaystyle S^{n-1}}$ in${\displaystyle X^{n-1}}$.
3. A subset${\displaystyle U}$ is open if and only if${\displaystyle U\cap X^n}$ is open for each${\displaystyle n}$.

For example, a sphere${\displaystyle S^n}$ has two cells: one 0-cell and one${\displaystyle n}$-cell, since${\displaystyle S^n}$ can be obtained by collapsing the boundary${\displaystyle S^{n-1}}$ of then-disk to a point. In general, every manifold has the homotopy type of a CW complex;^[3] in fact,Morse theory implies that a compact manifold has the homotopy type of a finite CW complex.^{[citation needed]}

Remarkably,Whitehead's theorem says that for CW complexes, a weak homotopy equivalence and a homotopy equivalence are the same thing.

Another important result is the approximation theorem. First, thehomotopy category of spaces is the category where an object is a space but a morphism is the homotopy class of a map. Then

Explicitly, the above approximation functor can be defined as the composition of thesingular chain functor${\displaystyle S_{\ast }}$ followed by the geometric realization functor; see§ Simplicial set.

The above theorem justifies a common habit of working only with CW complexes. For example, given a space${\displaystyle X}$, one can just define the homology of${\displaystyle X}$ to the homology of the CW approximation of${\displaystyle X}$ (the cell structure of a CW complex determines the natural homology, thecellular homology and that can be taken to be the homology of the complex.)

A map${\displaystyle f:A\to X}$ is called acofibration if given:

1. A map${\displaystyle h_0:X\to Z}$, and
2. A homotopy${\displaystyle g_t:A\to Z}$

such that${\displaystyle h_0\circ f=g_0}$, there exists a homotopy${\displaystyle h_t:X\to Z}$ that extends${\displaystyle h_0}$ and such that${\displaystyle h_t\circ f=g_t}$. An example is aneighborhood deformation retract; that is,${\displaystyle X}$ contains amapping cylinder neighborhood of a closed subspace${\displaystyle A}$ and${\displaystyle f}$ the inclusion (e.g., atubular neighborhood of a closed submanifold).^[7] In fact, a cofibration can be characterized as a neighborhood deformation retract pair.^[8] Another basic example is aCW pair${\displaystyle (X,A)}$; many often work only with CW complexes and the notion of a cofibration there is then often implicit.

Afibration in the sense of Hurewicz is the dual notion of a cofibration: that is, a map${\displaystyle p:X\to B}$ is a fibration if given (1) a map${\displaystyle h_0:Z\to X}$ and (2) a homotopy${\displaystyle g_t:Z\to B}$ such that${\displaystyle p\circ h_0=g_0}$, there exists a homotopy${\displaystyle h_t:Z\to X}$ that extends${\displaystyle h_0}$ and such that${\displaystyle p\circ h_t=g_t}$.

While a cofibration is characterized by the existence of a retract, a fibration is characterized by the existence of a section called thepath lifting as follows. Let${\displaystyle p^{\prime }:Np\to B^I}$ be the pull-back of a map${\displaystyle p:E\to B}$ along${\displaystyle \chi \mapsto \chi (1):B^I\to B}$, called themapping path space of${\displaystyle p}$.^[9] Viewing${\displaystyle p^{\prime }}$ as a homotopy${\displaystyle Np\times I\to B}$ (see§ Homotopy), if${\displaystyle p}$ is a fibration, then${\displaystyle p^{\prime }}$ gives a homotopy^[10]

${\displaystyle s:Np\to E^I}$

such that${\displaystyle s(e,\chi )(0)=e,(p^I\circ s)(e,\chi )=\chi }$ where${\displaystyle p^I:E^I\to B^I}$ is given by${\displaystyle p}$.^[11] This${\displaystyle s}$ is called the path lifting associated to${\displaystyle p}$. Conversely, if there is a path lifting${\displaystyle s}$, then${\displaystyle p}$ is a fibration as a required homotopy is obtained via${\displaystyle s}$.

A basic example of a fibration is acovering map as it comes with a unique path lifting. If${\displaystyle E}$ is aprincipalG-bundle over a paracompact space, that is, a space with afree and transitive (topological)group action of a (topological) group, then the projection map${\displaystyle p:E\to X}$ is a fibration, because a Hurewicz fibration can be checked locally on a paracompact space.^[12]

While a cofibration is injective with closed image,^[13] a fibration need not be surjective.

There are also based versions of a cofibration and a fibration (namely, the maps are required to be based).^[14]

A pair of maps${\displaystyle i:A\to X}$ and${\displaystyle p:E\to B}$ is said to satisfy thelifting property^[15] if for each commutative square diagram

there is a map${\displaystyle \lambda }$ that makes the above diagram still commute. (The notion originates in the theory ofmodel categories.)

Let${\displaystyle \mathfrak{c}}$ be a class of maps. Then a map${\displaystyle p:E\to B}$ is said to satisfy theright lifting property or the RLP if${\displaystyle p}$ satisfies the above lifting property for each${\displaystyle i}$ in${\displaystyle \mathfrak{c}}$. Similarly, a map${\displaystyle i:A\to X}$ is said to satisfy theleft lifting property or the LLP if it satisfies the lifting property for each${\displaystyle p}$ in${\displaystyle \mathfrak{c}}$.

For example, a Hurewicz fibration is exactly a map${\displaystyle p:E\to B}$ that satisfies the RLP for the inclusions${\displaystyle i_0:A\to A\times I}$. ASerre fibration is a map satisfying the RLP for the inclusions${\displaystyle i:S^{n-1}\to D^n}$ where${\displaystyle S^{-1}}$ is the empty set. A Hurewicz fibration is a Serre fibration and the converse holds for CW complexes.^[16]

On the other hand, a cofibration is exactly a map satisfying the LLP for evaluation maps${\displaystyle p:B^I\to B}$ at${\displaystyle 0}$.

On the category of pointed spaces, there are two important functors: theloop functor${\displaystyle \mathrm{\Omega }}$ and the (reduced)suspension functor${\displaystyle \mathrm{\Sigma }}$, which are in theadjoint relation. Precisely, they are defined as^[17]

• ${\displaystyle \mathrm{\Omega }X=\mathrm{Map}(S^1,X)}$, and
• ${\displaystyle \mathrm{\Sigma }X=X\wedge S^1}$.

Because of the adjoint relation between a smash product and a mapping space, we have:

${\displaystyle \mathrm{Map}(\mathrm{\Sigma }X,Y)=\mathrm{Map}(X,\mathrm{\Omega }Y).}$

These functors are used to constructfiber sequences andcofiber sequences. Namely, if${\displaystyle f:X\to Y}$ is a map, the fiber sequence generated by${\displaystyle f}$ is the exact sequence^[18]

${\displaystyle \cdots \to {\mathrm{\Omega }}^{2}Ff\to {\mathrm{\Omega }}^{2}X\to {\mathrm{\Omega }}^{2}Y\to \mathrm{\Omega }Ff\to \mathrm{\Omega }X\to \mathrm{\Omega }Y\to Ff\to X\to Y}$

where${\displaystyle Ff}$ is thehomotopy fiber of${\displaystyle f}$; i.e., a fiber obtained after replacing${\displaystyle f}$ by a (based) fibration. The cofibration sequence generated by${\displaystyle f}$ is${\displaystyle X\to Y\to Cf\to \mathrm{\Sigma }X\to \cdots ,}$ where${\displaystyle Cf}$ is the homotopy cofiber of${\displaystyle f}$ constructed like a homotopy fiber (use a quotient instead of a fiber.)

The functors${\displaystyle \mathrm{\Omega },\mathrm{\Sigma }}$ restrict to the category of CW complexes in the following weak sense: a theorem of Milnor says that if${\displaystyle X}$ has the homotopy type of a CW complex, then so does its loop space${\displaystyle \mathrm{\Omega }X}$.^[19]

Given a topological groupG, theclassifying space forprincipalG-bundles ("the" up to equivalence) is a space${\displaystyle BG}$ such that, for each spaceX,

${\displaystyle [X,BG]=}$ {principalG-bundle onX} / ~${\displaystyle ,[f]\mapsto [f^{\ast }EG]}$

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.

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

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

where${\displaystyle K(A,n)}$ is theEilenberg–MacLane space. The above equation leads to the notion of a generalized cohomology theory; i.e., acontravariant functor from the category of spaces to thecategory of abelian groups that satisfies the axioms generalizing ordinary cohomology theory. As it turns out, such a functor may not berepresentable 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. AK-theory is an example of a generalized cohomology theory.

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

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• Seifert–van Kampen theorem
• Homotopy excision theorem
• Freudenthal suspension theorem (a corollary of the excision theorem)
• Landweber exact functor theorem
• Dold–Kan correspondence
• Eckmann–Hilton argument - this shows for instance higher homotopy groups areabelian.
• Universal coefficient theorem
• Dold–Thom theorem

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See also:Characteristic class,Postnikov tower,Whitehead torsion

This sectionneeds expansion. You can help byadding to it.(May 2020)

There are several specific theories

• simple homotopy theory
• stable homotopy theory
• chromatic homotopy theory
• rational homotopy theory
• p-adic homotopy theory
• equivariant homotopy theory
• simplicial homotopy theory

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

If one prefers to work with a space instead of a pointed space, there is the notion of afundamental groupoid (and higher variants): by definition, the fundamental groupoid of a spaceX is thecategory where theobjects are the points ofX and themorphisms are paths.

Abstract homotopy theory is an axiomatic approach to homotopy theory. Such axiomatization is useful for non-traditional applications of homotopy theory. One approach to axiomatization is by Quillen'smodel categories. A model category is a category with a choice of three classes of maps called weak equivalences, cofibrations and fibrations, subject to the axioms that are reminiscent of facts in algebraic topology. For example, the category of (reasonable) topological spaces has a structure of a model category where a weak equivalence is a weak homotopy equivalence, a cofibration a certain retract and a fibration a Serre fibration.^[20] Another example is the category of non-negatively graded chain complexes over a fixed base ring.^[21]

Asimplicial set is an abstract generalization of asimplicial complex and can play a role of a "space" in some sense. Despite the name, it is not a set but is a sequence of sets together with the certain maps (face and degeneracy) between those sets.

For example, given a space${\displaystyle X}$, for each integer${\displaystyle n\ge 0}$, let${\displaystyle S_{n}X}$ be the set of all maps from then-simplex to${\displaystyle X}$. Then the sequence${\displaystyle S_{n}X}$ of sets is a simplicial set.^[22] Each simplicial set${\displaystyle K=\{K_n{\}}_{n\ge 0}}$ has a naturally associated chain complex and the homology of that chain complex is the homology of${\displaystyle K}$. Thesingular homology of${\displaystyle X}$ is precisely the homology of the simplicial set${\displaystyle S_{\ast }X}$. Also, thegeometric realization${\displaystyle |\cdot |}$ of a simplicial set is a CW complex and the composition${\displaystyle X\mapsto |S_{\ast }X|}$ is precisely the CW approximation functor.

Another important example is a category or more precisely thenerve of a category, which is a simplicial set. In fact, a simplicial set is the nerve of some category if and only if it satisfies theSegal conditions (a theorem of Grothendieck). Each category is completely determined by its nerve. In this way, a category can be viewed as a special kind of a simplicial set, and this observation is used to generalize a category. Namely, an${\displaystyle \mathrm{\infty }}$-category or an${\displaystyle \mathrm{\infty }}$-groupoid is defined as particular kinds of simplicial sets.

Since simplicial sets are sort of abstract spaces (if not topological spaces), it is possible to develop the homotopy theory on them, which is called thesimplicial homotopy theory.^[22]

• Highly structured ring spectrum
• Homotopy type theory
• Pursuing Stacks
• Shape theory
• Moduli stack of formal group laws
• Crossed module
• Milnor's theorem on Kan complexes
• Fibration of simplicial sets

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• Cisinski, Denis-Charles (March 2015)."Higher Categories And Topos Theory(in french)"(PDF).Math -University of Toulouse.
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• Homotopy theory

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