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=== Spaces and maps ===
In homotopy theory and algebraic topology, the word "space" denotes a [[topological space]]. In order to avoid [[Pathological (mathematics)|pathologies]], one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being [[
In the same vein as above, a "[[Map (mathematics)|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.
The Cartesian product of two pointed spaces <math>X, Y</math> are not naturally pointed. A substitute is the [[smash product]] <math>X \wedge Y</math> which is characterized by the [[adjoint functor|adjoint relation]]
:<math>\operatorname{Map}(X \wedge Y, Z) = \operatorname{Map}(X, \operatorname{Map}(Y, Z))</math>,
that is, a smash product is an analog of a [[tensor product]] in abstract algebra (see [[tensor-hom adjunction]]). Explicitly, <math>X \wedge Y</math> is the quotient of <math>X \times Y</math> by the [[wedge sum]] <math>X \vee Y</math>.
=== Homotopy ===
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Let ''I'' denote the unit interval <math>[0, 1]</math>. A map
:<math>h: X \times I \to Y</math>
is called a homotopy from the map <math>h_0</math> to the map <math>h_1</math>, where <math>h_t(x) = h(x, t)</math>. Intuitively, we may think of <math>h</math> as a path from the map <math>h_0</math> to the map <math>h_1</math>. Indeed, a homotopy can be shown to be an [[equivalence relation]]. When ''X'', ''Y'' are pointed spaces, the maps <math>h_t</math> are required to preserve the
Given a pointed space ''X'' and an [[integer]] <math>n \ge 0</math>, let <math>\pi_n(X) = [S^n, X]_*</math> be the homotopy classes of based maps <math>S^n \to X</math> from a (pointed) ''n''-sphere <math>S^n</math> to ''X''. As it turns out, for <math>n > 0</math>, <math>\pi_n(X)</math> are [[group (mathematics)|group]]s called [[homotopy group]]s; in particular, <math>\pi_1(X)</math> is called the [[fundamental group]] of ''X'', while <math>\pi_0(X)</math> can be identified with the set of path-connected components in <math>X</math>. Every group is the fundamental group of some space.<ref>{{harvnb|May|loc=Ch 4. § 5.}}</ref>
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 (mathematics)|category]] where the [[object (category theory)|objects]] are the points of ''X'' and the [[morphism]]s are paths.
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# A subset <math>U</math> is open if and only if <math>U \cap X^n</math> is open for each <math>n</math>.
For example, a sphere <math>S^n</math> has two cells: one 0-cell and one <math>n</math>-cell, since <math>S^n</math> can be obtained by collapsing the boundary <math>S^{n-1}</math> of the ''n''-disk to a point.
Remarkably, [[Whitehead's theorem]] says that for CW complexes, a weak homotopy equivalence and a homotopy equivalence are the same thing.
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where <math>i : \operatorname{Ho}(\textrm{CW}) \hookrightarrow \operatorname{Ho}(\textrm{spaces})</math>, such that each <math>\theta_X : i(\Theta(X)) \to X</math> is a weak homotopy equivalence.
Similar statements also hold for pairs and excisive triads.<ref>{{harvnb|May|loc=Ch. 10., § 6}}</ref><ref>{{harvnb|May|loc=Ch. 10., § 7}}</ref>}}
Explicitly, the above approximation functor can be defined as the composition of the [[singular chain]] functor <math>S_*</math> followed by the geometric realization functor; see {{section link||Simplicial set}}.
The above theorem justifies a common habit of working only with CW complexes. For example, given a space <math>X</math>, one can just define the homology of <math>X</math> to the homology of the CW approximation of <math>X</math> (the cell structure of a CW complex determines the natural homology, the [[cellular homology]] and that can be taken to be the homology of the complex.)
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A [[fibration]] in the sense of Serre is the dual notion of a cofibration: that is, a map <math>p : X \to B</math> is a fibration if given (1) a map <math>h_0 : Z \to X</math> and (2) a homotopy <math>g_t : Z \to B</math> such that <math>p \circ h_0 = g_0</math>, there exists a homotopy <math>h_t: Z \to X</math> that extends <math>h_0</math> and such that <math>p \circ h_t = g_t</math>. A basic example is a [[covering map]] (in fact, a fibration is a generalization of a covering map). If <math>E</math> is a [[principal bundle|principal ''G''-bundle]], that is, a space with a [[Group action#Remarkable properties of actions|free and transitive]] (topological) [[group action]] of a ([[topological group|topological]]) group, then the projection map <math>p: E \to X</math> is an example of a fibration.
There are also based versions of a cofibration and a fibration (namely, the maps are required to be based).<ref>{{harvnb|May|loc=Ch 8. § 3. and § 5.}}</ref>
=== Loop and suspension ===
On the category of pointed spaces, there are two important functors: the [[loop functor]] <math>\Omega</math> and the (reduced) [[suspension functor]] <math>\Sigma</math>, which are in the [[adjoint functor|adjoint relation]].
*<math>\Omega X = \operatorname{Map}(S^1, X)</math>, and
▲On the category of spaces, there are two important functors: the [[loop functor]] <math>\Omega</math> and the [[suspension functor]] <math>\Sigma</math> which are in the [[adjoint functor|adjoint relation]] up to homotopy.
*<math>\Sigma X = X \wedge S^1</math>.
Because of the adjoint relation between a smash product and a mapping space, we have:
:<math>\operatorname{Map}(\Sigma X, Y) = \operatorname{Map}(X, \Omega Y).</math>
These functors are used to construct [[fiber sequence]]s and [[cofiber sequence]]s. Namely, if <math>f : X \to Y</math> is a map, the fiber sequence generated by <math>f</math> is the exact sequence<ref>{{harvnb|May|loc=Ch. 8, § 6.}}</ref>
:<math>\cdots \to \Omega^2 Ff \to \Omega^2 X \to \Omega^2 Y \to \Omega Ff \to \Omega X \to \Omega Y \to Ff \to X \to Y</math>
where <math>Ff</math> is the [[homotopy fiber]] of <math>f</math>; i.e., a fiber obtained after replacing <math>f</math> by a (based) fibration. The cofibration sequence generated by <math>f</math> is <math>X \to Y \to C f \to \Sigma X \to \cdots,</math> where <math>Cf</math> is the homotooy cofiber of <math>f</math> constructed like a homotopy fiber (use a quotient instead of a fiber.)
The functors <math>\Omega, \Sigma</math> restrict to the category of CW complexes in the following sense: a theorem of Milnor says that if <math>X</math> has the homotopy type of a CW complex, then so does its loop space <math>\Omega X</math>.<ref>{{harvnb|Milnor|1959|loc=Theorem 3.}}</ref>
=== Classifying spaces and homotopy operations ===
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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's [[model category|model 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.<ref>{{harvnb|Dwyer|Spalinski|loc=Example 3.5.}}</ref> Another example is the category of non-negatively graded chain complexes over a fixed base ring.<ref>{{harvnb|Dwyer|Spalinski|loc=Example 3.7.}}</ref>
See also: [[Algebraic homotopy]]▼
=== Simplicial
A [[simplicial set]] is an abstract generalization of a [[simplicial 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.
▲See also: [[Algebraic homotopy]]
For example, given a space <math>X</math>, for each integer <math>n \ge 0</math>, let <math>S_n X</math> be the set of all maps from the ''n''-simplex to <math>X</math>. Then the sequence <math>S_n X</math> of sets is a simplicial set.<ref name="May simplicial">{{harvnb|May|loc=Ch. 16, § 4.}}</ref> Each simplicial set <math>K = \{ K_n \}_{n \ge 0}</math> has a naturally associated chain complex and the homology of that chain complex is the homology of <math>K</math>. The [[singular homology]] of <math>X</math> is precisely the homology of the simplicial set <math>S_* X</math>. Also, the [[Simplicial_set#Geometric_realization|geometric realization]] <math>| \cdot |</math> of a simplicial set is a CW complex and the composition <math>X \mapsto |S_* X|</math> is precisely the CW approximation functor.
Another important example is a category or more precisely the [[nerve 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 the [[Segal condition]]s (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 [[infinity category|<math>\infty</math>-category]] or an [[infinity groupoid|<math>\infty</math>-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 the [[simplicial homotopy theory]].<ref name="May simplicial" />
== See also ==
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* Homotopy Theories and Model Categories by W.G. Dwyer and J. Spalinski in [https://books.google.com/books?id=xoM5DxQZihQC&printsec=copyright#v=onepage&q&f=false Handbook of Algebraic Topology] edited by I.M. James
*{{cite web |first=Allen |last=Hatcher |url=http://www.math.cornell.edu/~hatcher/AT/ATpage.html |title=Algebraic topology}}
*{{cite journal |last1=Milnor |first1=John |title=On spaces having the homotopy type of 𝐶𝑊-complex |journal=Transactions of the American Mathematical Society |date=1959 |volume=90 |issue=2 |pages=272–280 |doi=10.1090/S0002-9947-1959-0100267-4 |url=https://www.semanticscholar.org/paper/On-spaces-having-the-homotopy-type-of-a-CW-complex-Milnor/905bb7242d4e2b7b7e168d12718b6595c98e98d9 |language=en |issn=0002-9947}}
* Edwin Spanier, Algebraic topology
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