Homotopy category of topological spaces
Encyclopedia
In mathematics
, a homotopy category is a category
whose objects are topological space
s and whose morphism
s are homotopy classes of continuous functions. The homotopy category of all topological spaces is often denoted hTop or Toph.
. The set of morphisms between spaces X and Y in a homotopy category is commonly denoted [X,Y] rather than Hom(X,Y).
The composition
is defined by
This is well-defined since the homotopy relation is compatible with function composition
. That is, if f1, g1 : X → Y are homotopic and f2, g2 : Y → Z are homotopic then their compositions f2 o f1, g2 o g1 X → Z are homotopic as well.
While the objects of a homotopy category are sets (with additional structure), the morphisms are not actual functions between them, but rather classes of functions. Indeed, hTop is an example of a category that is not concretizable
, meaning there does not exist a faithful forgetful functor
to the category of sets
. Homotopy categories are examples of quotient categories
. The category hTop is a quotient of Top, the ordinary category of topological spaces
.
of topological space is, properly speaking, dependent on the basepoint chosen. A topological space with a distinguished basepoint is called a pointed space
. The pointed homotopy category hTop• is defined to be the category whose objects are pointed topological space and whose morphisms are equivalence classes of pointed maps (i.e., sending the distinguished base point to the base point) modulo pointed homotopy (i.e., the homotopy fixes the base points, as well). The set of maps between pointed spaces X and Y in hTop• is commonly denoted [X,Y]•.
The need to use basepoints has a significant effect on the products (and other limit
s) appropriate to use. For example, in homotopy theory, the smash product
X ∧ Y of spaces X and Y is used.
they are homotopy equivalent (i.e. have the same homotopy type).
of homotopy classes of maps from Sn to some topological space X is the same as the n-th homotopy group πn(X) (for n ≥ 1, the set of connected path-components for n = 0).
Even immediate examples, such as the homotopy groups of spheres
,
are hard to compute.
K(G,n) is a topological space satisfying, for any CW-complex X,
where the right hand side denotes the n-th singular cohomology group
of X with coefficients in G. In this sense, singular cohomology is representable
by the representing space K(G,n). The Brown representability theorem is concerned with the representability of more general functors
es is the appropriate choice. In the opinion of some experts the homotopy category of CW complexes is the best, if not the only, candidate for the homotopy category. One basic result is that the representable functor
s on the homotopy category of CW complexes have a simple characterization (the Brown representability theorem).
The category of CW complexes is deficient in the sense that the space of maps
between two CW complexes is not always a CW complex. A more well-behaved category commonly used in homotopy theory is the category of compactly generated Hausdorff spaces (also called k-spaces). This category includes all CW complexes, locally compact space
s, and first-countable space
s (such as metric space
s).
One important later development was that of spectra
in homotopy theory, essentially the derived category
idea in a form useful for topologists. Spectra have also been defined in various cases using the model category
approach, generalizing the topological case. Many theorists interested in the classical topological theory consider this more axiomatic approach less useful for their purposes. Finding good replacements for CW complexes in the purely algebraic case is a subject of current research.
. Roughly speaking, a model category is a category
C with three distinguished types of morphisms called fibration
s, cofibration
s and weak equivalence
s. Localizing
C with respect to the weak equivalences yields the homotopy category.
This construction, applied to the model category of topological spaces, gives back the homotopy category outlined above. Applied to the model category of chain complex
es over some commutative ring R, for example, yields the derived category
of R-modules. The homotopy category of chain complexes
can also be interpreted along these lines.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a homotopy category is a category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
whose objects are topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s and whose morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s are homotopy classes of continuous functions. The homotopy category of all topological spaces is often denoted hTop or Toph.
Definition and examples
The homotopy category hTop of topological spaces is the category whose objects are topological spaces. Instead of taking continuous functions as morphisms between two such spaces, the morphisms in hTop between two spaces X and Y are given by the equivalence classes of all continuous functions X → Y with respect to the relation of homotopy. That is to say, two continuous functions are considered the same morphism in hTop if they can be deformed into one another via a (continuous) homotopyHomotopy
In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
. The set of morphisms between spaces X and Y in a homotopy category is commonly denoted [X,Y] rather than Hom(X,Y).
The composition
- [X, Y] × [Y, Z] → [X, Z]
is defined by
- [f] o [g] = [f o g].
This is well-defined since the homotopy relation is compatible with function composition
Function composition
In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
. That is, if f1, g1 : X → Y are homotopic and f2, g2 : Y → Z are homotopic then their compositions f2 o f1, g2 o g1 X → Z are homotopic as well.
While the objects of a homotopy category are sets (with additional structure), the morphisms are not actual functions between them, but rather classes of functions. Indeed, hTop is an example of a category that is not concretizable
Concrete category
In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions...
, meaning there does not exist a faithful forgetful functor
Forgetful functor
In mathematics, in the area of category theory, a forgetful functor is a type of functor. The nomenclature is suggestive of such a functor's behaviour: given some object with structure as input, some or all of the object's structure or properties is 'forgotten' in the output...
- U : hTop → Set
to the category of sets
Category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...
. Homotopy categories are examples of quotient categories
Quotient category
In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. The notion is similar to that of a quotient group or quotient space, but in the categorical setting.-Definition:Let C be a category...
. The category hTop is a quotient of Top, the ordinary category of topological spaces
Category of topological spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous...
.
Pointed version
For the purposes of homotopy theory it is usually necessary to keep track of basepoints in each space: for example the fundamental groupFundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
of topological space is, properly speaking, dependent on the basepoint chosen. A topological space with a distinguished basepoint is called a pointed space
Pointed space
In mathematics, a pointed space is a topological space X with a distinguished basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i.e. a continuous map f : X → Y such that f = y0...
. The pointed homotopy category hTop• is defined to be the category whose objects are pointed topological space and whose morphisms are equivalence classes of pointed maps (i.e., sending the distinguished base point to the base point) modulo pointed homotopy (i.e., the homotopy fixes the base points, as well). The set of maps between pointed spaces X and Y in hTop• is commonly denoted [X,Y]•.
The need to use basepoints has a significant effect on the products (and other limit
Limit (category theory)
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits....
s) appropriate to use. For example, in homotopy theory, the smash product
Smash product
In mathematics, the smash product of two pointed spaces X and Y is the quotient of the product space X × Y under the identifications ∼ for all x ∈ X and y ∈ Y. The smash product is usually denoted X ∧ Y...
X ∧ Y of spaces X and Y is used.
Homotopy equivalences
A continuous map f : X → Y is called a homotopy equivalence, if there is another continuous map g : Y → X such that the two compositions f o g and g o f are homotopic to the respective identity maps. Equivalently, the classes of [f o g] and [g o f] agree with the ones of the identity map of Y and X, respectively. Yet in other words, a map of topological spaces becomes an isomorphism if and only if it is a homotopy equivalence. That is, two topological spaces are isomorphic in hTop if and only ifIf and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
they are homotopy equivalent (i.e. have the same homotopy type).
Homotopy groups
Given the n-sphere Sn, the set- [Sn, X]
of homotopy classes of maps from Sn to some topological space X is the same as the n-th homotopy group πn(X) (for n ≥ 1, the set of connected path-components for n = 0).
Even immediate examples, such as the homotopy groups of spheres
Homotopy groups of spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting...
,
- πn(Sm) = [Sn, Sm]
are hard to compute.
Eilenberg-MacLane spaces
Given an abelian group G and n ≥ 0, the Eilenberg-MacLane spaceEilenberg-MacLane space
In mathematics, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" , and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. In mathematics, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without...
K(G,n) is a topological space satisfying, for any CW-complex X,
- [X, K(G,n)] = Hn(X; G),
where the right hand side denotes the n-th singular cohomology group
Singular homology
In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n....
of X with coefficients in G. In this sense, singular cohomology is representable
Representable functor
In mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures In mathematics, particularly category theory, a...
by the representing space K(G,n). The Brown representability theorem is concerned with the representability of more general functors
- hTopop → SetCategory of setsIn the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...
.
Homotopy theory
Many of the elementary results in homotopy theory can be formulated for arbitrary topological spaces, but as one goes deeper into the theory it is often necessary to work with a more restrictive category of spaces. For most purposes, the homotopy category of CW complexCW complex
In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...
es is the appropriate choice. In the opinion of some experts the homotopy category of CW complexes is the best, if not the only, candidate for the homotopy category. One basic result is that the representable functor
Representable functor
In mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures In mathematics, particularly category theory, a...
s on the homotopy category of CW complexes have a simple characterization (the Brown representability theorem).
The category of CW complexes is deficient in the sense that the space of maps
Function space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...
between two CW complexes is not always a CW complex. A more well-behaved category commonly used in homotopy theory is the category of compactly generated Hausdorff spaces (also called k-spaces). This category includes all CW complexes, locally compact space
Locally compact space
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.-Formal definition:...
s, and first-countable space
First-countable space
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis...
s (such as metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
s).
One important later development was that of spectra
Spectrum (homotopy theory)
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. There are several different constructions of categories of spectra, any of which gives a context for the same stable homotopy theory....
in homotopy theory, essentially the derived category
Derived category
In mathematics, the derived category D of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C...
idea in a form useful for topologists. Spectra have also been defined in various cases using the model category
Model category
In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes...
approach, generalizing the topological case. Many theorists interested in the classical topological theory consider this more axiomatic approach less useful for their purposes. Finding good replacements for CW complexes in the purely algebraic case is a subject of current research.
Generalizations
The above definition of the homotopy of topological spaces is a special case of the more general construction of the homotopy category of a model categoryModel category
In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes...
. Roughly speaking, a model category is a category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
C with three distinguished types of morphisms called fibration
Fibration
In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...
s, cofibration
Cofibration
In mathematics, in particular homotopy theory, a continuous mappingi\colon A \to X,where A and X are topological spaces, is a cofibration if it satisfies the homotopy extension property with respect to all spaces Y. The name is because the dual condition, the homotopy lifting property, defines...
s and weak equivalence
Weak equivalence
-Mathematics:In mathematics, a weak equivalence is a notion from homotopy theory which in some sense identifies objects that have the same basic "shape"...
s. Localizing
Localization of a category
In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. This is formally similar to the process of localization of a ring; it in general makes objects isomorphic that were not so before...
C with respect to the weak equivalences yields the homotopy category.
This construction, applied to the model category of topological spaces, gives back the homotopy category outlined above. Applied to the model category of chain complex
Chain complex
In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...
es over some commutative ring R, for example, yields the derived category
Derived category
In mathematics, the derived category D of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C...
of R-modules. The homotopy category of chain complexes
Homotopy category of chain complexes
In homological algebra in mathematics, the homotopy category K of chain complexes in an additive category A is a framework for working with chain homotopies and homotopy equivalences...
can also be interpreted along these lines.