In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the category of topological spaces, often denoted Top, is the
category whose
object
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ...
s are
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
s and whose
morphisms are
continuous maps. This is a category because the
composition of two continuous maps is again continuous, and the identity function is continuous. The study of Top and of properties of
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
s using the techniques of
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
is known as categorical topology.
N.B. Some authors use the name Top for the categories with
topological manifolds, with
compactly generated spaces as objects and continuous maps as morphisms or with the
category of compactly generated weak Hausdorff spaces In mathematics, the category of compactly generated weak Hausdorff spaces CGWH is one of typically used categories in algebraic topology as a substitute for the category of topological spaces, as the latter lacks some of the pleasant properties one ...
.
As a concrete category
Like many categories, the category Top is a
concrete category, meaning its objects are
sets with additional structure (i.e. topologies) and its morphisms are
functions preserving this structure. There is a natural
forgetful functor
:''U'' : Top → Set
to the
category of sets which assigns to each topological space the underlying set and to each continuous map the underlying
function.
The forgetful functor ''U'' has both a
left adjoint
:''D'' : Set → Top
which equips a given set with the
discrete topology, and a
right adjoint
:''I'' : Set → Top
which equips a given set with the
indiscrete topology. Both of these functors are, in fact,
right inverses to ''U'' (meaning that ''UD'' and ''UI'' are equal to the
identity functor on Set). Moreover, since any function between discrete or between indiscrete spaces is continuous, both of these functors give
full embeddings of Set into Top.
Top is also ''fiber-complete'' meaning that the
category of all topologies on a given set ''X'' (called the ''
fiber
Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorporate ...
'' of ''U'' above ''X'') forms a
complete lattice
In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' ...
when ordered by
inclusion. The
greatest element in this fiber is the discrete topology on ''X'', while the
least element is the indiscrete topology.
Top is the model of what is called a
topological category. These categories are characterized by the fact that every
structured source has a unique
initial lift
In a written or published work, an initial capital, also referred to as a drop capital or simply an initial cap, initial, initcapital, initcap or init or a drop cap or drop, is a letter at the beginning of a word, a chapter (books), chapter, or ...
. In Top the initial lift is obtained by placing the
initial topology on the source. Topological categories have many properties in common with Top (such as fiber-completeness, discrete and indiscrete functors, and unique lifting of limits).
Limits and colimits
The category Top is both
complete and cocomplete, which means that all small
limits and colimits exist in Top. In fact, the forgetful functor ''U'' : Top → Set uniquely lifts both limits and colimits and preserves them as well. Therefore, (co)limits in Top are given by placing topologies on the corresponding (co)limits in Set.
Specifically, if ''F'' is a
diagram
A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three ...
in Top and (''L'', ''φ'' : ''L'' → ''F'') is a limit of ''UF'' in Set, the corresponding limit of ''F'' in Top is obtained by placing the
initial topology on (''L'', ''φ'' : ''L'' → ''F''). Dually, colimits in Top are obtained by placing the
final topology on the corresponding colimits in Set.
Unlike many ''algebraic'' categories, the forgetful functor ''U'' : Top → Set does not create or reflect limits since there will typically be non-universal
cones in Top covering universal cones in Set.
Examples of limits and colimits in Top include:
*The
empty set (considered as a topological space) is the
initial object of Top; any
singleton topological space is a
terminal object. There are thus no
zero objects in Top.
*The
product in Top is given by the
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seem ...
on the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\t ...
. The
coproduct is given by the
disjoint union of topological spaces.
*The
equalizer of a pair of morphisms is given by placing the
subspace topology on the set-theoretic equalizer. Dually, the
coequalizer is given by placing the
quotient topology on the set-theoretic coequalizer.
*
Direct limits and
inverse limits are the set-theoretic limits with the
final topology and
initial topology respectively.
*
Adjunction spaces are an example of
pushouts in Top.
Other properties
*The
monomorphisms in Top are the
injective continuous maps, the
epimorphisms are the
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
continuous maps, and the
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
s are the
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...
s.
*The
extremal monomorphisms are (up to isomorphism) the
subspace embeddings. In fact, in Top all extremal monomorphisms happen to satisfy the stronger property of being
regular.
*The extremal epimorphisms are (essentially) the
quotient maps. Every extremal epimorphism is regular.
*The split monomorphisms are (essentially) the inclusions of
retracts into their ambient space.
*The split epimorphisms are (up to isomorphism) the continuous surjective maps of a space onto one of its retracts.
*There are no
zero morphisms in Top, and in particular the category is not
preadditive.
*Top is not
cartesian closed
In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mat ...
(and therefore also not a
topos) since it does not have
exponential objects for all spaces. When this feature is desired, one often restricts to the full subcategory of
compactly generated Hausdorff spaces CGHaus or the
category of compactly generated weak Hausdorff spaces In mathematics, the category of compactly generated weak Hausdorff spaces CGWH is one of typically used categories in algebraic topology as a substitute for the category of topological spaces, as the latter lacks some of the pleasant properties one ...
. However, Top is contained in the exponential category of
pseudotopologies, which is itself a subcategory of the (also exponential) category of
convergence space
In mathematics, a convergence space, also called a generalized convergence, is a set together with a relation called a that satisfies certain properties relating elements of ''X'' with the family of filters on ''X''. Convergence spaces generaliz ...
s.
Relationships to other categories
*The category of
pointed topological spaces Top
• is a
coslice category over Top.
* The
homotopy category hTop has topological spaces for objects and
homotopy equivalence classes of continuous maps for morphisms. This is a
quotient category
In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally small) categories, analogous to a quotient group or quotient space, but ...
of Top. One can likewise form the pointed homotopy category hTop
•.
*Top contains the important category Haus of
Hausdorff spaces as a
full subcategory
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, ...
. The added structure of this subcategory allows for more epimorphisms: in fact, the epimorphisms in this subcategory are precisely those morphisms with
dense images in their
codomains, so that epimorphisms need not be
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
.
*Top contains the full subcategory CGHaus of
compactly generated Hausdorff spaces, which has the important property of being a
Cartesian closed category while still containing all of the typical spaces of interest. This makes CGHaus a particularly ''convenient category of topological spaces'' that is often used in place of Top.
* The forgetful functor to Set has both a left and a right adjoint, as described above in the concrete category section.
* There is a functor to the category of
locales Loc sending a topological space to its locale of open sets. This functor has a right adjoint that sends each locale to its topological space of points. This adjunction restricts to an equivalence between the category of
sober spaces and spatial locales.
*The
homotopy hypothesis
In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states that the ∞-groupoids are spaces. If we model our ∞-groupoids as Kan complexes, then the homotopy types of the geometric realizations of these sets give m ...
relates Top with ∞Grpd, the category of
∞-groupoids. The conjecture states that ∞-groupoids are equivalent to topological spaces modulo
weak homotopy equivalence.
See also
*
*
*
*
*
Citations
References
* Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990)
''Abstract and Concrete Categories''(4.2MB PDF). Originally publ. John Wiley & Sons. . (now free on-line edition).
*
*
*
*
Herrlich, Horst:
Topologische Reflexionen und Coreflexionen'. Springer Lecture Notes in Mathematics 78 (1968).
* Herrlich, Horst: ''Categorical topology 1971–1981''. In: General Topology and its Relations to Modern Analysis and Algebra 5, Heldermann Verlag 1983, pp. 279–383.
* Herrlich, Horst & Strecker, George E.
Categorical Topology – its origins, as exemplified by the unfolding of the theory of topological reflections and coreflections before 1971 In: Handbook of the History of General Topology (eds. C.E.Aull & R. Lowen), Kluwer Acad. Publ. vol 1 (1997) pp. 255–341.
{{refend
Topological spaces
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
General topology