Difference between revisions of "Category of topological spaces"

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In [[mathematics]], the '''category of topological spaces''', often denoted '''Top''', is the [[category (category theory)|category]] whose [[object (category theory)|object]]s are [[topological space]]s and whose [[morphism]]s are [[continuous map]]s. This is a category because the [[function composition|composition]] of two continuous maps is again continuous. The study of '''Top''' and of properties of [[topological space]]s using the techniques of [[category theory]] is known as '''categorical topology'''.
In [[mathematics]], the '''category of topological spaces''', often denoted '''Top''', is the [[category (category theory)|category]] whose [[object (category theory)|object]]s are [[topological space]]s and whose [[morphism]]s are [[continuous map]]s or some other variant; for example, objects are often assumed to be [[compactly generated space|compactly generated]]. This is a category because the [[function composition|composition]] of two continuous maps is again continuous. The study of '''Top''' and of properties of [[topological space]]s using the techniques of [[category theory]] is known as '''categorical topology'''.


N.B. Some authors use the name '''Top''' for the category with [[topological manifold]]s as objects and continuous maps as morphisms.
N.B. Some authors use the name '''Top''' for the category with [[topological manifold]]s as objects and continuous maps as morphisms.
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The construct '''Top''' is also ''fiber-complete'' meaning that the [[lattice of topologies|category of all topologies]] on a given set ''X'' (called the ''[[fiber (mathematics)|fiber]]'' of ''U'' above ''X'') forms a [[complete lattice]] when ordered by [[set inclusion|inclusion]]. The [[greatest element]] in this fiber is the discrete topology on ''X'' while the [[least element]] is the indiscrete topology.
The construct '''Top''' is also ''fiber-complete'' meaning that the [[lattice of topologies|category of all topologies]] on a given set ''X'' (called the ''[[fiber (mathematics)|fiber]]'' of ''U'' above ''X'') forms a [[complete lattice]] when ordered by [[set inclusion|inclusion]]. The [[greatest element]] in this fiber is the discrete topology on ''X'' while the [[least element]] is the indiscrete topology.


The construct '''Top''' is the model of what is called a [[topological category]]. These categories are characterized by the fact that every [[structured source]] <math>(X \to UA_i)_I</math> has a unique [[initial lift]] <math>( A \to A_i)_I</math>. In '''Top''' the initial lift is obtained by placing the [[initial topology]] on the source. Topological categories have many nice properties in common with '''Top''' (such as fiber-completeness, discrete and indiscrete functors, and unique lifting of limits).
The construct '''Top''' is the model of what is called a [[topological category]]. These categories are characterized by the fact that every [[structured source]] <math>(X \to UA_i)_I</math> has a unique [[initial lift]] <math>( A \to A_i)_I</math>. 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==
==Limits and colimits==
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*The [[empty set]] (considered as a topological space) is the [[initial object]] of '''Top'''; any [[singleton (mathematics)|singleton]] topological space is a [[terminal object]]. There are thus no [[zero object]]s in '''Top'''.
*The [[empty set]] (considered as a topological space) is the [[initial object]] of '''Top'''; any [[singleton (mathematics)|singleton]] topological space is a [[terminal object]]. There are thus no [[zero object]]s in '''Top'''.
*The [[product (category theory)|product]] in '''Top''' is given by the [[product topology]] on the [[Cartesian product]]. The [[coproduct (category theory)|coproduct]] is given by the [[disjoint union (topology)|disjoint union]] of topological spaces.
*The [[product (category theory)|product]] in '''Top''' is given by the [[product topology]] on the [[Cartesian product]]. The [[coproduct (category theory)|coproduct]] is given by the [[disjoint union (topology)|disjoint union]] of topological spaces.
*The [[equaliser_(mathematics)#In_category_theory |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.
*The [[equaliser (mathematics)#In category theory|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 limit]]s and [[inverse limit]]s are the set-theoretic limits with the [[final topology]] and [[initial topology]] respectively.
*[[Direct limit]]s and [[inverse limit]]s are the set-theoretic limits with the [[final topology]] and [[initial topology]] respectively.
*[[Adjunction space]]s are an example of [[pushout (category theory)|pushouts]] in '''Top'''.
*[[Adjunction space]]s are an example of [[pushout (category theory)|pushouts]] in '''Top'''.
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==Other properties==
==Other properties==
*The [[monomorphism]]s in '''Top''' are the [[injective]] continuous maps, the [[epimorphism]]s are the [[surjective]] continuous maps, and the [[isomorphism]]s are the [[homeomorphism]]s.
*The [[monomorphism]]s in '''Top''' are the [[injective]] continuous maps, the [[epimorphism]]s are the [[surjective]] continuous maps, and the [[isomorphism]]s are the [[homeomorphism]]s.
*The extremal monomorphisms are (up to isomorphism) the [[subspace topology|subspace]] embeddings. Every extremal monomorphism is [[regular morphism|regular]].
*The extremal monomorphisms are (up to isomorphism) the [[subspace topology|subspace]] embeddings. Every extremal monomorphism is [[regular morphism (topology)|regular]].
*The extremal epimorphisms are (essentially) the [[quotient map]]s. Every extremal epimorphism is regular.
*The extremal epimorphisms are (essentially) the [[quotient map]]s. Every extremal epimorphism is regular.
*The split monomorphisms are (essentially) the inclusions of [[retract]]s 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 morphism]]s in '''Top''', and in particular the category is not [[preadditive category|preadditive]].
*There are no [[zero morphism]]s in '''Top''', and in particular the category is not [[preadditive category|preadditive]].
*'''Top''' is not [[cartesian closed category|cartesian closed]] (and therefore also not a [[topos]]) since it does not have [[exponential object]]s for all spaces.
*'''Top''' is not [[cartesian closed category|cartesian closed]] (and therefore also not a [[topos]]) since it does not have [[exponential object]]s for all spaces.
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* Herrlich, Horst: ''Topologische Reflexionen und Coreflexionen''. Springer Lecture Notes in Mathematics 78 (1968).
* 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: ''Categorical topology 1971 - 1981''. In: General Topology and its Relations to Modern Analysis and Algebra 5, Heldermann Verlag 1983, pp.&nbsp;279 383.


* Herrlich, Horst & Strecker, George E.: Categorical Topology - its origins, as examplified 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.
* Herrlich, Horst & Strecker, George E.: Categorical Topology - its origins, as examplified 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.&nbsp;255 341.


* Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990). [http://katmat.math.uni-bremen.de/acc/acc.pdf ''Abstract and Concrete Categories''] (4.2MB PDF). Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition).
* Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990). [http://katmat.math.uni-bremen.de/acc/acc.pdf ''Abstract and Concrete Categories''] (4.2MB PDF). Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition).
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[[Category:Category-theoretic categories|Topological spaces]]
[[Category:Category-theoretic categories|Topological spaces]]
[[Category:General topology]]
[[Category:General topology]]
[[es:Categoría de espacios topológicos]]
[[nl:Categorie van topologische ruimten]]
[[pl:Kategoria przestrzeni topologicznych]]
[[zh:拓撲空間範疇]]

Latest revision as of 07:07, 30 November 2014

In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps or some other variant; for example, objects are often assumed to be compactly generated. This is a category because the composition of two continuous maps is again continuous. The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology.

N.B. Some authors use the name Top for the category with topological manifolds as objects and continuous maps as morphisms.

As a concrete category

Like many categories, the category Top is a concrete category (also known as a construct), 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 : TopSet

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 : SetTop

which equips a given set with the discrete topology and a right adjoint

I : SetTop

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 indiscrete spaces is continuous, both of these functors give full embeddings of Set into Top.

The construct Top is also fiber-complete meaning that the category of all topologies on a given set X (called the fiber of U above X) forms a 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.

The construct 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 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 : TopSet 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 in Top and (L, φ) is a limit of UF in Set, the corresponding limit of F in Top is obtained by placing the initial topology on (L, φ). 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 : TopSet 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:

Other properties

Relationships to other categories

References

  • 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 examplified 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.
  • Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990). Abstract and Concrete Categories (4.2MB PDF). Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition).