# 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 | 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 [[ | *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 | * 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 | * 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). [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]] | ||

## 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*:**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 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* : **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 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* : **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 on the Cartesian product. 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 continuous maps, and the isomorphisms are the homeomorphisms. - The extremal monomorphisms are (up to isomorphism) the subspace embeddings. Every extremal monomorphism is 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 (and therefore also not a topos) since it does not have exponential objects for all spaces.

## 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 of**Top**. One can likewise form the pointed homotopy category**hTop**_{•}. **Top**contains the important category**Haus**of topological spaces with the Hausdorff property as a full subcategory. 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.

## 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).