Editing Category of topological spaces

{{Short description|Category whose objects are topological spaces and whose morphisms are continuous maps}}
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, and the identity function is 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 categories with [[topological manifold]]s, with 
[[compactly generated space|compactly generated spaces]] as objects and continuous maps as morphisms or with the [[category of compactly generated weak Hausdorff spaces]].

==As a concrete category==

Like many categories, the category '''Top''' is a [[concrete category]], meaning its objects are [[Set (mathematics)|sets]] with additional structure (i.e. topologies) and its morphisms are [[function (mathematics)|function]]s preserving this structure. There is a natural [[forgetful functor]]

{{block indent|''U'' : '''Top''' &rarr; '''Set'''}}

to the [[category of sets]] which assigns to each topological space the underlying set and to each continuous map the underlying [[function (mathematics)|function]].

The forgetful functor ''U'' has both a [[left adjoint]]

{{block indent|''D'' : '''Set''' &rarr; '''Top'''}}

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

{{block indent|''I'' : '''Set''' &rarr; '''Top'''}}

which equips a given set with the [[indiscrete topology]]. Both of these functors are, in fact, [[Inverse function#Left and right inverses|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 embedding]]s of '''Set''' into '''Top'''.

'''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.

'''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==

The category '''Top''' is both [[complete category|complete and cocomplete]], which means that all small [[limit (category theory)|limits and colimit]]s 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 (category theory)|diagram]] 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 [[cone (category theory)|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 (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 [[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.
*[[Adjunction space]]s are an example of [[pushout (category theory)|pushouts]] in '''Top'''.

==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 [[extremal  monomorphism|extremal ]] monomorphisms are (up to isomorphism) the [[subspace topology|subspace]] embeddings. In fact, in '''Top''' all extremal monomorphisms happen to satisfy the stronger property of being [[regular monomorphism|regular]].
*The extremal epimorphisms are (essentially) the [[quotient map (topology)|quotient map]]s. Every extremal epimorphism is regular.
*The split monomorphisms are (essentially) the inclusions of [[retraction (topology)|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 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. When this feature is desired, one often restricts to the full subcategory of [[compactly generated Hausdorff space]]s '''CGHaus''' or the [[category of compactly generated weak Hausdorff spaces]]. However, '''Top''' is contained in the exponential category of [[pseudotopologies]], which is itself a subcategory of the (also exponential) category of [[convergence space]]s.<ref name="Dolecki 2009 Init. Conv. 1-51">{{harvnb|Dolecki|2009|pages=1-51}}</ref>

==Relationships to other categories==

*The category of [[pointed topological space]]s '''Top'''<sub>•</sub> is a [[coslice category]] over '''Top'''. 
* The [[homotopy category of topological spaces|homotopy category]] '''hTop''' has topological spaces for objects and [[homotopy equivalent|homotopy equivalence classes]] of continuous maps for morphisms. This is a [[quotient category]] of '''Top'''. One can likewise form the pointed homotopy category '''hTop'''<sub>•</sub>.
*'''Top''' contains the important category '''Haus''' of [[Hausdorff space|Hausdorff spaces]] 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 set|dense]] [[image (mathematics)|images]] in their [[codomain]]s, so that epimorphisms need not be [[surjective]].
*'''Top''' contains the full subcategory '''CGHaus''' of [[compactly generated Hausdorff space]]s, 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 [[Locale (mathematics)|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 space]]s and spatial locales.
*The [[homotopy hypothesis]] relates '''Top''' with '''∞Grpd''', the category of [[∞-groupoid|∞-groupoids]]. The conjecture states that ∞-groupoids are equivalent to topological spaces modulo [[Weak equivalence (homotopy theory)|weak homotopy equivalence]].

== See also ==

* {{annotated link|Category of groups}}
* {{annotated link|Category of metric spaces}}
* {{annotated link|Category of sets}}
* {{annotated link|Category of topological spaces with base point}}
* {{annotated link|Category of topological vector spaces}}
* [[Category of measurable spaces]]

== Citations ==
{{reflist}}

== References ==

{{refbegin}}
* Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990). [http://katmat.math.uni-bremen.de/acc/acc.pdf ''Abstract and Concrete Categories''] {{Webarchive|url=https://web.archive.org/web/20150421081851/http://katmat.math.uni-bremen.de/acc/acc.pdf |date=2015-04-21 }} (4.2MB PDF). Originally publ. John Wiley & Sons. {{ISBN|0-471-60922-6}}. (now free on-line edition).
* {{Dolecki Mynard Convergence Foundations Of Topology}} <!-- {{sfn|Dolecki|Mynard|2016|p=}} -->
* {{cite book |last=Dolecki |first=Szymon |title=Beyond Topology |chapter=An initiation into convergence theory |series=Contemporary Mathematics |date=2009 |chapter-url=http://dolecki.perso.math.cnrs.fr/init_IX07.pdf |editor1-last=Mynard |editor1-first=Frédéric |editor2-last=Pearl |editor2-first=Elliott |volume=486 |pages=115–162 |doi=10.1090/conm/486/09509 |isbn=9780821842799 |access-date=14 January 2021 }}
* {{cite journal |last1=Dolecki |first1=Szymon |last2=Mynard |first2=Frédéric |date=2014 |title=A unified theory of function spaces and hyperspaces: local properties |url=http://dolecki.perso.math.cnrs.fr/18dolecki.pdf |journal=Houston J. Math. |volume=40 |issue=1 |pages=285–318 |doi= |access-date=14 January 2021 }}
* [[Horst Herrlich|Herrlich, Horst]]: ''[https://books.google.com/books?id=Q1J7CwAAQBAJ 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.&nbsp;279–383.
* Herrlich, Horst & Strecker, George E.: [https://link.springer.com/chapter/10.1007/978-94-017-0468-7_15 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.&nbsp;255–341.
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[[Category:Categories in category theory|Topological spaces]]
[[Category:General topology]]