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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'''.
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| 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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| ==As a concrete category==
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| Like many categories, the category '''Top''' is a [[concrete category]] (also known as a ''construct''), 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]]
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| :''U'' : '''Top''' → '''Set'''
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| to the [[category of sets]] which assigns to each topological space the underlying set and to each continuous map the underlying [[function (mathematics)|function]].
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| The forgetful functor ''U'' has both a [[left adjoint]]
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| :''D'' : '''Set''' → '''Top'''
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| which equips a given set with the [[discrete topology]] and a [[right adjoint]]
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| :''I'' : '''Set''' → '''Top'''
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| which equips a given set with the [[indiscrete topology]]. Both of these functors are, in fact, [[right inverse]]s 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 embedding]]s of '''Set''' into '''Top'''.
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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. | |
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| 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).
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| ==Limits and colimits==
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| 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'''.
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| Specifically, if ''F'' is a [[diagram (category theory)|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'''.
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| 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'''.
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| Examples of limits and colimits in '''Top''' include:
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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'''.
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| *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.
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| *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.
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| *[[Direct limit]]s and [[inverse limit]]s are the set-theoretic limits with the [[final topology]] and [[initial topology]] respectively.
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| *[[Adjunction space]]s are an example of [[pushout (category theory)|pushouts]] in '''Top'''.
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| ==Other properties==
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| *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.
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| *The extremal monomorphisms are (up to isomorphism) the [[subspace topology|subspace]] embeddings. Every extremal monomorphism is [[regular morphism|regular]].
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| *The extremal epimorphisms are (essentially) the [[quotient map]]s. Every extremal epimorphism is regular.
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| *There are no [[zero morphism]]s in '''Top''', and in particular the category is not [[preadditive category|preadditive]].
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| *'''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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| ==Relationships to other categories==
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| *The category of [[pointed topological space]]s '''Top'''<sub>•</sub> is a [[coslice category]] over '''Top'''.
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| * 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>.
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| *'''Top''' contains the important category '''Haus''' of topological spaces with the [[Hausdorff space|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 set|dense]] [[image (mathematics)|images]] in their [[codomain]]s, so that epimorphisms need not be [[surjective]].
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| == References ==
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| * Herrlich, Horst: ''Topologische Reflexionen und Coreflexionen''. Springer Lecture Notes in Mathematics 78 (1968).
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| * Herrlich, Horst: ''Categorical topology 1971 - 1981''. In: General Topology and its Relations to Modern Analysis and Algebra 5, Heldermann Verlag 1983, pp. 279 - 383.
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| * 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.
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| * 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]]
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| [[Category:General topology]]
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The writer is known as Wilber Pegues. My spouse and I live in Mississippi but now I'm considering other options. The preferred pastime for him and his children is to play lacross and he would by no means give it up. My day occupation is an invoicing officer but I've already applied for an additional one.
Feel free to surf to my blog ... free psychic readings - arthritisreduction.com -