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In [[category theory]], a discipline in [[mathematics]], a '''topological category''' is a category that is enriched over the category of compactly generated [[Hausdorff space]]s. They can be used as a foundation for [[higher category theory]].{{harv|Lurie|2009}}


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A topological category is a category <math>A</math> along with a grounding functor <math>T: A \to X</math> that maps to the category of sets <math>X</math> such that <math>A</math> admits weak or initial structures with respect to <math>T</math>. This means that constant functions in <math>X</math> lift to <math>A</math>-morphisms and that fibers <math>T^{-1} x, x \in X</math> are small (they are sets and not proper classes).<ref name=Brümmer1984>{{cite journal|last=Brümmer|first=G. C. L.|title=Topological categories|journal=Topology and its Applications|date=1984|month=September|volume=18|issue=1|pages=27–41|url=http://www.sciencedirect.com/science/article/pii/0166864184900294|accessdate=1 October 2013}}</ref> 
 
==See also==
 
*[[Infinity category]]
*[[Simplicial category]]
 
==References==
{{reflist}}
*{{Citation | last1=Lurie | first1=Jacob | title=Higher topos theory | arxiv=math.CT/0608040 | publisher=[[Princeton University Press]] | series=Annals of Mathematics Studies | isbn=978-0-691-14049-0; 978-0-691-14049-0 | mr=2522659 | year=2009 | volume=170}}
 
[[Category:Category theory]]

Revision as of 07:02, 28 January 2014

In category theory, a discipline in mathematics, a topological category is a category that is enriched over the category of compactly generated Hausdorff spaces. They can be used as a foundation for higher category theory.Template:Harv

A topological category is a category A along with a grounding functor T:AX that maps to the category of sets X such that A admits weak or initial structures with respect to T. This means that constant functions in X lift to A-morphisms and that fibers T1x,xX are small (they are sets and not proper classes).[1]

See also

References

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