View source for Category of compactly generated weak Hausdorff spaces

In mathematics, the '''[[Category (mathematics)|category]] of [[compactly generated space|compactly generated]] [[weak Hausdorff space]]s,''' '''CGWH,''' is a category used in [[algebraic topology]] as an alternative to the [[category of topological spaces]], '''Top''', as the latter lacks some properties that are common in practice and often convenient to use in proofs. There is also such a category for the '''CGWH''' analog of [[Pointed space|pointed topological spaces]], defined by requiring maps to preserve base points.<ref>{{harvnb|Strickland|2009|loc=Definition 4.1.}}</ref>

The articles [[compactly generated space]] and [[weak Hausdorff space]] define the respective topological properties. For the historical motivation behind these conditions on spaces, see [[Compactly generated space#Motivation]]. This article focuses on the properties of the category.

== Properties ==
'''CGWH''' has the following properties:
*It is [[complete category|complete]]<ref name="complete">{{harvnb|Strickland|2009|loc=Proposition 2.30.}}</ref> and cocomplete.<ref>{{harvnb|Strickland|2009|loc=Corollary 2.23.}}</ref>
*The [[forgetful functor]] to the sets preserves small limits.<ref name="complete" />
*It contains all the [[Locally compact Hausdorff space|locally compact Hausdorff spaces]]<ref>{{harvnb|Strickland|2009|loc=Proposition 1.7.}}</ref>  and all the [[CW complexes]].<ref>{{harvnb|Frankland|2013|loc= Proposition 3.2.}}</ref>
*An [[internal Hom]] exists for any pairs of spaces ''X'' and ''Y'';<ref>{{harvnb|Strickland|2009|loc=Proposition 2.24.}}</ref><ref>{{harvnb|Frankland|2013|loc= Proposition 2.10.}}</ref> it is denoted by <math>\operatorname{Map}(X, Y)</math> or <math>Y^X</math> and is called the (free) '''mapping space''' from ''X'' to ''Y''.  Moreover, there is a homeomorphism
*:<math>\operatorname{Map}(X \times Y, Z) \simeq \operatorname{Map}(X, \operatorname{Map}(Y, Z))</math>
:that is natural in ''X'', ''Y'', and ''Z''.<ref>{{harvnb|Strickland|2009|loc=Proposition 2.12.}}</ref> In short, the category is [[Cartesian closed category|Cartesian closed]] in an [[Enriched category|enriched]] sense.
*A finite product of CW complexes is a CW complex.<ref>{{harvnb|Frankland|2013|loc= Proposition 4.2.}}</ref>
*If <math>(X, *)</math> and <math>(Y, \circ)</math> are pointed spaces, then the [[smash product]] of them exists.<ref>{{harvnb|Strickland|2009|loc=§ 5.}}</ref> The (based) '''mapping space''' <math>\operatorname{Map}( (X, *), (Y, \circ))</math> from <math>(X, *)</math> to <math>(Y, \circ)</math> consists of all base-point-preserving maps from <math>(X, *)</math> to <math>(Y, \circ)</math> and is a closed subspace of the mapping space between the underlying spaces without base points.<ref>{{harvnb|Strickland|2009|loc=Remark 5.6.}}</ref> It is a based space with the base point the unique constant map. Moreover, for based spaces <math>(X, *)</math>, <math>(Y, \circ)</math>, and <math>(Z, \star)</math>, there is a homeomorphism
*:<math>\operatorname{Map}((X, *) \wedge (Y, \circ), (Z, \star)) \simeq \operatorname{Map}((X, *), \operatorname{Map}((Y, \circ), (Z, \star)))</math>
:that is natural in <math>(X, *)</math>, <math>(Y, \circ)</math>, and <math>(Z, \star)</math>.<ref>{{harvnb|Strickland|2009|loc=Proposition 5.7.}}</ref>

==Notes==
{{reflist}}

==References==
* {{cite web |last=Frankland |first=Martin |title=Math 527 - Homotopy Theory – Compactly generated spaces |url=https://uregina.ca/~franklam/Math527/Math527_0204.pdf |date=February 4, 2013}}
* {{cite journal |last1=Steenrod |first1=N. E. |title=A convenient category of topological spaces. |journal=Michigan Mathematical Journal |date=1 May 1967 |volume=14 |issue=2 |pages=133–152 |doi=10.1307/mmj/1028999711|doi-access=free }}
* {{cite web |last=Strickland |first=Neil |title=The category of CGWH spaces |url=https://ncatlab.org/nlab/files/StricklandCGHWSpaces.pdf |date=2009}}
* {{cite book |doi=10.1017/CBO9780511983948.007 |chapter=Appendix |title=Cellular Structures in Topology |year=1990 |pages=241–305 |isbn=9780521327848 }}

== Further reading ==
* [https://web.stanford.edu/~dkim04/blog/cgwh-spaces/ The CGWH category, Dongryul Kim 2017]

[[Category:Algebraic topology]]
[[Category:Categories in category theory]]
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