Category of compactly generated weak Hausdorff spaces

In mathematics, the category of compactly generated weak Hausdorff spaces CGWH is one of typically used categories in algebraic topology as a substitute for the category of topological spaces, as the latter lacks some of the pleasant properties one would desire. There is also such a category for based spaces, defined by requiring maps to preserve the base points.^[1]

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^[2] and cocomplete.^[3]
The forgetful functor to the sets preserves small limits.^[2]
It contains all the locally compact Hausdorff spaces^[4] and all the CW complexes.^[5]
The internal Hom exists for any pairs of spaces X, Y;^[6]^[7] it is denoted by $\operatorname {Map} (X,Y)$ or $Y^{X}$ and is called the (free) mapping space from X to Y. Moreover, there is a homeomorphism
$\operatorname {Map} (X\times Y,Z)\simeq \operatorname {Map} (X,\operatorname {Map} (Y,Z))$

that is natural in X, Y, Z.^[8] In short, the category is Cartesian closed in an enriched sense.

A finite product of CW complexes is a CW complex.^[9]
If X, Y are based spaces, then the smash product of them exists.^[10] The (based) mapping space $\operatorname {Map} (X,Y)$ from X to Y consists of all base-point-preserving maps from X to Y and is a closed subspace of the mapping space between the underlying unbased spaces.^[11] It is a based space with the base point the unique constant map. For based spaces X, Y, Z, there is a homeomorphism
$\operatorname {Map} (X\wedge Y,Z)\simeq \operatorname {Map} (X,\operatorname {Map} (Y,Z))$

that is natural in X, Y, Z.^[12]

Notes

^ Strickland 2009, Definition 4.1.
^ ^a ^b Strickland 2009, Proposition 2.30.
^ Strickland 2009, Corollary 2.23.
^ Strickland 2009, Proposition 1.7.
^ Frankland 2013, Proposition 3.2.
^ Strickland 2009, Proposition 2.24.
^ Frankland 2013, Proposition 2.10.
^ Strickland 2009, Proposition 2.12.
^ Frankland 2013, Proposition 4.2.
^ Strickland 2009, § 5.
^ Strickland 2009, Remark 5.6.
^ Strickland 2009, Proposition 5.7.

References

Frankland, Martin (February 4, 2013). "Math 527 - Homotopy Theory – Compactly generated spaces" (PDF).
Steenrod, N. E. (1 May 1967). "A convenient category of topological spaces". Michigan Mathematical Journal. 14 (2): 133–152. doi:10.1307/mmj/1028999711.
Strickland, Neil (2009). "The category of CGWH spaces" (PDF).
"Appendix". Cellular Structures in Topology. 1990. pp. 241–305. doi:10.1017/CBO9780511983948.007. ISBN 9780521327848.

Properties

Notes

References

Further reading