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 or and is called the (free) mapping space from X to Y. Moreover, there is a homeomorphism
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 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
that is natural in X, Y, Z.[12]

Notes
  1. Strickland 2009, Definition 4.1.
  2. Strickland 2009, Proposition 2.30.
  3. Strickland 2009, Corollary 2.23.
  4. Strickland 2009, Proposition 1.7.
  5. Frankland 2013, Proposition 3.2.
  6. Strickland 2009, Proposition 2.24.
  7. Frankland 2013, Proposition 2.10.
  8. Strickland 2009, Proposition 2.12.
  9. Frankland 2013, Proposition 4.2.
  10. Strickland 2009, § 5.
  11. Strickland 2009, Remark 5.6.
  12. 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.

Further reading
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