Category of compactly generated weak Hausdorff spaces

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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. 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 [math]\displaystyle{ \operatorname{Map}(X, Y) }[/math] or [math]\displaystyle{ Y^X }[/math] and is called the (free) mapping space from X to Y. Moreover, there is a homeomorphism
    [math]\displaystyle{ \operatorname{Map}(X \times Y, Z) \simeq \operatorname{Map}(X, \operatorname{Map}(Y, Z)) }[/math]
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 [math]\displaystyle{ \operatorname{Map}(X, Y) }[/math] 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
    [math]\displaystyle{ \operatorname{Map}(X \wedge Y, Z) \simeq \operatorname{Map}(X, \operatorname{Map}(Y, Z)) }[/math]
that is natural in X, Y, Z.[12]

Notes

  1. ↑ Strickland 2009, Definition 4.1.
  2. ↑ 2.0 2.1 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

Further reading



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