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I have read Peter Johnstone's “The Point of Pointless Topology” and the idea that topological spaces are not quite the right abstraction for topology seems, at least philosophically, rather appealing. Certainly, however, to convince any topologist of its use one would need to provide analogues to basic concrete objects and constructions used in general topology.

I am looking for references which transcribe the following notions (or others with the same concrete flavour) in the category of locales:

  • Homotopy and homotopy equivalence;

  • The fundamental group/groupoid;

  • Homology and cohomology;

  • $S^1$, $[0,1]$, $\mathbb R^n$;

  • Classical constructions such as $CX, \Sigma X, X \vee Y, \beta X$;

  • CW complexes.

Many thanks.

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    $\begingroup$ To some extent this is related to shape theory - in an arbitrary topos $X$ (in particular the sheaf topos of a locale) you can describe cohomology with constant coefficients (which, in the case of nice spaces, recovers singular cohomology); you can also describe a (pro-)fundamental groupoid and in fact a (pro-)fundamental $\infty$-groupoid. I don't think looking for CW-complexes and ordinary spaces such as $S^1$ leads to interesting things in the sense that these are sober spaces and so are completely characterized by their locales $\endgroup$
    – Maxime Ramzi
    Commented Jun 10, 2022 at 8:14
  • $\begingroup$ The category of locales has products, and the "forgetful" functor $\textbf{Top} \to \textbf{Loc}$ preserves all colimits, the terminal object and also binary products where at least one of the factors is locally compact Hausdorff. The "forgetful" functor is also fully faithful on the full subcategory of Hausdorff spaces. Thus, as Maxime says, if you want to focus on the homotopy theory of CW-complexes or other pleasant spaces then there is no theoretical obstruction to using locales. $\endgroup$
    – Zhen Lin
    Commented Jun 10, 2022 at 11:53
  • $\begingroup$ Some references can be found at nlab. $\endgroup$
    – Tyrone
    Commented Jun 10, 2022 at 12:26

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