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My question is essentially in the title: is there a well-developped theory of algebraic pointless topology, that is algebraic topology on locales ?

If not, would it make sense, i.e. would it be relevant (for instance the usual fundamental group construction makes no sense for spaces like the spectrum of a ring, because such spaces are usually totally disconnected or at least, really not connected; so one may wonder -if one does not know much about locales, as I do- whether it would be relevant to study algebraic topology on locales) ?

Added later : In particular (although more general information is also welcome, as in Hurkyl's answer) I would specifically like to know if there is a way of making sense of a "fundamental groupoid" of a locale (which would be defined differently from usual spaces, otherwise pointless locales would have an empty fundamental groupoid) that would be interesting ?

Maxime Ramzi
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One way to answer the question is the observation that sheaves are defined on locales (or more generally, sites or localic groupoids) rather than on topological spaces, so anything that can be expressed in terms of sheaf theory can be interpreted as "algebraic topology on locales".

A category of sheaves of sets is a Grothendieck topos. You can do a lot of algebraic topology in topos theory — as a simple example, you can define sheaf cohomology. This is one of the main lines of development in algebraic geometry.

Regarding the specific interest in the fundamental groupoid, I don't know anything but found these links that may be of interest:

  • I see, thank you for your answer. One thing I didn't specifically ask for but would like to know (I'll add it to my question) is whether one could make sense of a fundamental groupoid of a locale. Obviously (I think) its definition couldn't be the same as for a space (otherwise any pointless locale would have an empty fundamental groupoid, which is sad); but perhaps there is a way of making sense of such a notion ? – Maxime Ramzi Jul 06 '18 at 09:30
  • Thanks, that's very interesting as well ! – Maxime Ramzi Jul 09 '18 at 21:35
  • @MaximeRamzi But badly behaved spaces do have badly behaved fundamental groupoids too. So I wouldn't be surprised if pointless locales have the empty groupoid. – Trebor May 10 '24 at 08:36