Post #1815097

2024-07-04 08:26 UTC

@hallasurvivor - this is great! Not at all where my head is at these days, so I won't be able to offer serious feedback... but here's some less serious feedback: "We think of maps 𝑋→2 being Decidable Propositions. These classify the clopen subsets of 𝑋, and thus are quite rare." I might change that to "and thus may be quite rare". Some people, and famously Johnstone, enjoy Stone spaces, which have lots of clopen sets (at least classically). As you probably know, a Stone space is the same as a profinite topological space, and any Boolean algebra gives a Stone space. So while they're not the kind of space most topologists like, they're great for logicians and even Galois theorists.

Replies (1)

@hallasurvivor@sunny.garden 2024-07-04 13:54

@johncarlosbaez Ah, good catch. Fixed! I've been thinking about this lately after reading a mathoverflow answer from Peter Scholze that talked about the three quite different ways "topology" gets used in math. 1. It's used to formalize geometric intuition for things like manifolds 2. It's used to "tame" infinitary algebraic objects, for things like galois groups/topological vector spaces/etc 3. It's used as a model for homotopy theory He said that, at least for him, these three different use cases demand three different tools, which are all "generalizations" of topology along the three directions. 1. For the mildest generalization, locales are obviously the right thing to consider if you're interested in what's usually called "topology", pointset or otherwise 2. In another direction, he proposes these "condensed sets" as being a better notion of topology for taming algebraic objects. I actually talk about this a bit in part 2 of the blog post. 3. Lastly, for pure homotopy theory we should be working with "anima" (read: oo-groupoids) directly.

Open ##1815098