Post #1815098

2024-07-04 13:54 UTC

@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.

Replies (2)

@hallasurvivor@sunny.garden 2024-07-04 13:56

@johncarlosbaez Anyways, it's funny that I made this mistake, since in a lot of ways the topological topos solves the same problems as condensed sets, so I should probably have profinite topologies closer to the front of my mind when thinking about examples!

Open ##1815099
@leemph@mathstodon.xyz 2024-07-04 15:16

@hallasurvivor I'd love to read that MO discussion. Can you share the title, or the link? In this picture, what role plays the concept of topos? Recently I listened to an old talk by Colin McLarty about Grothendieck. Something that really got stuck in my head was the idea that topoi are the real right notion of space from Grothendieck pov... and that the topoi Grothendieck thinks of are not really the big categories satisfying the Giraud axioms but some kind of "ghosts" (my term) that manifest as the big categories that we know... he goes on explaining this by interpreting the product of two topoi (as categories) as somehow the avatar (my term) of the union of their underlying true topoi.

Open ##1815107