Post #1815107
2024-07-04 15:16 UTC
@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.
Replies (1)
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@hallasurvivor@sunny.garden 2024-07-04 15:27
@leemph It looks like I was combining two different mathoverflow posts in my mind. First, the comment from Peter Scholze on his answer here, saying that in his world "topology" is either anima, topoi, or condensed sets: https://mathoverflow.net/q/441610/145247 Then an answer here by Maxime Ramzi that says something similar: https://mathoverflow.net/q/459101/145247 As for what role plays the concept of "topos", remember that there's a very close connection between topoi and locales. Indeed, there's a fully faithful 2-embedding from locales to topoi, and this embedding admits a left adjoint ("localic reflection").