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Ruby Khondaker (she/her) (Dec 04 2025 at 12:49):https://youtu.be/vmcbm5FxRJE?si=AN9HzdedoLbeQl3J
I was watching this video recently, and there’s a remark about 33 mins in that intrigued me. The speaker suggests that it’s useful to think of arrows into an object X of a Topos as “etale spaces” over X, and identify X itself with the corresponding slice category.
The slice category is equivalently the category of elements of Hom(-, X). So is this some instance of yoneda?
Morgan Rogers (he/him) (Dec 04 2025 at 19:41):This intuition is a natural extension of what happens for spaces. Sh(X) is equivalent to the category of etale spaces over X (given an etale space, the sheaf of local sections is the corresponding sheaf; in the other direction we can glue together copies of the opens of X indexed by the elements of a sheaf to get an etale space). The slice over an etale space Z is equivalent to Sh(Z).
Peva Blanchard (Dec 05 2025 at 01:23):John explains the interplay between étale spaces and sheaves in one post in his series about topos theory.