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Finiteness in Sheaf Topoi

The notion of “finiteness” is constructively subtle in ways that can be tricky for people new to the subject to understand. For a while now I’ve wanted to figure out what’s going on with the different versions of “finite” in a way that felt concrete and obvious (I mentioned this in a few older blog posts here and here), and for me that means I want to understand them inside a sheaf topos $\mathsf{Sh}(X)$. I’ve thought about this a few times, but I wasn’t able to really see what was happening until a few days ago when I realized I had a serious misconception about picturing bundles and etale spaces! In this post, we’ll talk about that misconception, and spend some time discussing constructive finiteness in its most important forms.

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