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• CommentRowNumber1.
• CommentAuthorDavid_Corfield
• CommentTimeJan 27th 2021

For a treatment in homotopy type theory see

• Dan Frumin, Herman Geuvers, Léon Gondelman, Niels van der Weide, Finite Sets in Homotopy Type Theory, (pdf)
1. Re-organised slightly. Added a brief introductory section, moved a couple of paragraphs of existing content into it. Removed the two context menus ’foundations’ and ’mathematics’ which I don’t think really fit here (the latter is arguably too general to be useful on any page).

Intend to add a new section in a subsequent edit.

2. Added a section explaining how to view a finite set as a scheme (over any base).

• CommentRowNumber4.
• CommentAuthorUlrik
• CommentTimeFeb 22nd 2021

Richard, maybe we can mention (in the “Viewing as schemes” section) that a finite coproduct of affine schemes $Spec R_i$, $i=1,\ldots,n$, is again affine, $Spec (R_1 \times \cdots \times R_n)$. Taking $R_i=\mathbb{Z}$, we can view the finite set $X$ as the (affine) scheme $Spec (\mathbb{Z}^X)$. This agrees with what you wrote, but seems more canonical.

• CommentRowNumber5.
• CommentAuthorRichard Williamson
• CommentTimeFeb 22nd 2021
• (edited Feb 22nd 2021)

Nice! Great if you can go ahead and make an edit if you have time, as I’ll be tied up until the evening European time!

Maybe keep the explicit description, but add the nice canonical one in addition?

• CommentRowNumber6.
• CommentAuthorUlrik
• CommentTimeFeb 22nd 2021

Alternative description of finite sets as affine schemes.

3. Thanks for the edit!

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTime3 days ago