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Every time I start to read about schemes from a birds-eye view (like in the introduction to The Geometry of Schemes by Eisenbud and Harris) I get really excited; they sound like a categorical approach to geometry, allowing for a rigorously formalized account of the infinitesimal which almost directly permits one to formalize their intuition about infinitesimal geometric concepts and then reason about these concepts precisely in an intuitively satisfying manner.

By the time I get to the formal introduction, however, we're immediately talking about prime ideals of commutative rings and topological spaces -- I have no problem with these concepts in a vacuum, but this approach feels far off base from the expectations outlined above. I have no doubt that category theory is hiding nearby in the background, but the approach through commutative algebra and topology is off-putting to me.

In the midst of my despair I came across Martin Brandenburg's answer to a question over at MSE where he says that 'schemes are categories fibered in setoids', however the page of the Stacks project linked in the answer makes no reference to schemes. In the comments on this same answer he outlines how the category of setoids is isomorphic to the category of thin groupoids, which leads to the following

Proposition. A scheme is a category fibered in thin groupoids; that is, a Grothendieck fibration $p:\mathcal{E}\to\mathcal{B}$ whose fibres are all thin groupoids.

Since Grothendieck fibrations aren't the fibrations in the model structure on $\mathfrak{Cat}$, we may want instead to offer the following

Proposition. A scheme is a category essentially fibered in thin groupoids; that is, a Street fibration $p:\mathcal{E}\to\mathcal{B}$ whose essential fibres are all thin groupoids.

Either of these points of view would be very appealing to me.

Question: Are either of these propositions correct, and if so are there any introductions to the theory of schemes viewing them in this light?

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    $\begingroup$ “Is a” is highly misleading here. Every scheme is a category fibred in setoids, sure. But so is any presheaf, not necessarily satisfying any sheaf condition, let alone representability by a scheme. $\endgroup$
    – Zhen Lin
    Commented Feb 15, 2022 at 2:42
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    $\begingroup$ Schemes are categories fibered in setoids . . . over the category of schemes (or affine schemes, if you want something “smaller”). $\endgroup$
    – Jason Starr
    Commented Feb 15, 2022 at 3:06
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    $\begingroup$ It doesn't matter. The two approaches define the same category up to equivalence if you are scrupulous about size issues. The functor of points approach is based on sheaves of sets on a site; presheaves of sets can equally well be viewed as discrete fibrations; and discrete fibrations are up to equivalence the same as fibrations whose fibres are setoids. This is what Martin's comment is about. If I may be frank: you should bite the bullet and pick some concrete definition and learn it. There is no compromise-free beautiful high-concept definition. $\endgroup$
    – Zhen Lin
    Commented Feb 15, 2022 at 3:37
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    $\begingroup$ I find your opening paragraphs very peculiar. It seems like you’re approaching this assuming that “a rigorously formalized account of the infinitesimal which almost directly permits one to formalize their intuition about infinitesimal geometric concepts and then reason about these concepts precisely in an intuitively satisfying manner” must a priori mean something built from pure category-theory, rather than topology or commutative algebra. Category theory gives a fantastic toolbox and philosophy, but it’s not the be-all and end-all of maths! $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Feb 15, 2022 at 9:56
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    $\begingroup$ @AlecRhea Well, take it from someone who lived underwater in the ocean of category theory and wrote a thesis on how to construct the category of schemes: it cannot be done. People who claim there is an elegant definition are cheats: either they throw away some class of schemes they say are not worth caring about, or they allow some things that are not really schemes but they do want to think about, or both at once. My thesis would be only a dozen pages long if it could really be done as easily as people say. $\endgroup$
    – Zhen Lin
    Commented Feb 15, 2022 at 10:47

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This is essentially the functor of points approach to schemes.

Define the site AffSch of affine schemes as the opposite category of commutative rings, equipped with the Zariski Grothendieck topology. Concretely, the poset (locale) of opens of the Zariski spectrum of a commutative ring R can be identified with the poset of radical ideals of R; open covers are given by collections of radical ideals in R that generate an ideal whose radical equals R.

Consider the category S of sheaves of sets on the site AffSch. Various objects considered in algebraic geometry, such as schemes and algebraic spaces, form full subcategories of S. Concretely, schemes can be characterized as objects of S that admit an atlas, i.e., a family of open immersions from affine schemes such that the induced map from their coproduct is an epimorphism. The nLab has another characterization of schemes, see Definition 2.4 there.

The 2-categories of thin groupoids and sets (the latter with identity 2-morphisms) are equivalent. Thus, sheaves of sets on AffSch (equivalently, categories fibered in sets over AffSch) can be replaced with the equivalent 2-category of categories fibered in thin groupoids over AffSch.

The cited answer by Martin Brandenburg gives an example when such an adjustment is (marginally) useful, since it allows us not to take isomorphism classes for certain constructions.

A lot of books cover the functor of points approach in some way, including Vakil's modern exposition. Among the more classical sources one can point to Demazure and Gabriel's Introduction to Algebraic Geometry and Algebraic Groups (North-Holland, 1980).

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  • $\begingroup$ Thank you for the references, I’ll give them a look. $\endgroup$
    – Alec Rhea
    Commented Feb 15, 2022 at 5:20
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    $\begingroup$ @Dmitri I think you are sweeping some subtleties under the rug with your definition of atlas. How do you exclude algebraic spaces that are not schemes? $\endgroup$
    – Zhen Lin
    Commented Feb 15, 2022 at 8:25
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    $\begingroup$ I don't know whether my understanding is correct. Let $\mathscr F$ be a sheaf on the big Zariski site of affine schemes. Then $\mathscr F$ is representable by a scheme if and only if the natural map $\operatorname{colim}_{h_A\to\mathscr F}h_A\to\mathscr F$ of Zariski sheaves is an isomorphism, where $h_A$ is the functor $\operatorname{Hom}(A,\cdot)$ represented by the ring $A$, and the colimit is taken along all open immersions. If we replace open immersions by étale maps representable by schemes, we get algebraic spaces? $\endgroup$
    – Z. M
    Commented Feb 15, 2022 at 19:19
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    $\begingroup$ I "learned" this from Scholze's Lecture notes on Analytic Geometry Definition 13.5, which seems a bit neater since it does not directly involve any special choice. In some sense, this index category of open immersions $h_A\to\mathscr F$ is an analogue of maximal atlas in differential geometry. $\endgroup$
    – Z. M
    Commented Feb 15, 2022 at 19:24
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    $\begingroup$ @DmitriPavlov That's better. Your original post did not mention the condition that the individual pieces of the atlas are open immersions, which is crucial if you want to define schemes but not algebraic spaces. $\endgroup$
    – Zhen Lin
    Commented Feb 15, 2022 at 22:15

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