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Every definition I have seen of algebraic spaces starts with the category of schemes (usually of $S$-schemes), and defines algebraic spaces as certain sheaves on this category, with respect to étale or fppf topology (depending on the definition), which is "close to being representable by a scheme".

But I have heard several people defend the idea that algebraic spaces are arguably more natural than (or at least as natural as) schemes as the "basic objects" of algebraic geometry. That makes me wonder if one could skip the middleman, and directly give a workable definition of algebraic spaces with no reference to schemes, and retrieve schemes as a suitable subcategory as an afterthought.

The most logical way to do that would be to define algebraic spaces as certain presheaves on $\mathbf{CRing^{op}}$, similar to the functor-of-points definition of schemes (where schemes are Zariski sheaves on $\mathbf{CRing^{op}}$ which are locally representable). I guess the first question is whether an algebraic space is characterized by its functor of points restricted to affine schemes (ie to rings).

Captain Lama
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  • I think it is possible, keeping in mind that an algebraic space is some étale-locally sheaf (for scheme, it is Zariski-locally) but maybe the price to pay is that we have to restrict to affine ones. You may look at http://web.archive.org/web/20150525073107/https://perso.math.univ-toulouse.fr/btoen/videos-lecture-notes-etc/ – Alexey Do Apr 26 '24 at 19:36

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I believe this follows from an appropriate version of the Comparison Lemma from Mac Lane and Moerdijk (p. 590):

The Comparison Lemma. For a subcanonical site $(\mathbf{C}, J)$, let $\mathbf {A}$ be a full subcategory $\mathbf C$ for which every object of $\mathbf C$ has a cover by objects from $\mathbf A$ [each object has a sieve such that all arrows factor through objects in $\mathbf{A}$]. Define a topology $J'$ on $\mathbf A$ by specifying that a sieve $S$ on $A$ is a $J'$-cover of $A$ iff the sieve $(S)$ which it generates in $\mathbf C$ is a $J$-cover of $A$. Then the restriction functor induces an equivalence of categories $$\text{Sh}(\mathbf{C}, J) \sim \text{Sh}(\mathbf{A}, J').$$ (There are sharper versions of this comparison lemma, for which $J$ need not be subcanonical and $\mathbf A$ need not be full...

The version for a dense subsite on nLab assumes the subsite is small

https://ncatlab.org/nlab/show/dense+sub-site https://ncatlab.org/nlab/show/category+of+sheaves

Also, the notes of Toën define schemes and algebraic spaces via $\text{Sh}(\text{Aff})$.

https://ncatlab.org/nlab/files/toen-master-course.pdf#page22

Ben
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