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Conclusion (Added 3/18/14): Even though $\mathbf{Sch}$ is not small-concretizable, it is concretizable more generally. However the construction of a faithful functor to $\mathbf{Set}$ is convoluted simple if you use Zhen Lin's approach!

The answers to this question jumped to using a modified notion of concreteness. But there are classic results about concretizability in the original sense of this question -- admitting a faithful functor to $\mathbf{Set}$. The most famous, of course is Freyd's result that the homotopy category of spaces is not concrete.

But Freyd wrote another paper on concreteness. In it he shows, (Thm 4.1.iii), that a finitely-complete category is concretizable iff it is regularly-well-powered. That is, iff every object has a small set of isomorphism classes of regular subobjects, where a regular subobject means an equalizer of two morphisms.

I know nothing about schemes, but I'll go out on a limb and assume that category is finitely complete and regularly-well-powered. Therefore it is concretizable.

EDIT (3/18/14): After discussing this issue with Zhen below, I decided it was worth its own MO question. It turns out that $\mathbf{Sch}$ is regularly wellpowered. A simple argument given by Martin Brandenberg in this question statement shows that regular monos are locally closed immersions, and Laurent Moret-Bailly showed, by a factorization into a closed and an open immersion, that the there can only be a small set of the latter into a given scheme.

(The following is superseded by Zhen's answer:) So there exists a faithful functor $\mathbf{Sch} \to \mathbf{Set}$. The obvious question is: what does it look like? Is it perhaps a certain \emph{large} colimit of representables? Unfortunately, Freyd's construction is very artificial -- it involves embedding $\mathbf{Sch}$ into an abelian category and then using a brute-force transfinite induction. Anyway, I suspect it might give some insight into the category of schemes to try to construct a more natural faithful functor to $\mathbf{Set}$.

Conclusion (Added 3/18/14): Even though $\mathbf{Sch}$ is not small-concretizable, it is concretizable more generally. However the construction of a faithful functor to $\mathbf{Set}$ is convoluted simple if you use Zhen Lin's approach!

The answers to this question jumped to using a modified notion of concreteness. But there are classic results about concretizability in the original sense of this question -- admitting a faithful functor to $\mathbf{Set}$. The most famous, of course is Freyd's result that the homotopy category of spaces is not concrete.

But Freyd wrote another paper on concreteness. In it he shows, (Thm 4.1.iii), that a finitely-complete category is concretizable iff it is regularly-well-powered. That is, iff every object has a small set of isomorphism classes of regular subobjects, where a regular subobject means an equalizer of two morphisms.

I know nothing about schemes, but I'll go out on a limb and assume that category is finitely complete and regularly-well-powered. Therefore it is concretizable.

EDIT (3/18/14): After discussing this issue with Zhen below, I decided it was worth its own MO question. It turns out that $\mathbf{Sch}$ is regularly wellpowered. A simple argument given by Martin Brandenberg in this question statement shows that regular monos are locally closed immersions, and Laurent Moret-Bailly showed, by a factorization into a closed and an open immersion, that the there can only be a small set of the latter into a given scheme.

(The following is superseded by Zhen's answer:) So there exists a faithful functor $\mathbf{Sch} \to \mathbf{Set}$. The obvious question is: what does it look like? Is it perhaps a certain \emph{large} colimit of representables? Unfortunately, Freyd's construction is very artificial -- it involves embedding $\mathbf{Sch}$ into an abelian category and then using a brute-force transfinite induction. Anyway, I suspect it might give some insight into the category of schemes to try to construct a more natural faithful functor to $\mathbf{Set}$.

Conclusion (Added 3/18/14): Even though $\mathbf{Sch}$ is not small-concretizable, it is concretizable more generally. However the construction of a faithful functor to $\mathbf{Set}$ is convoluted.

The answers to this question jumped to using a modified notion of concreteness. But there are classic results about concretizability in the original sense of this question -- admitting a faithful functor to $\mathbf{Set}$. The most famous, of course is Freyd's result that the homotopy category of spaces is not concrete.

But Freyd wrote another paper on concreteness. In it he shows, (Thm 4.1.iii), that a finitely-complete category is concretizable iff it is regularly-well-powered. That is, iff every object has a small set of isomorphism classes of regular subobjects, where a regular subobject means an equalizer of two morphisms.

I know nothing about schemes, but I'll go out on a limb and assume that category is finitely complete and regularly-well-powered. Therefore it is concretizable.

EDIT (3/18/14): After discussing this issue with Zhen below, I decided it was worth its own MO question. It turns out that $\mathbf{Sch}$ is regularly wellpowered. A simple argument given by Martin Brandenberg in this question statement shows that regular monos are locally closed immersions, and Laurent Moret-Bailly showed, by a factorization into a closed and an open immersion, that the there can only be a small set of the latter into a given scheme.

So there exists a faithful functor $\mathbf{Sch} \to \mathbf{Set}$. The obvious question is: what does it look like? Is it perhaps a certain \emph{large} colimit of representables? Unfortunately, Freyd's construction is very artificial -- it involves embedding $\mathbf{Sch}$ into an abelian category and then using a brute-force transfinite induction. Anyway, I suspect it might give some insight into the category of schemes to try to construct a more natural faithful functor to $\mathbf{Set}$.

Conclusion (Added 3/18/14): Even though $\mathbf{Sch}$ is not small-concretizable, it is concretizable more generally. However the construction of a faithful functor to $\mathbf{Set}$ is convoluted simple if you use Zhen Lin's approach!

The answers to this question jumped to using a modified notion of concreteness. But there are classic results about concretizability in the original sense of this question -- admitting a faithful functor to $\mathbf{Set}$. The most famous, of course is Freyd's result that the homotopy category of spaces is not concrete.

But Freyd wrote another paper on concreteness. In it he shows, (Thm 4.1.iii), that a finitely-complete category is concretizable iff it is regularly-well-powered. That is, iff every object has a small set of isomorphism classes of regular subobjects, where a regular subobject means an equalizer of two morphisms.

I know nothing about schemes, but I'll go out on a limb and assume that category is finitely complete and regularly-well-powered. Therefore it is concretizable.

EDIT (3/18/14): After discussing this issue with Zhen below, I decided it was worth its own MO question. It turns out that $\mathbf{Sch}$ is regularly wellpowered. A simple argument given by Martin Brandenberg in this question statement shows that regular monos are locally closed immersions, and Laurent Moret-Bailly showed, by a factorization into a closed and an open immersion, that the there can only be a small set of the latter into a given scheme.

(The following is superseded by Zhen's answer:) So there exists a faithful functor $\mathbf{Sch} \to \mathbf{Set}$. The obvious question is: what does it look like? Is it perhaps a certain \emph{large} colimit of representables? Unfortunately, Freyd's construction is very artificial -- it involves embedding $\mathbf{Sch}$ into an abelian category and then using a brute-force transfinite induction. Anyway, I suspect it might give some insight into the category of schemes to try to construct a more natural faithful functor to $\mathbf{Set}$.

The answers to this question jumped to using a modified notion of concreteness. But there are classic results about concretizability in the original sense of this question -- admitting a faithful functor to $\mathbf{Set}$. The most famous, of course is Freyd's result that the homotopy category of spaces is not concrete.

But Freyd wrote another paper on concreteness. In it he shows, (Thm 4.1.iii), that a finitely-complete category is concretizable iff it is regularly-well-powered. That is, iff every object has a small set of isomorphism classes of regular subobjects, where a regular subobject means an equalizer of two morphisms.

I know nothing about schemes, but I'll go out on a limb and assume that category is finitely complete and regularly-well-powered. Therefore it is concretizable.

Conclusion (Added 3/18/14): Even though $\mathbf{Sch}$ is not small-concretizable, it is concretizable more generally. However the construction of a faithful functor to $\mathbf{Set}$ is convoluted.

The answers to this question jumped to using a modified notion of concreteness. But there are classic results about concretizability in the original sense of this question -- admitting a faithful functor to $\mathbf{Set}$. The most famous, of course is Freyd's result that the homotopy category of spaces is not concrete.

But Freyd wrote another paper on concreteness. In it he shows, (Thm 4.1.iii), that a finitely-complete category is concretizable iff it is regularly-well-powered. That is, iff every object has a small set of isomorphism classes of regular subobjects, where a regular subobject means an equalizer of two morphisms.

I know nothing about schemes, but I'll go out on a limb and assume that category is finitely complete and regularly-well-powered. Therefore it is concretizable.

EDIT (3/18/14): After discussing this issue with Zhen below, I decided it was worth its own MO question. It turns out that $\mathbf{Sch}$ is regularly wellpowered. A simple argument given by Martin Brandenberg in this question statement shows that regular monos are locally closed immersions, and Laurent Moret-Bailly showed, by a factorization into a closed and an open immersion, that the there can only be a small set of the latter into a given scheme.

So there exists a faithful functor $\mathbf{Sch} \to \mathbf{Set}$. The obvious question is: what does it look like? Is it perhaps a certain \emph{large} colimit of representables? Unfortunately, Freyd's construction is very artificial -- it involves embedding $\mathbf{Sch}$ into an abelian category and then using a brute-force transfinite induction. Anyway, I suspect it might give some insight into the category of schemes to try to construct a more natural faithful functor to $\mathbf{Set}$.