In his paper

*Basically bounded functors and flat sheaves*(Pacific Math. J, vol. 57, no. 2, 1975, p. 597-610), William C. Waterhouse gives a nice example of a presheaf that has no associated sheaf. This is Theorem 5.5 (page 605). I thank François Loeser for having indicated this paper to me, and for his suggestion of explaining it here!
Of course, such a beast is reputed not to exist, since it is well known that

*any*presheaf has an associated sheaf, see for example Godement's book Topologie algébrique et théorie des faisceaux, pages 110-111.
That is, for any presheaf $F$ on a topological space, there is a sheaf $G$ with a morphism of presheaves $\alpha\colon F\to G$ which satisfies a universal property: any morphism from $F$ to a sheaf factors uniquely through $\alpha$.

Waterhouse's presheaf is a more sophisticated example of a presheaf, since it is

*a presheaf on the category of affine schemes for the flat topology*. Thus, a*presheaf*$F$ on the category of affine schemes is the datum,- of a set $F(A)$ for every ring $A$,
- and of a map $\phi_*\colon F(A)\to F(B)$ for every morphism of rings $\phi\colon A\to B$,

subject to the following conditions:

- if $\phi\colon A\to B$ and $\psi\colon B\to C$ are morphism of rings, then $(\psi\circ\phi)_*=\psi_*\circ\phi_*$;
- one has ${\rm id}_A)_*={\rm id}_{F(A)}$ for every ring $A$.

Any morphism of rings $\phi\colon A\to B$ gives rise to two morphisms $\psi_1,\psi_2\colon B\to B\otimes B$ respectively defined by $\psi_1(b)=b\otimes 1$ and $\psi_2(b)=1\otimes b$, and the two compositions $A\to B\to B\otimes_A B$ are equal. Consequently, for any presheaf $F$, the two associated maps $F(A) \to F(B) \to F(B\otimes_A B)$ are equal.

By definition, a presheaf $F$ is a

*sheaf*for the flat topology if for any faithfully flat morphism of rings, the map ${\phi_*} \colon F(A)\to F(B)$ is injective and its image is the set of elements $g\in F(B)$ at which the two natural maps $(\psi_1)_*$ and $(\psi_2)_*$ from $F(B)$ to $F(B\otimes_A B)$ coincide.
Here is Waterhouse's example.

For every ring $A$, let $F(A)$ be the set of all locally constant functions $f$ from $\mathop{\rm Spec}(A)$ to some von Neumann cardinal such that $f(\mathfrak p)<\mathop{\rm Card}(\kappa(\mathfrak p))$ for every $\mathfrak p\in\mathop{\rm Spec}(A)$.

*This is a presheaf.*Indeed, let $\phi\colon A\to B$ is a ring morphism, let $\phi^a\colon\mathop{\rm Spec}(B)\to \mathop{\rm Spec}(A)$ be the associated continuous map on spectra. For $f\in F(A)$, then $f\circ\phi^a$ is a locally constant map from ${\rm Spec}(B)$ to some von Neumann cardinal. Moreover, for every prime ideal $\mathfrak q$ in $B$, with inverse image $\mathfrak p=\phi^{-1}(\mathfrak q)=\phi^a(\mathfrak q)$, the morphism $\phi$ induces an injection from the residue field $\kappa(\mathfrak q)$ into $\kappa(\mathfrak p)$, so that $f\circ\phi^a$ satisfies the additional condition on $F$, hence $f\circ\phi^a\in F(B)$.

*However, this presheaf has no associated sheaf for the flat topology.*The proof is by contradiction. So assume that $G$ is a sheaf and $\alpha\colon F\to G$ satisfies the universal property.

First of all, we prove that the morphism

*$\alpha$ is injective:*for any ring $A$, the map $\alpha_A\colon F(A)\to G(A)$ is injective. For any cardinal $c$ and any ring $A$, let $L_c(A)$ be the set of locally constant maps from ${\rm Spec}(A)$ to $c$. Then $L_c$ is a presheaf, and in fact a sheaf. There is a natural morphism of presheaves $\beta_c\colon F\to L_c$, given by $\beta_c(f)(\mathfrak p)=f(\mathfrak p)$ if $f(\mathfrak p)\in c$, that is, $f(\mathfrak p)<c$, and $\beta_c(f)(\mathfrak p)=0$ otherwise. Consequently, there is a unique morphism of sheaves $\gamma_c\colon G\to L_c$ such that $\beta_c=\gamma_c\circ\alpha$. For any ring $A$, and any large enough cardinal $c$, the map $\beta_c(A)\colon F(A)\to L_c(A)$ is injective. In particular, the map $\alpha(A)$ must be injective.
Let $B$ be a ring and $\phi\colon A\to B$ be a faithfully flat morphism. Let $\psi_1,\psi_2\colon B\to B\otimes_A B$ be the two natural morphisms of rings defined above. Then, the equalizer $E(A,B)$ of the two maps $(\psi_1)_*$ and $(\psi_2)_*$ from $F(B)$ to $F(B\otimes_A B)$ must inject into the equalizer of the two corresponding maps from $G(B)$ to $G(B\otimes_A B)$. Consequently, one has an injection from $E(A,B)$ to $G(A)$.

The contradiction will become apparent once one can find rings $B$ for which $E(A,B)$ has a cardinality as large as desired. If ${\rm Spec}(B)$ is a point $\mathfrak p$, then $F(B)$ is just the set of functions $f$ from the point $\mathfrak p$ to some von Neumann cardinal $c$ such that $f(\mathfrak p)<{\rm Card}(\kappa(\mathfrak p))$. That is, $F(B)$ is the cardinal ${\rm Card}(\kappa(\mathfrak p))$ itself. And since ${\rm Spec}(B)$ is a point, the coincidence condition is necessarily satisfied, so that $E(A,B)= {\rm Card}(\kappa(\mathfrak p))\leq G(A)$.

To conclude, it suffices to take a faithfully flat morphism $A\to B$ such that $B$ is field of cardinality strictly greater than $G(A)$. For example, one can take $A$ to be a field and $B$ the field of rational functions in many indeterminates (strictly more than the cardinality of $G(A)$).

*What does this example show? Why isn't there a contradiction in mathematics (yet)?*

Because the definition of sheaves and presheaves for the flat topology that I gave above was definitely defective: it neglects in a too dramatic way the set theoretical issues that one must tackle to define sheaves on categories. In the standard setting of set theory provided by ZFC, everything is a set. In particular, categories, presheaves, etc. are sets or maps between sets (themselves represented by sets). But the presheaf $F$ that Waterhouse defines does not exist as a set, since there does not exist a set $\mathbf{Ring}$ of all rings, nor a set $\mathbf{card}$ of all von Neumann cardinals.

The usual way (as explained in SGA 4) to introduce sheaves for the flat topology consists in adding the axiom of universes — there exists a set $\mathscr U$ which is a model of set theory. Then, one does not consider the (inexistent) set of all rings, or cardinals, but only those belonging to the universe $\mathscr U$—one talks of $\mathscr U$-categories, $\mathscr U$-(pre)sheaves, etc.. In that framework, the $\mathscr U$-presheaf $F$ defined by Waterhouse (where one restricts oneself to algebras and von Neumann cardinals in $\mathscr U$) has an associated sheaf $G_{\mathscr U}$. But this sheaf depends on the chosen universe: if $\mathscr V$ is an universe containing $\mathscr U$, the restriction of $G_{\mathscr V}$ to algebras in $\mathscr U$ will no longer be a $\mathscr U$-presheaf.

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