# Étale sheaves as colimits of representable sheaves

If $X$ is a scheme, then by a representable étale sheaf one means the following: for a scheme $Y\to X$ over $X$, we may consider the presheaf of sets $$U \mapsto \operatorname{Hom}_X (U,Y),$$ and it is actually a sheaf on $X_\text{ét}$.

I've seen several times the claim that every sheaf on $X_\text{ét}$ is a colimit of representable sheaves. Could someone give a reference for that? (I guess it must be proved and used in SGA 4.)

Can we assume that the representable sheaves in question are represented by étale schemes $Y\to X$?

Thank you.

• The etale site is not special; the fact is true for any subcanonical site. In fact, it's even true for all sites, with the modification that the colimit is over the sheafification of representables. – Hurkyl Oct 30 '17 at 6:12

## 1 Answer

Every presheaf is a colimit of representable presheaves, which are in fact sheaves as the \'etale topology is subcanonical (that is, every representable functor is a sheaf).

Let $\mathscr{F}$ be any sheaf on $X_{et}.$ If $\iota : \mathsf{Sh}(X_{et})\to\mathsf{Psh}(X_{et})$ is the forgetful functor including sheaves on $X_{et}$ into presheaves on $X_{et},$ we may write by the above $\iota(\mathscr{F}) \cong \operatorname{colim}h_U.$ Now, recall that we have an adjoint pair $$(-)^{++}:\mathsf{Psh}(X_{et})\leftrightarrows\mathsf{Sh}(X_{et}): \iota$$ ($(-)^{++}$ denotes sheafification) and that left adjoints preserve colimits and right adjoints preserve limits. Thus, it follows that (because there is a natural isomorphism $(-)^{++}\circ\iota\to\operatorname{id}_{\mathsf{Sh}(X_{et})}$) $$\mathscr{F}\cong (\iota(\mathscr{F}))^{++}\cong(\operatorname{colim}h_U)^{++}\cong\operatorname{colim} (h_U)^{++}\cong\operatorname{colim} h_U.$$

• Of course! I'm not sure about your question about the sheaf being representable by \'etale schemes - this is by definition true in the small \'etale site, but the colimit of representables defining a presheaf in the big \'etale site will be over lots of objects, and I don't see an easy reason for it to be cofinal to a colimit only over \'etale objects. – Stahl Nov 1 '17 at 2:14
• Just a question: take a presheaf $P$ and write it as colimit of representable presheaves $P=\rm colim h_U$. Since the site is subcanonical, we have $P=\operatorname{colim} h_U=\operatorname{colim} h_U^{++}=(\operatorname{colim} h_U)^{++}$ so it would seem that we have proved that $P$ is a sheaf. Of course this cannot be true, so where am I mistaken? Thanks. – W. Rether Jan 2 at 16:34
• @w.rether: it seems the issue in your "proof" is that you implicitly switch the categories in which you're taking the colimits. $\operatorname{colim} h_U^{++}$ is a colimit of sheaves, and is performed in the category of sheaves. But $P$ is a presheaf, and the colimit $\operatorname{colim} h_U$ computing $P$ is performed in the presheaf category. If you try to write your chain of isomorphisms such that the end objects are always in the same category (as they should be) you'll need some forgetful functors/sheafifications to make everything coherent, and this should resolve the issue. – Stahl Jan 3 at 16:34