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It is well-known that any presheaf (for simplicity say we're talking about presheaves of sets on a topological space $X$) is a colimit of representable presheaves in a canonical way. This has been brought up over at MathOverflow a couple of times, e.g. here, but the answers and references tend to give a very abstract version of this claim: one considers the Yoneda embedding and forms the so-called comma category over the given presheaf, and so on.

I have reason to believe (namely, a smart person who I do not currently have access to told me) that one can write down a fairly concrete diagram of which the given presheaf $F$ is the colimit: if we write $h_U$ for the presheaf represented by an open set $U \subset X$, I think it looks something like \begin{equation} \coprod_? h_V \rightrightarrows \coprod_? h_U \to F. \end{equation} Perhaps the disjoint union on the right is $\coprod_{U \subset X} \coprod_{F(U)} h_U$: in any case, there is a natural surjection from this presheaf to $F$. Does anyone know what I'm talking about or have a reference?

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up vote 5 down vote accepted

I found what I was looking for in the Stacks Project. Write $F_1 = \coprod_{U \subset X} \coprod_{F(U)} h_U$, so there is a natural surjection $F_1 \to F$. Set $G = F_1 \times_F F_1$, so again we can construct $G_1 \to G$. There are two natural maps $G_1 \rightrightarrows F_1$, and $F$ is the coequalizer.

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