Density theorem

Remarks

Proof idea: colimits in $\widehat{\mathcal{C}}$ are computed objectwise in $\mathbf{Set}$. At an object $d$, the colimit of the sets $\Hom(d, c)$ over $(c, x) \in \int P$ is a quotient of the set of pairs $(g \colon d \to c, x \in P(c))$, and each pair is identified with $(\id_d, P(g)(x))$. So the colimit at $d$ is $P(d)$, and the identifications are compatible with the maps of $P$.

Example: if $P = y(a)$ is itself representable, then $\int P$ is the slice category over $a$, which has the terminal object $(a, \id_a)$, and the colimit is just $y(a)$.

In short: every presheaf is a colimit of representable presheaves. Smallness of $\mathcal{C}$ makes $\int P$ small, so the colimit is a small colimit.

Going deeper

For a student

The theorem is used to extend a functor $F \colon \mathcal{C} \to \mathcal{E}$ to all presheaves when $\mathcal{E}$ has colimits: send $P$ to the colimit of $F \circ \pi$. This extension is the left Kan extension of $F$ along $y$, and it is left adjoint to the functor $E \mapsto \Hom_{\mathcal{E}}(F(-), E)$. Geometric realisation of simplicial sets is built this way, from the standard simplices. The theorem also shows that $\widehat{\mathcal{C}}$ is the free cocompletion of $\mathcal{C}$.