The Stacks project

Lemma 6.21.3. Let $f : X \to Y$ be a continuous map. There exists a functor $f_ p : \textit{PSh}(Y) \to \textit{PSh}(X)$ which is left adjoint to $f_*$. For a presheaf $\mathcal{G}$ it is determined by the rule

\[ f_ p\mathcal{G}(U) = \mathop{\mathrm{colim}}\nolimits _{f(U) \subset V} \mathcal{G}(V) \]

where the colimit is over the collection of open neighbourhoods $V$ of $f(U)$ in $Y$. The colimits are over directed partially ordered sets. (The restriction mappings of $f_ p\mathcal{G}$ are explained in the proof.)

Proof. The colimit is over the partially ordered set consisting of open subsets $V \subset Y$ which contain $f(U)$ with ordering by reverse inclusion. This is a directed partially ordered set, since if $V, V'$ are in it then so is $V \cap V'$. Furthermore, if $U_1 \subset U_2$, then every open neighbourhood of $f(U_2)$ is an open neighbourhood of $f(U_1)$. Hence the system defining $f_ p\mathcal{G}(U_2)$ is a subsystem of the one defining $f_ p\mathcal{G}(U_1)$ and we obtain a restriction map (for example by applying the generalities in Categories, Lemma 4.14.8).

Note that the construction of the colimit is clearly functorial in $\mathcal{G}$, and similarly for the restriction mappings. Hence we have defined $f_ p$ as a functor.

A small useful remark is that there exists a canonical map $\mathcal{G}(U) \to f_ p\mathcal{G}(f^{-1}(U))$, because the system of open neighbourhoods of $f(f^{-1}(U))$ contains the element $U$. This is compatible with restriction mappings. In other words, there is a canonical map $i_\mathcal {G} : \mathcal{G} \to f_* f_ p \mathcal{G}$.

Let $\mathcal{F}$ be a presheaf of sets on $X$. Suppose that $\psi : f_ p\mathcal{G} \to \mathcal{F}$ is a map of presheaves of sets. The corresponding map $\mathcal{G} \to f_*\mathcal{F}$ is the map $f_*\psi \circ i_\mathcal {G} : \mathcal{G} \to f_* f_ p \mathcal{G} \to f_* \mathcal{F}$.

Another small useful remark is that there exists a canonical map $c_\mathcal {F} : f_ p f_* \mathcal{F} \to \mathcal{F}$. Namely, let $U \subset X$ open. For every open neighbourhood $V \supset f(U)$ in $Y$ there exists a map $f_*\mathcal{F}(V) = \mathcal{F}(f^{-1}(V))\to \mathcal{F}(U)$, namely the restriction map on $\mathcal{F}$. And this is compatible with the restriction mappings between values of $\mathcal{F}$ on $f^{-1}$ of varying opens containing $f(U)$. Thus we obtain a canonical map $f_ p f_* \mathcal{F}(U) \to \mathcal{F}(U)$. Another trivial verification shows that these maps are compatible with restriction maps and define a map $c_\mathcal {F}$ of presheaves of sets.

Suppose that $\varphi : \mathcal{G} \to f_*\mathcal{F}$ is a map of presheaves of sets. Consider $f_ p\varphi : f_ p \mathcal{G} \to f_ p f_* \mathcal{F}$. Postcomposing with $c_\mathcal {F}$ gives the desired map $c_\mathcal {F} \circ f_ p\varphi : f_ p\mathcal{G} \to \mathcal{F}$. We omit the verification that this construction is inverse to the construction in the other direction given above. $\square$

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