The Stacks project

7.19 More functoriality of presheaves

In this section we revisit the material of Section 7.5. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor between categories. Recall that

\[ u^ p : \textit{PSh}(\mathcal{D}) \longrightarrow \textit{PSh}(\mathcal{C}) \]

is the functor that associates to $\mathcal{G}$ on $\mathcal{D}$ the presheaf $u^ p\mathcal{G} = \mathcal{G} \circ u$. It turns out that this functor not only has a left adjoint (namely $u_ p$) but also a right adjoint.

Namely, for any $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ we define a category ${}_ V\mathcal{I} = {}_ V^ u\mathcal{I}$. Its objects are pairs $(U, \psi : u(U) \to V)$. Note that the arrow is in the opposite direction from the arrow we used in defining the category $\mathcal{I}_ V^ u$ in Section 7.5. A morphism $(U, \psi ) \to (U', \psi ')$ is given by a morphism $\alpha : U \to U'$ such that $\psi = \psi ' \circ u(\alpha )$. In addition, given any presheaf of sets $\mathcal{F}$ on $\mathcal{C}$ we introduce the functor ${}_ V\mathcal{F} : {}_ V\mathcal{I}^{opp} \to \textit{Sets}$, which is defined by the rule ${}_ V\mathcal{F}(U, \psi ) = \mathcal{F}(U)$. We define

\[ {}_ pu(\mathcal{F})(V) := \mathop{\mathrm{lim}}\nolimits _{{}_ V\mathcal{I}^{opp}} {}_ V\mathcal{F} \]

As a limit there are projection maps $c(\psi ) : {}_ pu(\mathcal{F})(V) \to \mathcal{F}(U)$ for every object $(U, \psi )$ of ${}_ V\mathcal{I}$. In fact,

\[ {}_ pu(\mathcal{F})(V) = \left\{ \begin{matrix} \text{collections } s_{(U, \psi )} \in \mathcal{F}(U) \\ \forall \beta : (U_1, \psi _1) \to (U_2, \psi _2) \text{ in }{}_ V\mathcal{I} \\ \text{ we have } \beta ^*s_{(U_2, \psi _2)} = s_{(U_1, \psi _1)} \end{matrix} \right\} \]

where the correspondence is given by $s \mapsto s_{(U, \psi )} = c(\psi )(s)$. We leave it to the reader to define the restriction mappings ${}_ pu(\mathcal{F})(V) \to {}_ pu(\mathcal{F})(V')$ associated to any morphism $V' \to V$ of $\mathcal{D}$. The resulting presheaf will be denoted ${}_ pu\mathcal{F}$.

Lemma 7.19.1. There is a canonical map ${}_ pu\mathcal{F}(u(U)) \to \mathcal{F}(U)$, which is compatible with restriction maps.

Proof. This is just the projection map $c(\text{id}_{u(U)})$ above. $\square$

Note that any map of presheaves $\mathcal{F} \to \mathcal{F}'$ gives rise to compatible systems of maps between functors ${}_ V\mathcal{F} \to {}_ V\mathcal{F}'$, and hence to a map of presheaves ${}_ pu\mathcal{F} \to {}_ pu\mathcal{F}'$. In other words, we have defined a functor

\[ {}_ pu : \textit{PSh}(\mathcal{C}) \longrightarrow \textit{PSh}(\mathcal{D}) \]

Lemma 7.19.2. The functor ${}_ pu$ is a right adjoint to the functor $u^ p$. In other words the formula

\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(u^ p\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{D})}(\mathcal{G}, {}_ pu\mathcal{F}) \]

holds bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.

Proof. This is proved in exactly the same way as the proof of Lemma 7.5.4. We note that the map $u^ p{}_ pu \mathcal{F} \to \mathcal{F}$ from Lemma 7.19.1 is the map that is used to go from the right to the left.

Alternately, think of a presheaf of sets $\mathcal{F}$ on $\mathcal{C}$ as a presheaf $\mathcal{F}'$ on $\mathcal{C}^{opp}$ with values in $\textit{Sets}^{opp}$, and similarly on $\mathcal{D}$. Check that $({}_ pu \mathcal{F})' = u_ p(\mathcal{F}')$, and that $(u^ p\mathcal{G})' = u^ p(\mathcal{G}')$. By Remark 7.5.5 we have the adjointness of $u_ p$ and $u^ p$ for presheaves with values in $\textit{Sets}^{opp}$. The result then follows formally from this. $\square$

Thus given a functor $u : \mathcal{C} \to \mathcal{D}$ of categories we obtain a sequence of functors

\[ u_ p, u^ p, {}_ pu \]

between categories of presheaves where in each consecutive pair the first is left adjoint to the second.

Lemma 7.19.3. Let $u : \mathcal{C} \to \mathcal{D}$ and $v : \mathcal{D} \to \mathcal{C}$ be functors of categories. Assume that $v$ is right adjoint to $u$. Then we have

  1. $u^ ph_ V = h_{v(V)}$ for any $V$ in $\mathcal{D}$,

  2. the category $\mathcal{I}^ v_ U$ has an initial object,

  3. the category ${}_ V^ u\mathcal{I}$ has a final object,

  4. ${}_ pu = v^ p$, and

  5. $u^ p = v_ p$.

Proof. Proof of (1). Let $V$ be an object of $\mathcal{D}$. We have $u^ ph_ V = h_{v(V)}$ because $u^ ph_ V(U) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {D}(u(U), V) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U, v(V))$ by assumption.

Proof of (2). Let $U$ be an object of $\mathcal{C}$. Let $\eta : U \to v(u(U))$ be the map adjoint to the map $\text{id} : u(U) \to u(U)$. Then we claim $(u(U), \eta )$ is an initial object of $\mathcal{I}_ U^ v$. Namely, given an object $(V, \phi : U \to v(V))$ of $\mathcal{I}_ U^ v$ the morphism $\phi $ is adjoint to a map $\psi : u(U) \to V$ which then defines a morphism $(u(U), \eta ) \to (V, \phi )$.

Proof of (3). Let $V$ be an object of $\mathcal{D}$. Let $\xi : u(v(V)) \to V$ be the map adjoint to the map $\text{id} : v(V) \to v(V)$. Then we claim $(v(V), \xi )$ is a final object of ${}_ V^ u\mathcal{I}$. Namely, given an object $(U, \psi : u(U) \to V)$ of ${}_ V^ u\mathcal{I}$ the morphism $\psi $ is adjoint to a map $\phi : U \to v(V)$ which then defines a morphism $(U, \psi ) \to (v(V), \xi )$.

Hence for any presheaf $\mathcal{F}$ on $\mathcal{C}$ we have

\begin{eqnarray*} v^ p\mathcal{F}(V) & = & \mathcal{F}(v(V)) \\ & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_{v(V)}, \mathcal{F}) \\ & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(u^ ph_ V, \mathcal{F}) \\ & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{D})}(h_ V, {}_ pu\mathcal{F}) \\ & = & {}_ pu\mathcal{F}(V) \end{eqnarray*}

which proves part (4). Part (5) follows by the uniqueness of adjoint functors. $\square$


Comments (4)

Comment #4065 by Frid on

In the last line of the proof of Lemma 7.19.3., it should say "part (4)" and "Part (5)" instead of "part (2)" and "Part (3)".

Comment #8826 by DIpankar Maity on

May be somewhere you can mention the following : Let be a left adjoint to (of functors of sites). Then is continuous iff is cocontinuous.

Comment #9264 by on

Thanks very much for this comment. With definitions as in the Stacks project, this "iff" is not true. When we say that a functor between sites is (co)continuous we are referring to the notions defined in Definitions 7.13.1 and 7.20.1; we do not apply our definitions to categories which do not have the structure of a site. Still, I decided to answer your question: in one direction the result is true, but in the other direction it is not. You can read this in these changes. Hopefully this will soon be online too (in this section).


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 00XF. Beware of the difference between the letter 'O' and the digit '0'.