The Stacks Project


Tag 08NG

7.22. Cocontinuous functors which have a left adjoint

It may happen that a cocontinuous functor $u$ has a left adjoint $w$.

Lemma 7.22.1. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $g : \mathop{\mathit{Sh}}\nolimits(\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits(\mathcal{D})$ be the morphism of topoi associated to a continuous and cocontinuous functor $u : \mathcal{C} \to \mathcal{D}$, see Lemmas 7.20.1 and 7.20.5.

  1. If $w : \mathcal{D} \to \mathcal{C}$ is a left adjoint to $u$, then
    1. $g_!\mathcal{F}$ is the sheaf associated to the presheaf $w^p\mathcal{F}$, and
    2. $g_!$ is exact.
  2. if $w$ is a continuous left adjoint, then $g_!$ has a left adjoint.
  3. If $w$ is a cocontinuous left adjoint, then $g_! = h^{-1}$ and $g^{-1} = h_*$ where $h : \mathop{\mathit{Sh}}\nolimits(\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits(\mathcal{C})$ is the morphism of topoi associated to $w$.

Proof. Recall that $g_!\mathcal{F}$ is the sheafification of $u_p\mathcal{F}$. Hence (1)(a) follows from the fact that $u_p = w^p$ by Lemma 7.18.3.

To see (1)(b) note that $g_!$ commutes with all colimits as $g_!$ is a left adjoint (Categories, Lemma 4.24.5). Let $i \mapsto \mathcal{F}_i$ be a finite diagram in $\mathop{\mathit{Sh}}\nolimits(\mathcal{C})$. Then $\mathop{\mathrm{lim}}\nolimits \mathcal{F}_i$ is computed in the category of presheaves (Lemma 7.10.1). Since $w^p$ is a right adjoint (Lemma 7.5.4) we see that $w^p \mathop{\mathrm{lim}}\nolimits \mathcal{F}_i = \mathop{\mathrm{lim}}\nolimits w^p\mathcal{F}_i$. Since sheafification is exact (Lemma 7.10.14) we conclude by (1)(a).

Assume $w$ is continuous. Then $g_! = (w^p )^\# = w^s$ but sheafification isn't necessary and one has the left adjoint $w_s$, see Lemmas 7.13.2 and 7.13.3.

Assume $w$ is cocontinuous. The equality $g_! = h^{-1}$ follows from (1)(a) and the definitions. The equality $g^{-1} = h_*$ follows from the equality $g_! = h^{-1}$ and uniqueness of adjoint functor. Alternatively one can deduce it from Lemma 7.21.1. $\square$

    The code snippet corresponding to this tag is a part of the file sites.tex and is located in lines 4398–4463 (see updates for more information).

    \section{Cocontinuous functors which have a left adjoint}
    \label{section-cocontinuous-left-adjoint}
    
    \noindent
    It may happen that a cocontinuous functor $u$ has a left adjoint $w$.
    
    \begin{lemma}
    \label{lemma-have-left-adjoint}
    Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let
    $g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be
    the morphism of topoi associated to a continuous and cocontinuous functor
    $u : \mathcal{C} \to \mathcal{D}$, see
    Lemmas \ref{lemma-cocontinuous-morphism-topoi} and
    \ref{lemma-when-shriek}.
    \begin{enumerate}
    \item If $w : \mathcal{D} \to \mathcal{C}$ is a left adjoint to $u$, then
    \begin{enumerate}
    \item $g_!\mathcal{F}$ is the sheaf associated to the presheaf
    $w^p\mathcal{F}$, and
    \item $g_!$ is exact.
    \end{enumerate}
    \item if $w$ is a continuous left adjoint, then $g_!$
    has a left adjoint.
    \item If $w$ is a cocontinuous left adjoint, then $g_! = h^{-1}$ and
    $g^{-1} = h_*$ where $h : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})$ is
    the morphism of topoi associated to $w$.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Recall that $g_!\mathcal{F}$ is the sheafification of $u_p\mathcal{F}$.
    Hence (1)(a) follows from the fact that $u_p = w^p$ by
    Lemma \ref{lemma-adjoint-functors}.
    
    \medskip\noindent
    To see (1)(b) note that $g_!$ commutes with all colimits as $g_!$
    is a left adjoint (Categories, Lemma \ref{categories-lemma-adjoint-exact}).
    Let $i \mapsto \mathcal{F}_i$ be a finite diagram in $\Sh(\mathcal{C})$.
    Then $\lim \mathcal{F}_i$ is computed in the category of presheaves
    (Lemma \ref{lemma-limit-sheaf}). Since $w^p$ is a right adjoint
    (Lemma \ref{lemma-adjoints-u})
    we see that $w^p \lim \mathcal{F}_i = \lim w^p\mathcal{F}_i$. Since
    sheafification is exact
    (Lemma \ref{lemma-sheafification-exact})
    we conclude by (1)(a).
    
    \medskip\noindent
    Assume $w$ is continuous. Then $g_! = (w^p\ )^\# = w^s$ but sheafification
    isn't necessary and one has the left adjoint $w_s$, see
    Lemmas \ref{lemma-pushforward-sheaf} and \ref{lemma-adjoint-sheaves}.
    
    \medskip\noindent
    Assume $w$ is cocontinuous. The equality $g_! = h^{-1}$ follows from (1)(a)
    and the definitions. The equality $g^{-1} = h_*$ follows from the equality
    $g_! = h^{-1}$ and uniqueness of adjoint functor. Alternatively one can deduce
    it from Lemma \ref{lemma-have-functor-other-way}.
    \end{proof}

    Comments (0)

    There are no comments yet for this tag.

    Add a comment on tag 08NG

    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 lower-right corner).

    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 box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?