Editing restriction and extension of sheaves in nLab

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See the <a href="/nlab/show/HowTo#diagrams">HowTo</a>.</li> <li style="font-size: 0.8em">As an alternative to the Markdown syntax for sections (headings), one can use the usual LaTeX syntax \section, \subsection, etc.</li> <li style="font-size: 0.8em">For further help, see the <a href="/nlab/show/HowTo">HowTo</a>, or you are very welcome to ask at the <a href="https://nforum.ncatlab.org/">nForum</a>.</li> </ol> </div> <form accept-charset="utf-8" action="/nlab/save/restriction+and+extension+of+sheaves" id="editForm" method="post"> <div style="display: none;"> <input name="see_if_human" id="see_if_human" style="tabindex: -1; autocomplete: off"/> </div> <div> <textarea name="content" id="content" style="height: 45em; width: 70%;">#Contents# * table of contents {:toc} ## Idea Recall that for [[presheaf|presheaves]] on a [[site]] $X$ (with underlying category $S_X$) with values in a category $A$ that admits small [[limit]]s and small colimits (so in particular for $A =$ [[Set]]), $PSh(X, A) = [S_X^{op}, A]$, every functor $f^t : S_Y \to S_X$ induces three functors of presheaf catgeories: | Notation | | Definition | | ---------------------------------------- |-| ----------------------------- | | $(f^t)_* : PSh(X,A) \to PSh(Y,A)$ | | direct image | | $(f^t)^\dagger : PSh(Y,A) \to PSh(X,A)$ | | left adjoint to direct image | | $(f^t)^\ddagger : PSh(Y,A) \to PSh(X,A)$ | | right adjoint to direct image | Recall moreover that for $f : X \to Y$ any [[site|morphism of sites]], the left adjoint to [[direct image]] followed by [[sheafification]] $\bar{(-)}$ is the [[inverse image]] map of sheaves: $$ f^{-1} : Sh(Y,A) \to Sh(X,A) \,. $$ Now, if the morphism of sites $f$ happens to be restriction to a sub-site $f : X \to U$ with $U \in PSh(X,A)$ with $U$ carrying the induced [[Grothendieck topology|topology]], then * the direct image is called **restriction** of sheaves; * the right adjoint takes sheaves to sheaves and is called **extension** of sheaves. ## Definition Given a [[site]] $X$ with underlying [[category]] $S_X$ and given a [[presheaf]] $U \in PSh(X)$ with the induced sub-site $j_{U \to X} : X \to U$ corresponding to the [[stuff, structure, property|forgetful]] functor $j^t_{U \to X} : (Y_{S_X}/U) \to S_X$ from the [[comma category]] $S_U = (Y_{S_X}/U) \to S_X$ underlying the [[site]] $U$ (as discussed at [[site]]) the right [[adjoint functor]] $$ j^{\ddagger}_{U \to X} : PSh(U) \to PSh(X) $$ to the [[direct image]] or, in this case, **restriction** functor $$ (j_{U \to X})_* : Sh(X) \to Sh(U) $$ whose action may suggestively be denoted $$ (j_{U \to X})_* : F \mapsto F|_U $$ happens to take sheaves to sheaves (when $U$ is equipped with the canonical induced topology as described at [[site]]): one calls $$ j^{\ddagger}_{U \to X} : Sh(U) \to Sh(X) $$ the **extension** of sheaves on $U$ to sheaves on $X$. To summarize notation and terminology: | Terminology | | Notation | | Definition | | ------------------- |-| -------------------------------------------------------- |-| ------------------------------------------------------- | | morphism of sites | | $j_{U \to X} : X \to U $ | | | | underlying functor | | $j^t_{U \to X} : (Y_{S_X}/U) \to S_X $ | | | | sheaf restriction | | $(j_{U \to X})_* : Sh(X) \to Sh(U)$ | | direct image | | sheaf extension | | $j^{\ddagger}_{U \to X} : Sh(U) \to Sh(X)$ | | right adjoint to direct image | | sheaf inverse image | | $\overline{(j^t)^{\dagger}_{U \to X}} : Sh(U) \to Sh(X)$ | | left adjoint to direct image followed by sheafification | ## Remarks Notice the difference to the [[inverse image]] operation $$ j^{-1}_{U \to X} : Sh(U) \to Sh(X) \,. $$ ## References {#References} For instance section 17.6 of * Kashiwara, Schapira, _[[Categories and Sheaves]]_ * [[Stacks Project]], [Tag 00VC](http://stacks.math.columbia.edu/tag/00VC) and [Tag 00XF](http://stacks.math.columbia.edu/tag/00XF).</textarea> <p> <input id="alter_title" name="alter_title" onchange="toggleVisibility();" type="checkbox" value="1" /> <label for="alter_title">Change page name.</label><br/> <span id="title_change" style="display:none"><label for="new_name">New name:</label> <input id="new_name" name="new_name" onblur="addRedirect();" type="text" value="restriction and extension of sheaves" /></span> </p> <div> <p style="font-size: 0.8em; width: 70%;"> For non-trivial edits, please briefly describe your changes below. 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Editing restriction and extension of sheaves in nLab