Writeup to Exercise 2.3.C
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chapter2.tex
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chapter2.tex
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%! TeX root: ./main.tex
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\begin{Lemma} \label{lem:subdiag-colim}
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Let $A\colon I \to \cC$ be a diagram in a category $\cC$,
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and let $B\colon J \to \cC$ be another diagram factoring through $A$.
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\[
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\begin{tikzcd}
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I \ar[r, "A"] & \cC \\
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J \ar[u, "\alpha"] \ar[ru, "B"']
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\end{tikzcd}
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\]
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Then the factoring $\alpha$ induces a map on the colimits, if they exist.
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\[
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\colim_{j \in J} B(j) \longrightarrow \colim_{i \in I} A(i)
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\]
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\end{Lemma}
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\begin{proof}
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Consider any co-cone $Z \in C$ over $A$ in $\cC$.
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We can write such a co-cone as a natural transformation
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$\eta$ from $A$ to the composition $I \to * \to \cC$,
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where $*$ is the discrete single object category,
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$* \to \cC$ selects the object $Z$,
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and $I \to *$ of course takes all objects to the single object.
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\[
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\begin{tikzcd}
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* \ar[rd, "Z"] \\
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I \ar[u] \ar[r, "A"'] & \cC
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\end{tikzcd}
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\]
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Note that the components of such a transformation are $\func{\eta_i}{A(i)}{Z}$,
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satisfying, for each $i \to j$ in $I$,
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\[
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\begin{tikzcd}
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A(i) \ar[r, "\eta_i"] \ar[d] & Z \ar[d, equals] \\
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A(j) \ar[r, "\eta_j"] & Z
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\end{tikzcd}
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\]
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This is precisely the definition of a co-cone.
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However, we can then take the composition $\nu = \eta * id_{\alpha}$,
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which naturally describes $Z$ as a co-cone over $B$.
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\[
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\begin{tikzcd}
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& * \ar[rd, "Z"] \\
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J \ar[r, "\alpha"'] \ar[ru] & I \ar[u] \ar[r, "A"'] & \cC
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\end{tikzcd}
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\]
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Thus, if $Z$ is any co-cone over $A$,
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and the colimit over $B$ exists,
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then there is a unique map from said colimit to $Z$ by universality.
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Taking $Z$ to be the co-cone over $A$ proves the lemma.
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\end{proof}
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\begin{Lemma} \label{lem:colim-composition}
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Let $A, B: I \to \cC$ be two diagrams in a category $\cC$,
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and let $\eta \colon A \implies B$ be a natural transformation between them.
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Then $\eta$ induces a map on the colimits, if they exist.
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\[
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\colim_{i \in I} B(i) \longrightarrow \colim_{i \in I} A(i)
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\]
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\end{Lemma}
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\begin{proof}
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Let $Z$ be any co-cone over the diagram $B$,
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written as a natural transformation as in the previous lemma.
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\[
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\begin{tikzcd}[row sep=large]
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& * \ar[rd, "Z"] \\
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I \ar[ru]
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\ar[rr, bend left, "A"{anchor=center, fill=white, name=A}]
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\ar[rr, bend right, "B"'{name=B}]
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\ar[from=B.north-|A, shorten=4pt, to=A, Rightarrow, "\eta"]
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\ar[from=A, ur, Rightarrow, shorten=2pt, "\varepsilon"]
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& & C\\
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\end{tikzcd}
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\]
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The composition $\varepsilon \circ \eta$ then describes $Z$
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as a co-cone over the diagram $B$.
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If we choose $Z$ to be the colimit over $A$,
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and assume the colimit of $B$ also exists,
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then we get the map desired by the lemma.
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\end{proof}
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\begin{qanda}
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\question
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@@ -375,4 +453,201 @@
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then the pushforward of $\cF$ over any continuous map $X \to Y$
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is also a sheaf on $Y$, as desired.
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% Question 2.2.I
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\question
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Let $\func{\pi}{X}{Y}$ be a continuous map of spaces,
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and $\cF$ be a sheaf on $X$.
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if $\pi(p) = q$ for some $p \in X$,
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describe the natural morphism $(\pi_*\cF)_q \to \cF_p$.
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\answer
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We first define the map explicitly.
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Take some germ $[f]_q$ in $(\pi_*\cF)_q$,
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with $f \in (\pi_*\cF)(V)$ a section over some open $V \ni q$.
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Then by the definition of the sections in the pushforward sheaf,
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$f \in (\pi_*\cF)(V) = \cF(\pi^{-1}(V))$,
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thus $f$ is also an element of the set of sections of $\cF$ over the open nbd
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$\pi^{-1}(V)$ of $p$. Thus, we can define
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$[f]_p \in \cF_p$, the germ of $f$ over $p$.
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The induced map on stalks may then be expressed simply as $[f]_q \mapsto [f]_p$.
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Alternatively, we can induce the map using the universal property of the stalks.
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\[
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\begin{tikzcd}
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\cTop_{Y}^{op}|_{q} \ar[r, "\pi^{-1}_q"] \ar[d] & \cTop_{X}^{op}|_{p} \ar[d] \\
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\cTop_{Y}^{op} \ar[r, "\pi^{-1}"] & \cTop_{X}^{op} \ar[r, "\cF"] & \Set
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\end{tikzcd}
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\]
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Here $\cTop_Y^{op}|_q$ denotes the full subcategory of $\cTop_Y^{op}$
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consisting of open sets containing $q$, and analagously for $\cTop_X^{op}|_p$.
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The stalk $\cF_p$ is then the colimit over the diagram
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$\cTop^{op}_X|_p \to \cTop^{op}_X \to \Set$,
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and the stalk $(\pi_*\cF)_q$ is the colimit over the diagram
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$\cTop_Y^{op}|_q \to \cTop_Y^{op} \to \cTop_X^{op} \to \Set$.
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Lemma \ref{lem:subdiag-colim} then induces the desired map by universal property of colimit.
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% Question 2.2.J
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\question
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If $(X, \cO_X)$ is a ringed space,
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and $\cF$ is an $\cO_X$-module, describe how for each $p \in X$,
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$\cF_p$ is an $\cO_{X, p}$-module.
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\answer
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We shall describe the action explicitly.
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Let $p \in X$, and consider $\cF_p$.
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We already know this retains the structure of an abelian group,
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since the category of abelian groups has colimits.
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To define the action of $\cO_{X, p}$,
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let $[f]_p \in \cO_{X, p}$ and $[a]_p \in \cF_p$
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where $f$ and $a$ are sections of their respective sheaves
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over some shared neighbourhood $U \ni p$ (restricting if necessary).
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Then define $[f]_p \cdot [a]_p$ to be $[f\cdot a]_p$
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using the action of $\cO_X(U)$ on $\cF(U)$.
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To see that this is well defined, let $[f]_p = [f']_p$ and $[a]_p = [a']_p$
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where $f', a'$ are defined over another neighbourhood $V$ of $p$.
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Since they determine the same germ, there are some $W_1, W_2$ around $p$
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where $f|_{W_1} = f'|_{W_2}$ and $a|_{W_1} = a'_{W_2}$.
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Defining $W = W_1 \cap W_2$, we can finally put everything into the same neighbourhood,
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so that $f|_W = f'|_W$ and $a|_W = a'|_W$.
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Then the compatibility condition for the $\cO_X$-module $\cF$ says that
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$(f \cdot a)|_W = f|_W \cdot a|_W = f'_W \cdot a'|_W = (f' \cdot a')|_W$.
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Thus $[f \cdot a]_p = [f'\cdot a']_p$, and this definition of action on stalks
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is well defined.
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Finally, note that all the axioms of module action may be determined with
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a finite number of elements, and we may thus check them
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by taking representatives on some shared neighbourhood of $p$,
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where the axioms hold by the definition of an $\cO_X$-module.
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% Question 2.3.A
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\question
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Let $\phi\colon \cF \to \cG$ be a morphism of pre-sheaves on $X$,
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and let $p \in X$.
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Then describe the induced morphism of stalks $\phi_p\colon \cF_p \to \cG_p$.
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This defines the \emph{stalkification functor at $p$}
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$\Sets_X \to \Sets$, where $\Sets_X$ is the category of
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sheaves of sets over $X$.
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\answer
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This follows immediately from Lemma \ref{lem:colim-composition},
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since $\phi$ induces a natural transformation between the stalk diagrams
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for $\cF_p$ and $\cG_p$.
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% Question 2.3.B
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\question
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Let $\pi\colon X \to Y$ be a continuous map of topological spaces.
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Show that the pushforward induces a functor
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$\pi_*\colon \Sets_X \to \Sets_Y$.
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\answer
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Recall that $\pi_*\cF$ is pre-composition with $\pi^{-1}_{op}$.
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Then we can describe the desired functor as
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\[
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\begin{tikzcd}
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\cF \ar[d, Rightarrow, "\eta"] \ar[r, mapsto] &
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\pi_*\cF = \cF \circ \pi^{-1}_{op} \ar[d, Rightarrow, "\eta \circ id_{\pi^{-1}_{op}}"] \\
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\cG \ar[r, mapsto] & \pi_*\cG = \cG \circ \pi^{-1}_{op}
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\end{tikzcd}
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\]
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That is, on objects it acts as pre-composition by $\pi^{-1}_{op}$,
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and on morphisms it acts as pre-composition by (the natural transformation) $id_{\pi^{-1}_{op}}$
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% Question 2.3.C
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\question
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Let $\cF$ and $\cG$ be two sheaves over $X$.
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Define a pre-sheaf of sets $\cHom(\cF, \cG)$
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by lettings the set of sections over each open $U$ be
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\[
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\cHom(\cF, \cG)(U) \ceq \Mor(\cF|_U, \cG|_U).
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\]
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Show that this is a sheaf of set on $X$.
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\answer
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We first observe that for any $U$ open in $X$,
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the sections of $\cHom(\cF, \cG)$ over $U$ are defined to be the set
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of natural transformations $\eta$ in the diagram below.
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The restriction of a section $\eta$ to some open $V \subseteq U$
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is then a composition of $\eta$ with the (identity transformation on the) inclusion
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of $\cTop_V^{op} \to \cTop_U^{op}$
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\[
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\begin{tikzcd}
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& & \cTop_X^{op} \ar[rd, "\cF"] & \\
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\cTop_V^{op} \ar[r] & \cTop_U^{op} \ar[ru] \ar[rd] & & \Set \\
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& & \cTop_X^{op} \ar[ru, "\cG"'] \ar[uu, Rightarrow, shorten=5pt, "\eta"] &
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\end{tikzcd}
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\]
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Pre-composition is functorial,
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and so $\cHom(\cF, \cG)$ does form a presheaf of sets.
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Next, we need to show that it is also a sheaf.
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We first consider identity, in the usual way.
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Let $U$ be open in $X$, and take an open cover $U = \bigcup_{i \in I} U_i$.
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Let $\eta, \varepsilon\colon \cF|_U \Rightarrow \cG_U$,
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such that $\eta|_{U_i} = \varepsilon|_{U_i}$ for all $i \in I$.
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We claim that $\eta = \varepsilon$.
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To see this, take any open $V \subseteq U$,
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and define the cover $V_i = V \cap U_i$ of $V$. Then:
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\[
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\eta_{V_i} = (\eta|_{U_i})_{V_i} = (\varepsilon|_{U_i})_{V_i} = \varepsilon_{V_i}.
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\]
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We have the following diagram.
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\[
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\begin{tikzcd}
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\cF|_U(V) \ar[r, "\eta_V"] \ar[d] & \cG|_U(V) \ar[d] \\
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\cF|_U(V_i) \ar[r, "\eta_{V_i}", "\varepsilon_{V_i}"'] & \cG|_U(V_i) \\
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\cF|_U(V) \ar[u] \ar[r, "\varepsilon_{V}"] & \cG|_U(V) \ar[u]
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\end{tikzcd}
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\]
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Here the vertical maps are restriction maps, and both squares commute.
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Chasing any section $f \in \cF|_U(V)$, we see that
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\[
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(\eta_V(f))|_{V_i} = \eta_{V_i}(f|_{V_i}) = \varepsilon_{V_i}(f|_{V_i}) = (\varepsilon_V(f))|_{V_i}.
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\]
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Since $V_i$ forms a cover of $V$, identity in the sheaf $\cG$ then tells us $\eta_V = \varepsilon_V$.
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Since $V$ was arbitrary, this means that $\eta = \varepsilon$ as desired.
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Finally, we demonstrate gluing.
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Let $U_i$ form a cover of $U$ as above,
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and let $\eta^i\colon \cF|_{U_i} \Rightarrow \cG|_{U_i}$ in $\cHom(\cF, \cG)(U_i)$,
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such that they agree when restricted to intersections, that is,
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\[
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\forall W \subseteq U_i \cap U_j, \qquad
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\eta^i_W = \big(\eta^i|_{U_i \cap U_j}\big)_W = \big(\eta^j|_{U_i \cap U_j}\big)_W
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= \eta^j_W.
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\]
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Take any $V$ open in $U$, and as before define the cover $V_i = V \cap U_i$ of $V$.
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We define a gluing $\eta\colon \cF|_U \Rightarrow \cG|_U$
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by defining it on the arbitrary component $V$.
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\[
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\begin{tikzcd}
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\cF(V) \ar[r, dashed, "\eta_V"] \ar[d] & \cG(V) \ar[d] \\
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\cF(V_i) \ar[r, "\eta^i_{V_i}"] \ar[d] & \cG(V_i) \ar[d] \\
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\cF(V_i \cap V_j) \ar[r, "\eta^i_{V_i \cap V_j}", "\eta^j_{V_i \cap V_j}"'] & \cG(V_i \cap V_j)
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\end{tikzcd}
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\]
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Taking any section $f \in \cF(V) = \cF|_{U}(V)$,
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define $g_i = \eta^i_{V_i}(f|_{V_i}) \in \cG(V_i)$.
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Then
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\[
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g_i|_{V_i \cap V_j} =
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\eta^i_{V_i}(f|_{V_i}) |_{V_i \cap V_j}
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= \eta^i_{V_i \cap V_j}(f|_{V_i \cap V_j})
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= \eta^j_{V_i \cap V_j}(f|_{V_i \cap V_j})
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= \eta^j_{V_j}(f|_{V_j}) | _{V_i \cap V_j}
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= g_j |_{V_i \cap V_j}.
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\]
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Thus $\{g_i\}_{i\in I}$ is a family of sections of $\cG$ over $V_i$
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which agree upon restriction to intersections.
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Since $\cG$ was assumed a sheaf, it satisfies gluing,
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and there is thus some $g \in \cG(V)$ such that $g|_{V_i} = g_i$.
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We define $\eta_V(f) = g$.
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Thus, $\cHom(\cF, \cG)$ is a presheaf satisfying both identity and gluing, and is thus a sheaf,
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as desired.
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\end{qanda}
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