Add output.

chapter2.tex

@@ -1,5 +1,83 @@
 %! TeX root: ./main.tex
 
+\begin{Lemma} \label{lem:subdiag-colim}
+  Let $A\colon I \to \cC$ be a diagram in a category $\cC$,
+  and let $B\colon J \to \cC$ be another diagram factoring through $A$.
+  \[
+    \begin{tikzcd}
+      I \ar[r, "A"] & \cC \\
+      J \ar[u, "\alpha"] \ar[ru, "B"'] 
+    \end{tikzcd}
+  \]
+  Then the factoring $\alpha$ induces a map on the colimits, if they exist.
+  \[
+    \colim_{j \in J} B(j) \longrightarrow \colim_{i \in I} A(i) 
+  \]
+\end{Lemma}
+\begin{proof}
+  Consider any co-cone $Z \in C$ over $A$ in $\cC$.
+  We can write such a co-cone as a natural transformation
+  $\eta$ from $A$ to the composition $I \to * \to \cC$,
+  where $*$ is the discrete single object category,
+  $* \to \cC$ selects the object $Z$,
+  and $I \to *$ of course takes all objects to the single object.
+  \[
+    \begin{tikzcd}
+      * \ar[rd, "Z"] \\
+      I \ar[u] \ar[r, "A"'] & \cC
+    \end{tikzcd}
+  \]
+  Note that the components of such a transformation are $\func{\eta_i}{A(i)}{Z}$,
+  satisfying, for each $i \to j$ in $I$,
+  \[
+    \begin{tikzcd}
+      A(i) \ar[r, "\eta_i"] \ar[d] & Z \ar[d, equals] \\
+      A(j) \ar[r, "\eta_j"] & Z
+    \end{tikzcd}
+  \]
+  This is precisely the definition of a co-cone.
+  However, we can then take the composition $\nu = \eta * id_{\alpha}$,
+  which naturally describes $Z$ as a co-cone over $B$.
+  \[
+    \begin{tikzcd}
+      & * \ar[rd, "Z"] \\
+      J \ar[r, "\alpha"'] \ar[ru] & I \ar[u] \ar[r, "A"'] & \cC
+    \end{tikzcd}
+  \]
+  Thus, if $Z$ is any co-cone over $A$,
+  and the colimit over $B$ exists,
+  then there is a unique map from said colimit to $Z$ by universality.
+
+  Taking $Z$ to be the co-cone over $A$ proves the lemma.
+\end{proof}
+\begin{Lemma} \label{lem:colim-composition}
+  Let $A, B: I \to \cC$ be two diagrams in a category $\cC$,
+  and let $\eta \colon A \implies B$ be a natural transformation between them.
+  Then $\eta$ induces a map on the colimits, if they exist.
+  \[
+    \colim_{i \in I} B(i) \longrightarrow \colim_{i \in I} A(i) 
+  \]
+\end{Lemma}
+\begin{proof}
+  Let $Z$ be any co-cone over the diagram $B$,
+  written as a natural transformation as in the previous lemma.
+  \[
+    \begin{tikzcd}[row sep=large]
+      & * \ar[rd, "Z"] \\
+      I \ar[ru]
+      \ar[rr, bend left, "A"{anchor=center, fill=white, name=A}]
+      \ar[rr, bend right, "B"'{name=B}]
+      \ar[from=B.north-|A, shorten=4pt, to=A, Rightarrow, "\eta"]
+      \ar[from=A, ur, Rightarrow, shorten=2pt, "\varepsilon"]
+      & & C\\
+    \end{tikzcd}
+  \]
+  The composition $\varepsilon \circ \eta$ then describes $Z$
+  as a co-cone over the diagram $B$.
+  If we choose $Z$ to be the colimit over $A$,
+  and assume the colimit of $B$ also exists,
+  then we get the map desired by the lemma.
+\end{proof}
 
 \begin{qanda}
   \question
@@ -375,4 +453,201 @@
     then the pushforward of $\cF$ over any continuous map $X \to Y$
     is also a sheaf on $Y$, as desired.
 
+  % Question 2.2.I
+  \question
+    Let $\func{\pi}{X}{Y}$ be a continuous map of spaces,
+    and $\cF$ be a sheaf on $X$.
+    if $\pi(p) = q$ for some $p \in X$,
+    describe the natural morphism $(\pi_*\cF)_q \to \cF_p$.
+  \answer
+    We first define the map explicitly.
+    Take some germ $[f]_q$ in $(\pi_*\cF)_q$,
+    with $f \in (\pi_*\cF)(V)$ a section over some open $V \ni q$.
+    Then by the definition of the sections in the pushforward sheaf,
+    $f \in (\pi_*\cF)(V) = \cF(\pi^{-1}(V))$,
+    thus $f$ is also an element of the set of sections of $\cF$ over the open nbd
+    $\pi^{-1}(V)$ of $p$. Thus, we can define
+    $[f]_p \in \cF_p$, the germ of $f$ over $p$.
+    The induced map on stalks may then be expressed simply as $[f]_q \mapsto [f]_p$.
+
+    Alternatively, we can induce the map using the universal property of the stalks.
+    \[
+      \begin{tikzcd}
+        \cTop_{Y}^{op}|_{q} \ar[r, "\pi^{-1}_q"] \ar[d] & \cTop_{X}^{op}|_{p} \ar[d] \\
+        \cTop_{Y}^{op} \ar[r, "\pi^{-1}"] & \cTop_{X}^{op} \ar[r, "\cF"] & \Set
+      \end{tikzcd}
+    \]
+    Here $\cTop_Y^{op}|_q$ denotes the full subcategory of $\cTop_Y^{op}$
+    consisting of open sets containing $q$, and analagously for $\cTop_X^{op}|_p$.
+
+    The stalk $\cF_p$ is then the colimit over the diagram
+    $\cTop^{op}_X|_p \to \cTop^{op}_X \to \Set$,
+    and the stalk $(\pi_*\cF)_q$ is the colimit over the diagram
+    $\cTop_Y^{op}|_q \to \cTop_Y^{op} \to \cTop_X^{op} \to \Set$.
+
+    Lemma \ref{lem:subdiag-colim} then induces the desired map by universal property of colimit.
+
+  % Question 2.2.J
+  \question
+    If $(X, \cO_X)$ is a ringed space,
+    and $\cF$ is an $\cO_X$-module, describe how for each $p \in X$,
+    $\cF_p$ is an $\cO_{X, p}$-module.
+  \answer
+    We shall describe the action explicitly.
+    Let $p \in X$, and consider $\cF_p$.
+    We already know this retains the structure of an abelian group,
+    since the category of abelian groups has colimits.
+    To define the action of $\cO_{X, p}$,
+    let $[f]_p \in \cO_{X, p}$ and $[a]_p \in \cF_p$
+    where $f$ and $a$ are sections of their respective sheaves
+    over some shared neighbourhood $U \ni p$ (restricting if necessary).
+
+    Then define $[f]_p \cdot [a]_p$ to be $[f\cdot a]_p$
+    using the action of $\cO_X(U)$ on $\cF(U)$.
+    To see that this is well defined, let $[f]_p = [f']_p$ and $[a]_p = [a']_p$
+    where $f', a'$ are defined over another neighbourhood $V$ of $p$.
+
+    Since they determine the same germ, there are some $W_1, W_2$ around $p$
+    where $f|_{W_1} = f'|_{W_2}$ and $a|_{W_1} = a'_{W_2}$.
+    Defining $W = W_1 \cap W_2$, we can finally put everything into the same neighbourhood,
+    so that $f|_W = f'|_W$ and $a|_W = a'|_W$.
+
+    Then the compatibility condition for the $\cO_X$-module $\cF$ says that
+    $(f \cdot a)|_W = f|_W \cdot a|_W = f'_W \cdot a'|_W = (f' \cdot a')|_W$.
+    Thus $[f \cdot a]_p = [f'\cdot a']_p$, and this definition of action on stalks
+    is well defined.
+
+    Finally, note that all the axioms of module action may be determined with 
+    a finite number of elements, and we may thus check them
+    by taking representatives on some shared neighbourhood of $p$,
+    where the axioms hold by the definition of an $\cO_X$-module.
+
+  % Question 2.3.A
+  \question
+    Let $\phi\colon \cF \to \cG$ be a morphism of pre-sheaves on $X$,
+    and let $p \in X$.
+    Then describe the induced morphism of stalks $\phi_p\colon \cF_p \to \cG_p$.
+
+    This defines the \emph{stalkification functor at $p$}
+    $\Sets_X \to \Sets$, where $\Sets_X$ is the category of
+    sheaves of sets over $X$.
+  \answer
+    This follows immediately from Lemma \ref{lem:colim-composition},
+    since $\phi$ induces a natural transformation between the stalk diagrams
+    for $\cF_p$ and $\cG_p$.
+
+  % Question 2.3.B
+  \question
+    Let $\pi\colon X \to Y$ be a continuous map of topological spaces.
+    Show that the pushforward induces a functor
+    $\pi_*\colon \Sets_X \to \Sets_Y$.
+  \answer
+  Recall that $\pi_*\cF$ is pre-composition with $\pi^{-1}_{op}$.
+  Then we can describe the desired functor as
+  \[
+    \begin{tikzcd}
+      \cF \ar[d, Rightarrow, "\eta"] \ar[r, mapsto] &
+      \pi_*\cF = \cF \circ \pi^{-1}_{op} \ar[d, Rightarrow, "\eta \circ id_{\pi^{-1}_{op}}"] \\
+      \cG \ar[r, mapsto] & \pi_*\cG = \cG \circ \pi^{-1}_{op}
+    \end{tikzcd}
+  \]
+  That is, on objects it acts as pre-composition by $\pi^{-1}_{op}$,
+  and on morphisms it acts as pre-composition by (the natural transformation) $id_{\pi^{-1}_{op}}$
+
+  % Question 2.3.C
+  \question
+    Let $\cF$ and $\cG$ be two sheaves over $X$.
+    Define a pre-sheaf of sets $\cHom(\cF, \cG)$
+    by lettings the set of sections over each open $U$ be
+    \[
+      \cHom(\cF, \cG)(U) \ceq \Mor(\cF|_U, \cG|_U).
+    \]
+    Show that this is a sheaf of set on $X$.
+  \answer
+    We first observe that for any $U$ open in $X$,
+    the sections of $\cHom(\cF, \cG)$ over $U$ are defined to be the set 
+    of natural transformations $\eta$ in the diagram below.
+    The restriction of a section $\eta$ to some open $V \subseteq U$
+    is then a composition of $\eta$ with the (identity transformation on the) inclusion
+    of $\cTop_V^{op} \to \cTop_U^{op}$
+    \[
+      \begin{tikzcd}
+        & & \cTop_X^{op} \ar[rd, "\cF"] & \\
+        \cTop_V^{op} \ar[r] & \cTop_U^{op} \ar[ru] \ar[rd] & & \Set \\
+        &  & \cTop_X^{op} \ar[ru, "\cG"'] \ar[uu, Rightarrow, shorten=5pt, "\eta"] & 
+      \end{tikzcd}
+    \]
+    Pre-composition is functorial,
+    and so $\cHom(\cF, \cG)$ does form a presheaf of sets.
+
+    Next, we need to show that it is also a sheaf.
+    We first consider identity, in the usual way.
+    Let $U$ be open in $X$, and take an open cover $U = \bigcup_{i \in I} U_i$.
+    Let $\eta, \varepsilon\colon \cF|_U \Rightarrow \cG_U$,
+    such that $\eta|_{U_i} = \varepsilon|_{U_i}$ for all $i \in I$.
+    We claim that $\eta = \varepsilon$.
+    To see this, take any open $V \subseteq U$,
+    and define the cover $V_i = V \cap U_i$ of $V$. Then:
+    \[
+      \eta_{V_i} = (\eta|_{U_i})_{V_i} = (\varepsilon|_{U_i})_{V_i} = \varepsilon_{V_i}.
+    \]
+    We have the following diagram.
+    \[
+      \begin{tikzcd}
+        \cF|_U(V) \ar[r, "\eta_V"] \ar[d] & \cG|_U(V) \ar[d] \\
+        \cF|_U(V_i) \ar[r, "\eta_{V_i}", "\varepsilon_{V_i}"'] & \cG|_U(V_i) \\
+        \cF|_U(V) \ar[u] \ar[r, "\varepsilon_{V}"] & \cG|_U(V) \ar[u]
+      \end{tikzcd}
+    \]
+    Here the vertical maps are restriction maps, and both squares commute.
+    Chasing any section $f \in \cF|_U(V)$, we see that
+    \[
+      (\eta_V(f))|_{V_i} = \eta_{V_i}(f|_{V_i}) = \varepsilon_{V_i}(f|_{V_i}) = (\varepsilon_V(f))|_{V_i}.
+    \]
+    Since $V_i$ forms a cover of $V$, identity in the sheaf $\cG$ then tells us $\eta_V = \varepsilon_V$.
+    Since $V$ was arbitrary, this means that $\eta = \varepsilon$ as desired.
+
+    Finally, we demonstrate gluing.
+    Let $U_i$ form a cover of $U$ as above,
+    and let $\eta^i\colon \cF|_{U_i} \Rightarrow \cG|_{U_i}$ in $\cHom(\cF, \cG)(U_i)$,
+    such that they agree when restricted to intersections, that is,
+    \[
+      \forall W \subseteq U_i \cap U_j, \qquad 
+      \eta^i_W = \big(\eta^i|_{U_i \cap U_j}\big)_W = \big(\eta^j|_{U_i \cap U_j}\big)_W
+      = \eta^j_W.
+    \]
+    Take any $V$ open in $U$, and as before define the cover $V_i = V \cap U_i$ of $V$.
+    We define a gluing $\eta\colon \cF|_U \Rightarrow \cG|_U$
+    by defining it on the arbitrary component $V$.
+    \[
+      \begin{tikzcd}
+        \cF(V) \ar[r, dashed, "\eta_V"] \ar[d] & \cG(V) \ar[d] \\
+        \cF(V_i) \ar[r, "\eta^i_{V_i}"] \ar[d] & \cG(V_i) \ar[d] \\
+        \cF(V_i \cap V_j) \ar[r, "\eta^i_{V_i \cap V_j}", "\eta^j_{V_i \cap V_j}"'] & \cG(V_i \cap V_j)
+      \end{tikzcd}
+    \]
+    Taking any section $f \in \cF(V) = \cF|_{U}(V)$,
+    define $g_i = \eta^i_{V_i}(f|_{V_i}) \in \cG(V_i)$.
+    Then
+    \[
+      g_i|_{V_i \cap V_j} = 
+      \eta^i_{V_i}(f|_{V_i}) |_{V_i \cap V_j} 
+      = \eta^i_{V_i \cap V_j}(f|_{V_i \cap V_j}) 
+      = \eta^j_{V_i \cap V_j}(f|_{V_i \cap V_j})
+      = \eta^j_{V_j}(f|_{V_j}) | _{V_i \cap V_j}
+      = g_j |_{V_i \cap V_j}.
+    \]
+    Thus $\{g_i\}_{i\in I}$ is a family of sections of $\cG$ over $V_i$
+    which agree upon restriction to intersections.
+    Since $\cG$ was assumed a sheaf, it satisfies gluing,
+    and there is thus some $g \in \cG(V)$ such that $g|_{V_i} = g_i$.
+
+    We define $\eta_V(f) = g$.
+
+    Thus, $\cHom(\cF, \cG)$ is a presheaf satisfying both identity and gluing, and is thus a sheaf,
+    as desired.
+
+
+
+
 \end{qanda}

main.tex

@@ -9,8 +9,11 @@
 \let\Im\relax
 \DeclareMathOperator{\Im}{Im}
 \DeclareMathOperator{\Mor}{Mor}
+\DeclareMathOperator*{\colim}{colim}
 
 \DeclareMathOperator{\cTop}{\mathcal{T}op}
+\def\Sets{\Set}
+\DeclareMathOperator{\cHom}{\mathcal{H}om}
 
 %\rawtitle{ }
 %\rawauthor{ }