Merge branch 'master' into output
This commit is contained in:
234
chapter2.tex
234
chapter2.tex
@@ -55,28 +55,31 @@
|
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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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\[
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\colim_{i \in I} B(i) \longrightarrow \colim_{i \in I} A(i)
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\colim_{i \in I} A(i) \longrightarrow \colim_{i \in I} B(i)
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\]
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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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written as a natural transformation $\varepsilon$ as in the previous lemma.
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\[
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||||
\begin{tikzcd}[row sep=large]
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& * \ar[rd, "Z"] \\
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I \ar[ru]
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\ar[rr, bend left, "A"{anchor=center, fill=white, name=A}]
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\ar[rr, bend right, "B"'{name=B}]
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\ar[from=B.north-|A, shorten=4pt, to=A, Rightarrow, "\eta"]
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\ar[from=A, ur, Rightarrow, shorten=2pt, "\varepsilon"]
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\ar[rr, bend left, "B"{anchor=center, fill=white, name=B}]
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\ar[rr, bend right, "A"'{name=A}]
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\ar[from=A.north-|B, shorten=4pt, to=B, Rightarrow, "\eta"]
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\ar[from=B, ur, Rightarrow, shorten=2pt, "\varepsilon"]
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& & C\\
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\end{tikzcd}
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||||
\]
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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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as a co-cone over the diagram $A$ as well.
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If the diagram $A$ has a colimit, then it must admit a map to
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this cocone $Z$ by universal property.
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Thus, if $Z$ is assumed to be the colimit of $B$,
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we have induced a map from the colimit of $A$ to the colimit of $B$
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as desired.
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\end{proof}
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\begin{qanda}
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@@ -647,7 +650,218 @@
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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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% Exercise 2.3.D
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\question
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\begin{enumerate}
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\item Let $\cF$ be a sheaf of sets on $X$.
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Show that $\cHom(\underline{\{p\}}, \cF) \isom \cF$,
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where $\underline{\{p\}}$ is the constant sheaf with values in $\{p\}$.
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\item Let $\cF$ be a sheaf of abelian groups on $X$.
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Show that $\cHom_{Ab_X}(\underline{\bZ}, \cF) \isom \cF$
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in the category of sheaves of abelian groups.
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\item Let $\cF$ be an $\cO_X$-module.
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Show that $\cHom_{Mod_{\cO_X}}(\cO_X, \cF) \isom \cF$
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in the category of $\cO_X$-modules.
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\end{enumerate}
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\answer
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\begin{enumerate}
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\item We first consider the sheaf $\constset{p}$.
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Take any $U$ open in $X$, then the sections $\constset{p}(U)$
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are the functions $f\colon U \to \{p\}$ such that $f^{-1}(p)$ is open.
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That is, $\constset{p}(U)$ is the single-element set $*$.
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The restrictions are then the trivial map $* \to *$.
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Now, consider $\cHom(\constset{p}, \cF)$.
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The sections of this sheaf hom over $U$ are the natural transformations
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$\constset{p}|_U \to \cF|_U$.
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However, such a natural transformation is precisely the choice of
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an element of $\cF(U)$.
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Note that although the natural transformation
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describes a map $* \to \cF(V)$ for every open set $V \subseteq U$,
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the natural transformation diagram ensures that
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the chosen section over $V$ is the restriction of the
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section over $U$, so the image of the $U$ component
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does fully determine the transformation.
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\[
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||||
\begin{tikzcd}
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* \ar[r] \ar[d] & \cF(U) \ar[d] \\
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* \ar[r] & \cF(V)
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||||
\end{tikzcd}
|
||||
\]
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This then defines a natural transformation $\cHom(\constset{p}, \cF) \Rightarrow \cF$,
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which is isomorphic on components and is thus a natural isomorphism
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as desired.
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\item
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Next we consider $\ul{\bZ}$, the constant sheaf on $\bZ$.
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For $U$ open in $X$, the sections over $U$ are functions
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$\func{f}{U}{\bZ}$ such that $f^{-1}(n)$ is open in $U$ for each $n \in \bZ$.
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|
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The group structure on $\ul{\bZ}(U)$ is given by pointwise addition,
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that is, $(f+g)(x) = f(x) + g(x) \in \bZ$.
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||||
This is well-defined since for any $n \in \bZ$,
|
||||
\[
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||||
(f+g)^{-1}(n) = \bigcup_{a+b = n} f^{-1}(a) \cap g^{-1}(b).
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||||
\]
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||||
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Unlike in the previous part, $\ul{\bZ}(U)$ now has more than a single element.
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However, we shall show that each $\eta \in \Hom(\ul{\bZ}|_U, \cF|_U)$
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is still fully determined by where a single function in $\ul{\bZ}(U)$ is sent.
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||||
For $n \in \bZ$, let $\ul{n}^V\colon V \to \bZ$ denote the function in $\ul{\bZ}(V)$
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sending every point to $n$. Let $\sigma = \eta_U(\ul{1}^U)$.
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Now, take any $f \in \ul{\bZ}(U)$.
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Then for each $n$, by definition, $U_n \ceq f^{-1}(n)$ is open in $U$.
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Restricting, we have the following two diagrams.
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||||
\[
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||||
\begin{tikzcd}
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||||
f \ar[r, mapsto] \ar[d, mapsto] & \eta_U(f) \ar[d, mapsto] \\
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||||
\ul{n}^{U_n} \ar[r, mapsto] & \eta_U(f)|_{U_n}
|
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\end{tikzcd}
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\qquad
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\begin{tikzcd}
|
||||
\ul{1}^U \ar[r, mapsto] \ar[d, mapsto] & \sigma \ar[d, mapsto] \\
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\ul{1}^{U_n} \ar[r, mapsto] & \sigma|_{U_n}
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\end{tikzcd}
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||||
\]
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||||
Since $\eta_{U_n}$ is a group homomorphism, we have
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\[
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\eta_U(f)|_{U_n} = \eta_{U_n}(\ul{n}^{U_n}) = n \cdot \eta_{U_n}(\ul{1}^{U_n})
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= n \cdot \sigma|_{U_n}
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||||
\]
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||||
Thus the restrictions of $\eta_U(f)$ to the cover $U_n$ of $U$
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||||
are fully determined by the values of $\sigma|_{U_n}$.
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||||
Since $\cF$ is a sheaf, we then have that $\eta_U(f)$ itself is fully determined by
|
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$\sigma$, and thus we have a correspondence between $\cHom(\ul{\bZ}|_U, \cF|_U)$
|
||||
and $\cF(U)$ as desired.
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||||
\end{enumerate}
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||||
|
||||
% Question 2.3.E
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||||
\question
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||||
If $\phi\colon \cF \to \cG$ is a morphism of presheaves,
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||||
the presheaf kernel $\ker_{pre}(\phi)$ is defined by
|
||||
$(\ker_{pre}\phi)(U) \ceq \ker \phi(U)$
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||||
|
||||
Show that the presheaf kernel defined in this way is a presheaf.
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||||
|
||||
% Question 2.3.F
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\question
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||||
Show that the presheaf cokernel satisfies the universal property of cokernel.
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||||
|
||||
% Question 2.3.G
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||||
\question
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||||
Show that $\cF \mapsto \cF(U)$
|
||||
is an exact functor $Ab^{pre}_X \to Ab$.
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||||
|
||||
% Question 2.3.H
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||||
\question
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||||
Show that a sequence of presheaves
|
||||
\[
|
||||
0 \to \cF_1 \to \cF_2 \to \dots \cdot \cF_n \to 0.
|
||||
\]
|
||||
is exact if and only if
|
||||
\[
|
||||
0 \to \cF_1(U) \to \cF_2(U) \to \dots \to \cF_n(U) \to 0.
|
||||
\]
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||||
is exact for all $U$.
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||||
|
||||
% Question 2.3.I
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||||
\question
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||||
Assume $\phi\colon \cF \to \cG$ is a morphism of sheaves.
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||||
Show $\ker_{pre}\phi$ is a sheaf,
|
||||
and satisfies the universal property of kernels in the category of sheaves.
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||||
|
||||
% Question 2.3.J
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||||
\question
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||||
Take $X$ to be $\cC$ with the standard topology,
|
||||
and let $\cO_X$ be the sheaf of holomorphic functions.
|
||||
Let $\cF$ be the presheaf of functions on $X$ admitting a holomorphic logarithm.
|
||||
Show that the follow sequence of preshaves is exact
|
||||
\[
|
||||
0 \to \ul{\bZ} \to \cO_X \to \cF \to 0.
|
||||
\]
|
||||
Here $\ul{\bZ} \to \cO_X$ is the natural inclusion, and $\cO_X \to \cF$
|
||||
is given by $f \mapsto \exp(2\pi i f)$.
|
||||
|
||||
Show that even though $\cF$ is a presheaf cokernel of a morphism of sheaves,
|
||||
$\cF$ is not a sheaf itself.
|
||||
|
||||
% Question 2.4.A
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||||
\question
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||||
Prove that a section of a sheaf of sets is determined by its germs, i.e.,
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||||
the natural map
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||||
\[
|
||||
\cF(U) \to \prod_{p\in U} \cF_p
|
||||
\]
|
||||
is injective.
|
||||
|
||||
% Question 2.4.B
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||||
\question
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||||
Call a family of germs over an open set $U$ \emph{compatible}
|
||||
if they are locally the germs of some cover of $U$.
|
||||
|
||||
Formally, say that $(s_p)_{p \in U}$ is a compatible family of germs
|
||||
if for each $p \in U$,
|
||||
there is some neighbourhood $U_p \ni p$
|
||||
where for some section $t \in \cF(U_p)$ over that neighbourhood,
|
||||
$t_q = s_q$ for all $q \in U_p$.
|
||||
|
||||
% Question 2.4.C
|
||||
\question
|
||||
If $\phi_1$ and $\phi_2$ are morphisms from a presheaf of sets $\cF$
|
||||
to a sheaf of sets $\cG$ that induce the same maps on each stalk,
|
||||
show that $\phi_1 = \phi_2$.
|
||||
|
||||
% Question 2.4.D
|
||||
\question
|
||||
Show that a morphism of sheaves of sets is an isomorphism
|
||||
if and only if it induces an isomorphism of all stalks.
|
||||
|
||||
% Question 2.4.E
|
||||
\question
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||||
\begin{enumerate}
|
||||
\item Show that $\cF(U) \to \prod_{p \in U} \cF_p$
|
||||
need not be injective if $\cF$ is not a sheaf.
|
||||
\item Show that morphisms are not determined by stalks
|
||||
for general presheaves.
|
||||
\item Show that isomorphisms are not determined by stalks
|
||||
for general presheaves.
|
||||
\end{enumerate}
|
||||
|
||||
% Question 2.4.F
|
||||
\question
|
||||
Show that sheafification is unique up to unique isomorphism,
|
||||
assuming it exists. Show that $\cF$ is a sheaf,
|
||||
then the sheafification is $id\colon \cF \to \cF$.
|
||||
|
||||
% Question 2.4.G
|
||||
\question
|
||||
Show that sheafification is a functor from presheaves on $X$ to sheaves on $X$.
|
||||
|
||||
% Question 2.4.H
|
||||
\question
|
||||
Show that $\cF^{sh}$ forms a sheaf.
|
||||
|
||||
% Question 2.4.I
|
||||
\question
|
||||
Describe a natural map of presheaves $sh\colon \cF \to \cF^{sh}$
|
||||
|
||||
% Question 2.4.K
|
||||
\question
|
||||
Show that the sheafification functor is left-adjoint
|
||||
to the forgetful functor from sheaves on $X$ to presheaves on $X$.
|
||||
|
||||
% Question 2.4.L
|
||||
\question
|
||||
Show $\cF \to \cF^{sh}$ induces an isomorphism of stalks.
|
||||
|
||||
% Question 2.4.M
|
||||
\question
|
||||
Suppose $\phi \colon \cF \to \cG$ is a morphism of sheaves of sets on $X$.
|
||||
Show that the following are equivalent.
|
||||
\begin{enumerate}
|
||||
\item $\phi$ is a monomorphism in the category of sheaves.
|
||||
\item $\phi$ is injective on the level of stalks, i.e.
|
||||
$\phi_p\colon \cF_p \to \cG_p$ injective
|
||||
\item $\phi$ is injective on the level open sets.
|
||||
\end{enumerate}
|
||||
\end{qanda}
|
||||
|
||||
306
fancycom.sty
Normal file
306
fancycom.sty
Normal file
@@ -0,0 +1,306 @@
|
||||
%LaTeX package providing some shorthands and quick commands, tailored to certain unspecified fields of maths.
|
||||
%Not checked for conflicts with other packages.
|
||||
%Mostly for personal use.
|
||||
%Will not always be backwards compatible, although every effort will be made so that this is so.
|
||||
%Written by Adam Walsh
|
||||
%Version 1.0 (will not upgrade version numbers for minor command editions)
|
||||
|
||||
\ProvidesPackage{fancycom}[2018/02/03 v1.0 Fancy maths common commands package]
|
||||
|
||||
\RequirePackage{amsfonts}
|
||||
\RequirePackage{amsthm}
|
||||
\RequirePackage{amsmath}
|
||||
\RequirePackage{amssymb}
|
||||
|
||||
\RequirePackage{mathtools}
|
||||
|
||||
\RequirePackage{calc}
|
||||
\RequirePackage{textcomp}
|
||||
\RequirePackage[nointegrals]{wasysym}
|
||||
\RequirePackage{xparse}
|
||||
|
||||
%\RequirePackage{mathrsfs}
|
||||
\RequirePackage[mathscr]{euscript}
|
||||
|
||||
\RequirePackage{tikz-cd}
|
||||
|
||||
%Textual shortcuts/aids
|
||||
\newcommand{\mbb}[1]{\mathbb{#1}} %Emblacken your symbols!
|
||||
\newcommand{\mbf}[1]{\mathbf{#1}} %Embolden your symbols!
|
||||
\newcommand{\mfk}[1]{\mathfrak{#1}} %Enfraken your symbols!
|
||||
\newcommand{\bsl}{\backslash} %Backslash
|
||||
\newcommand{\cndall}{\,\,\,\forall} %forall given as condition
|
||||
\newcommand{\st}{\,\mid\,} %Such that symbol. : is also used.
|
||||
\newcommand{\what}[1]{& \qquad &\begin{sloppy}\textit{\small\mbox{#1}\small}\end{sloppy}} %Little comment for an align block. Memento of 2007.
|
||||
|
||||
%Basic symbol extensions
|
||||
\def\bA{\mathbb{A}}
|
||||
\def\bF{\mathbb{F}}
|
||||
\def\bN{\mathbb{N}}
|
||||
\def\bR{\mathbb{R}}
|
||||
\def\bZ{\mathbb{Z}}
|
||||
\def\bQ{\mathbb{Q}}
|
||||
\def\bC{\mathbb{C}}
|
||||
\def\bO{\mathbb{O}}
|
||||
\def\bP{\mathbb{P}}
|
||||
\def\bS{\mathbb{S}}
|
||||
\def\fR{\mathfrak{R}}
|
||||
\def\fZ{\mathfrak{Z}}
|
||||
\def\fQ{\mathfrak{Q}}
|
||||
\def\fC{\mathfrak{C}}
|
||||
\def\fO{\mathfrak{O}}
|
||||
\def\sC{\mathscr{C}}
|
||||
\def\sD{\mathscr{D}}
|
||||
\def\sP{\mathscr{P}}
|
||||
\def\sF{\mathscr{F}}
|
||||
\def\sH{\mathscr{H}}
|
||||
\def\sM{\mathscr{M}}
|
||||
\def\sT{\mathscr{T}}
|
||||
\def\cA{\mathcal{A}}
|
||||
\def\cB{\mathcal{B}}
|
||||
\def\cO{\mathcal{O}}
|
||||
\def\cF{\mathcal{F}}
|
||||
\def\cG{\mathcal{G}}
|
||||
\def\cC{\mathcal{C}}
|
||||
\def\cD{\mathcal{D}}
|
||||
\def\cT{\mathcal{T}}
|
||||
\def\cP{\mathcal{P}}
|
||||
\def\cS{\mathcal{S}}
|
||||
|
||||
\def\bfz{\mathbf{0}}
|
||||
|
||||
\def\fkp{\mathfrak{p}}
|
||||
\def\fkq{\mathfrak{q}}
|
||||
\def\fkm{\mathfrak{m}}
|
||||
|
||||
|
||||
%Symbol abbreviations
|
||||
\def\lm{\lambda}
|
||||
\def\eps{\epsilon}
|
||||
\def\veps{\varepsilon}
|
||||
\def\vphi{\varphi}
|
||||
|
||||
\def\ot{\otimes}
|
||||
\def\ptl{\partial}
|
||||
\def\bdry{\partial}
|
||||
|
||||
\def\nin{\not\in}
|
||||
|
||||
\newcommand{\isom}{\cong} %Isomorphism
|
||||
\newcommand{\ceq}{\coloneqq} %Defined to be equal to
|
||||
\newcommand{\eqc}{\eqqcolon} %Equal to be defined to
|
||||
\newcommand{\ra}{\rangle} %Right angle bracket
|
||||
\newcommand{\la}{\langle} %Left angle bracket
|
||||
\newcommand{\wto}{\rightharpoonup} %Weak convergence, or just an arrow with the bottom half missing
|
||||
\renewcommand{\to}{\longrightarrow} %A long right arrow, for functions and such
|
||||
\newcommand{\lmto}{\longmapsto} %A right arrow with a vertical tail, for mapping definitions
|
||||
\newcommand{\llim}{\lim\limits} %limit set up so that it always writes the limits underneath
|
||||
|
||||
%Function constructors
|
||||
\newcommand{\func}[3]{#1\colon#2\to #3} %Generic function definition without mapping
|
||||
\newcommand{\mfunc}[3]{#1\colon#2\mapsto #3} %Generic function definition with only mapping
|
||||
\newcommand{\afunc}[3]{#1\colon#2&\to #3} %Generic function with alignment on arrow
|
||||
\newcommand{\vfunc}[5]{\func{#1}{#2}{#3},\quad#4\longmapsto #5} %Generic function definition with mapping
|
||||
\newcommand{\vafunc}[5]{\afunc{#1}{#2}{#3}\\#4&\longmapsto #5} %Generic function over two lines aligned on arrow
|
||||
|
||||
|
||||
%More complex operator constructors
|
||||
\DeclareDocumentCommand\amod{ m g g }
|
||||
{%
|
||||
\IfNoValueTF{#2}
|
||||
{%
|
||||
\pmod{#1}
|
||||
}{%
|
||||
\IfNoValueTF{#3}
|
||||
{%
|
||||
{#1}\pmod{#2}
|
||||
}{%
|
||||
{#1} \,\equiv\, #2\pmod{#3}
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
\DeclareDocumentCommand\ppp{ m g }
|
||||
{%
|
||||
\IfNoValueTF{#2}
|
||||
{%
|
||||
\frac{\partial}{\partial #1}
|
||||
}{%
|
||||
\frac{\partial #1}{\partial #2}
|
||||
}
|
||||
}
|
||||
|
||||
\newcommand{\pp}[1]{\frac{\partial}{\partial #1}}
|
||||
|
||||
|
||||
%For theorems and proofs
|
||||
|
||||
\newtheorem{Theorem}{Theorem}
|
||||
\newtheorem{Proposition}{Proposition}
|
||||
\newtheorem{Conjecture}[Theorem]{Conjecture}
|
||||
\newtheorem{Lemma}[Theorem]{Lemma}
|
||||
\newtheorem{Corollary}[Theorem]{Corollary}
|
||||
|
||||
\theoremstyle{remark}
|
||||
\newtheorem{Example}{Example}
|
||||
\newtheorem{Note}{Note}
|
||||
\newtheorem{Question}{Question}
|
||||
\newtheorem{Remark}{Remark}
|
||||
|
||||
\theoremstyle{definition}
|
||||
\newtheorem{Result}[Theorem]{Result}
|
||||
\newtheorem{Definition}[Theorem]{Definition}
|
||||
%Shorthands
|
||||
\DeclareDocumentCommand\sdef{m}{\begin{Definition}#1\end{Definition}}
|
||||
\DeclareDocumentCommand\sex{m}{\begin{Example}#1\end{Example}}
|
||||
\DeclareDocumentCommand\snote{m}{\begin{Note}#1\end{Note}}
|
||||
\DeclareDocumentCommand\slem{m}{\begin{Lemma}#1\end{Lemma}}
|
||||
\DeclareDocumentCommand\sresult{m}{\begin{Result}#1\end{Result}}
|
||||
\DeclareDocumentCommand\sres{m}{\begin{Result}#1\end{Result}}
|
||||
\DeclareDocumentCommand\sques{m}{\begin{Question}#1\end{Question}}
|
||||
\DeclareDocumentCommand\srem{m}{\begin{Remark}#1\end{Remark}}
|
||||
|
||||
|
||||
|
||||
%Mathematical Operators
|
||||
\DeclareMathOperator{\Ell}{Ell} %Elliptic genera
|
||||
\DeclareMathOperator{\id}{id}
|
||||
\DeclareMathOperator{\Id}{id}
|
||||
\DeclareMathOperator{\Set}{\mathscr{S}et}
|
||||
\DeclareMathOperator{\Hom}{\mathscr{H}om}
|
||||
\DeclareMathOperator{\Top}{\mathscr{T}op}
|
||||
\DeclareMathOperator{\Mod}{\mathbf{Mod}}
|
||||
\DeclareMathOperator{\Comm}{\mathbf{Comm}}
|
||||
\DeclareMathOperator{\Grp}{\mathbf{Grp}}
|
||||
\DeclareMathOperator{\GL}{GL}
|
||||
\DeclareMathOperator{\Obj}{Obj}
|
||||
\DeclareMathOperator{\Spec}{Spec}
|
||||
\DeclareMathOperator{\Specm}{Specm}
|
||||
\DeclareMathOperator{\Min}{min}
|
||||
\DeclareMathOperator{\Ker}{Ker}
|
||||
\DeclareMathOperator{\im}{Im}
|
||||
\DeclareMathOperator{\CoIm}{CoIm}
|
||||
\DeclareMathOperator{\CoKer}{CoKer}
|
||||
\DeclareMathOperator{\Der}{Der}
|
||||
\DeclareMathOperator{\End}{End}
|
||||
|
||||
|
||||
% Brackets and braces
|
||||
\DeclareDocumentCommand\braces{}{{\ifnum\z@=`}\fi\@braces}
|
||||
\DeclareDocumentCommand\@braces{ s t\big t\Big t\bigg t\Bigg m m m }
|
||||
{ % General braces with automatic and manual sizing
|
||||
\IfBooleanTF{#1}
|
||||
{\left#6\smash{#8}\right#7\vphantom{#8}}
|
||||
{
|
||||
\IfBooleanTF{#2}{\bigl#6{#8}\bigr#7}{
|
||||
\IfBooleanTF{#3}{\Bigl#6{#8}\Bigr#7}{
|
||||
\IfBooleanTF{#4}{\biggl#6{#8}\biggr#7}{
|
||||
\IfBooleanTF{#5}{\Biggl#6{#8}\Biggr#7}{\left#6{#8}\right#7}
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
\ifnum\z@=`{\fi}
|
||||
}
|
||||
|
||||
|
||||
\DeclareDocumentCommand\quantity{}{{\ifnum\z@=`}\fi\@quantity}
|
||||
\DeclareDocumentCommand\@quantity{ t\big t\Big t\bigg t\Bigg g o d() d|| }
|
||||
{ % Flexible automatic bracketing of an expression in () or [] or {} or ||
|
||||
% Handles manual override of sizing
|
||||
\IfBooleanTF{#1}{\let\ltag\bigl \let\rtag\bigr}{
|
||||
\IfBooleanTF{#2}{\let\ltag\Bigl \let\rtag\Bigr}{
|
||||
\IfBooleanTF{#3}{\let\ltag\biggl \let\rtag\biggr}{
|
||||
\IfBooleanTF{#4}
|
||||
{\let\ltag\Biggl \let\rtag\Biggr}
|
||||
{\let\ltag\left \let\rtag\right}
|
||||
}
|
||||
}
|
||||
}
|
||||
% Handles actual bracketing
|
||||
\IfNoValueTF{#5}{
|
||||
\IfNoValueTF{#6}{
|
||||
\IfNoValueTF{#7}{
|
||||
\IfNoValueTF{#8}
|
||||
{()}
|
||||
{\ltag\lvert{#8}\rtag\rvert}
|
||||
}
|
||||
{\ltag(#7\rtag) \IfNoValueTF{#8}{}{|#8|}}
|
||||
}
|
||||
{\ltag[#6\rtag] \IfNoValueTF{#7}{}{(#7)} \IfNoValueTF{#8}{}{|#8|}}
|
||||
}
|
||||
{\ltag\lbrace#5\rtag\rbrace \IfNoValueTF{#6}{}{[#6]} \IfNoValueTF{#7}{}{(#7)} \IfNoValueTF{#8}{}{|#8|}}
|
||||
\ifnum\z@=`{\fi}
|
||||
}
|
||||
\DeclareDocumentCommand\qty{}{\quantity} % Shorthand for \quantity
|
||||
\DeclareDocumentCommand\pqty{ l m }{\braces#1{\lparen}{\rparen}{#2}}
|
||||
\DeclareDocumentCommand\bqty{ l m }{\braces#1{\lbrack}{\rbrack}{#2}}
|
||||
\DeclareDocumentCommand\Bqty{ l m }{\braces#1{\lbrace}{\rbrace}{#2}}
|
||||
\DeclareDocumentCommand\vqty{ l m }{\braces#1{\lvert}{\rvert}{#2}}
|
||||
|
||||
%Matrix constructors
|
||||
\DeclareDocumentCommand\pmqty{m}{\begin{pmatrix}#1\end{pmatrix}}
|
||||
\DeclareDocumentCommand\Pmqty{m}{\left\lgroup\begin{matrix}#1\end{matrix}\right\rgroup}
|
||||
\DeclareDocumentCommand\bmqty{m}{\begin{bmatrix}#1\end{bmatrix}}
|
||||
\DeclareDocumentCommand\Bmqty{m}{\begin{Bmatrix}#1\end{Bmatrix}}
|
||||
\DeclareDocumentCommand\sbmqty{m}{\left[\begin{smallmatrix}#1\end{smallmatrix}\right]}
|
||||
\DeclareDocumentCommand\vmqty{m}{\begin{vmatrix}#1\end{vmatrix}}
|
||||
\DeclareDocumentCommand\matrixquantity{}{{\ifnum\z@=`}\fi\@matrixquantity}
|
||||
\DeclareDocumentCommand\@matrixquantity{ s g o d() d|| }
|
||||
{
|
||||
\mathord{
|
||||
\IfNoValueTF{#2}
|
||||
{
|
||||
\IfNoValueTF{#3}
|
||||
{
|
||||
\IfNoValueTF{#4}
|
||||
{
|
||||
\IfNoValueTF{#5}
|
||||
{()}
|
||||
{\vmqty{#5}}
|
||||
}
|
||||
{
|
||||
\IfBooleanTF{#1}
|
||||
{\Pmqty{#4}}
|
||||
{\pmqty{#4}}
|
||||
\IfNoValueTF{#5}{}{|#5|}
|
||||
}
|
||||
}
|
||||
{\bmqty{#3} \IfNoValueTF{#4}{}{(#4)} \IfNoValueTF{#5}{}{|#5|}}
|
||||
}
|
||||
{\begin{matrix}#2\end{matrix} \IfNoValueTF{#3}{}{[#3]} \IfNoValueTF{#4}{}{(#4)} \IfNoValueTF{#5}{}{|#5|}}
|
||||
}
|
||||
\ifnum\z@=`{\fi}
|
||||
}
|
||||
\DeclareDocumentCommand\mqty{}{\matrixquantity} % Shorthand for \matrixquantity
|
||||
\DeclareDocumentCommand\matrixdeterminant{m}{\vmqty{#1}} % Matrix determinant
|
||||
\DeclareDocumentCommand\mdet{}{\matrixdeterminant} % Shorthand for matrix determinant
|
||||
|
||||
|
||||
% Quick quad text (math-mode text with \quad spacing)
|
||||
\DeclareDocumentCommand\qqtext{ s m }{\IfBooleanTF{#1}{}{\quad}\text{#2}\quad}
|
||||
\DeclareDocumentCommand\qq{}{\qqtext}
|
||||
|
||||
\DeclareDocumentCommand\qcomma{}{,\quad}
|
||||
\DeclareDocumentCommand\qc{}{\qcomma}
|
||||
|
||||
\DeclareDocumentCommand\qif{s}{\IfBooleanTF{#1}{}{\quad}\text{if}\quad}
|
||||
\DeclareDocumentCommand\qthen{s}{\IfBooleanTF{#1}{}{\quad}\text{then}\quad}
|
||||
\DeclareDocumentCommand\qelse{s}{\IfBooleanTF{#1}{}{\quad}\text{else}\quad}
|
||||
\DeclareDocumentCommand\qotherwise{s}{\IfBooleanTF{#1}{}{\quad}\text{otherwise}\quad}
|
||||
\DeclareDocumentCommand\qunless{s}{\IfBooleanTF{#1}{}{\quad}\text{unless}\quad}
|
||||
\DeclareDocumentCommand\qgiven{s}{\IfBooleanTF{#1}{}{\quad}\text{given}\quad}
|
||||
\DeclareDocumentCommand\qusing{s}{\IfBooleanTF{#1}{}{\quad}\text{using}\quad}
|
||||
\DeclareDocumentCommand\qassume{s}{\IfBooleanTF{#1}{}{\quad}\text{assume}\quad}
|
||||
\DeclareDocumentCommand\qsince{s}{\IfBooleanTF{#1}{}{\quad}\text{since}\quad}
|
||||
\DeclareDocumentCommand\qlet{s}{\IfBooleanTF{#1}{}{\quad}\text{let}\quad}
|
||||
\DeclareDocumentCommand\qfor{s}{\IfBooleanTF{#1}{}{\quad}\text{for}\quad}
|
||||
\DeclareDocumentCommand\qall{s}{\IfBooleanTF{#1}{}{\quad}\text{all}\quad}
|
||||
\DeclareDocumentCommand\qeven{s}{\IfBooleanTF{#1}{}{\quad}\text{even}\quad}
|
||||
\DeclareDocumentCommand\qodd{s}{\IfBooleanTF{#1}{}{\quad}\text{odd}\quad}
|
||||
\DeclareDocumentCommand\qinteger{s}{\IfBooleanTF{#1}{}{\quad}\text{integer}\quad}
|
||||
\DeclareDocumentCommand\qand{s}{\IfBooleanTF{#1}{}{\quad}\text{and}\quad}
|
||||
\DeclareDocumentCommand\qor{s}{\IfBooleanTF{#1}{}{\quad}\text{or}\quad}
|
||||
\DeclareDocumentCommand\qas{s}{\IfBooleanTF{#1}{}{\quad}\text{as}\quad}
|
||||
\DeclareDocumentCommand\qin{s}{\IfBooleanTF{#1}{}{\quad}\text{in}\quad}
|
||||
\DeclareDocumentCommand\qcc{s}{\IfBooleanTF{#1}{}{\quad}\text{c.c.}\quad}
|
||||
371
fancymath.sty
Normal file
371
fancymath.sty
Normal file
@@ -0,0 +1,371 @@
|
||||
%LaTeX package for writing fancy mathematics fast, specifically tailored to writing assignments and course notes.
|
||||
%Requires and integrates with the fancycom package, which defines many mathematical command shortcuts.
|
||||
%Not carefully checked for integration with other packages.
|
||||
%Written by Adam Walsh
|
||||
%Version 1.0
|
||||
|
||||
%----------Identification------------
|
||||
\NeedsTeXFormat{LaTeX2e}
|
||||
\ProvidesPackage{fancymath}[2016/06/17 v1.0 Fancy Maths Package]
|
||||
|
||||
%----------Initial code--------------
|
||||
|
||||
\RequirePackage{ifthen}
|
||||
|
||||
%Layout colour
|
||||
%% 0=Full (default)
|
||||
%% 1=Black (no colour)
|
||||
%% 2=Grey (Greyscale colours)
|
||||
\newcounter{colour}
|
||||
\newboolean{fullcolour}
|
||||
\setboolean{fullcolour}{true}
|
||||
\newcommand{\colour}{}
|
||||
|
||||
%Default Layouts
|
||||
%% 0=plain (default)
|
||||
%% 1=Assignment
|
||||
%% 2=General Submission
|
||||
%% 3=Course Notes
|
||||
%% 4=
|
||||
\newcounter{layout}
|
||||
|
||||
|
||||
%Separate questions on different pages (for question and answer environment)
|
||||
%%0= multiple questions per page (default)
|
||||
%%1= One question and answer per page
|
||||
\newboolean{sepquest}
|
||||
|
||||
\newboolean{qthm}
|
||||
|
||||
%----------Option Declaration------------
|
||||
|
||||
%Colour selection
|
||||
\DeclareOption{grey}{%
|
||||
\setcounter{colour}{2}%
|
||||
\setboolean{fullcolour}{false}%
|
||||
\renewcommand{\colour}{\color{Gray}}}
|
||||
|
||||
\DeclareOption{black}{%
|
||||
\setcounter{colour}{1}%
|
||||
\setboolean{fullcolour}{false}%
|
||||
\renewcommand{\colour}{\color{Black}}}
|
||||
|
||||
%Default Layout selection
|
||||
\DeclareOption{ass}{\setcounter{layout}{1}}
|
||||
\DeclareOption{general}{\setcounter{layout}{2}}
|
||||
\DeclareOption{notes}{\setcounter{layout}{3}}
|
||||
|
||||
%Seperate questions option
|
||||
\DeclareOption{qsep}{\setboolean{sepquest}{true}}
|
||||
|
||||
\DeclareOption{qthm}{\setboolean{qthm}{true}}
|
||||
|
||||
%---------Process Options----------------
|
||||
\ProcessOptions
|
||||
|
||||
%------------Package Loading---------------
|
||||
|
||||
\RequirePackage[table,dvipsnames]{xcolor} %To use coloured layout
|
||||
\RequirePackage{fancyhdr} %To make the headers and footers fancy
|
||||
\RequirePackage{geometry} %To manipulate the geometry of the pages
|
||||
|
||||
\RequirePackage{calc} %For manipulating values
|
||||
|
||||
|
||||
%------------Main Code---------------
|
||||
|
||||
|
||||
%Some formatting lengths not usually useful in mathematical submissions.
|
||||
\setlength{\parindent}{0mm}
|
||||
\setlength{\parskip}{0mm}
|
||||
|
||||
|
||||
|
||||
%General values used in titles and styling
|
||||
\newcommand{\authorname}{}
|
||||
\newcommand{\authornumber}{}
|
||||
\newcommand{\coursename}{}
|
||||
\newcommand{\coursenumber}{}
|
||||
\newcommand{\submissiontitle}{}
|
||||
\newcommand{\assignmentnumber}{}
|
||||
\newcommand{\thetitle}{\submissiontitle}
|
||||
\newcommand{\theauthor}{\authorname}
|
||||
\newcommand{\therightfoot}{}
|
||||
\newcommand{\theleftfoot}{}
|
||||
\newcommand{\thelefthead}{}
|
||||
\newcommand{\therighthead}{}
|
||||
|
||||
|
||||
|
||||
%Control sequences for basic style values
|
||||
\newcommand{\aname}[1]{%
|
||||
\renewcommand{\authorname}{#1}}
|
||||
\newcommand{\anum}[1]{%
|
||||
\renewcommand{\authornumber}{#1}}
|
||||
\newcommand{\cname}[1]{%
|
||||
\renewcommand{\coursename}{#1}}
|
||||
\newcommand{\cnum}[1]{%
|
||||
\renewcommand{\coursenumber}{#1}}
|
||||
\newcommand{\stitle}[1]{%
|
||||
\renewcommand{\submissiontitle}{#1}}
|
||||
\newcommand{\assnum}[1]{%
|
||||
\renewcommand{\assignmentnumber}{#1}}
|
||||
|
||||
%Advanced control sequences for controlling styling
|
||||
\newcommand{\rawtitle}[1]{%
|
||||
\renewcommand{\thetitle}{#1}}
|
||||
\newcommand{\rawauthor}[1]{%
|
||||
\renewcommand{\theauthor}{#1}}
|
||||
\newcommand{\rawrfoot}[1]{%
|
||||
\renewcommand{\therightfoot}{#1}}
|
||||
\newcommand{\rawlfoot}[1]{%
|
||||
\renewcommand{\theleftfoot}{#1}}
|
||||
\newcommand{\rawlhead}[1]{%
|
||||
\renewcommand{\thelefthead}{#1}}
|
||||
\newcommand{\rawrhead}[1]{%
|
||||
\renewcommand{\therighthead}{#1}}
|
||||
|
||||
|
||||
%Styling common to all layouts, including plain
|
||||
|
||||
%The page geometry
|
||||
\geometry{%
|
||||
a4paper,%
|
||||
left=20mm,%
|
||||
right=20mm,%
|
||||
top=20mm,%
|
||||
bottom=20mm,%
|
||||
heightrounded}
|
||||
|
||||
|
||||
|
||||
%Styling common to general non-plain layouts
|
||||
\ifthenelse{\not \value{layout}=0}{%
|
||||
%Base colours, determined from colour option
|
||||
\newcommand{\titlecolour}{%
|
||||
\ifthenelse{\boolean{fullcolour}}{\color{Brown}}{\colour}}
|
||||
\newcommand{\headcolour}{%
|
||||
\ifthenelse{\boolean{fullcolour}}{\color{Blue}}{\colour}}
|
||||
\newcommand{\headrulecolour}{%
|
||||
\ifthenelse{\boolean{fullcolour}}{\color{Brown}}{\colour}}
|
||||
\newcommand{\enumcolour}{%
|
||||
\ifthenelse{\boolean{fullcolour}}{\color{Brown}}{\colour}}
|
||||
|
||||
%Some default values for the footers
|
||||
\rawlfoot{\authorname}
|
||||
\rawrfoot{\today}
|
||||
\rawlhead{\submissiontitle}
|
||||
\rawrhead{\thepage}
|
||||
|
||||
%Page style
|
||||
\pagestyle{fancy}
|
||||
|
||||
%Set the headers and footers
|
||||
\fancyhead{}
|
||||
\fancyfoot{}
|
||||
\fancyfoot[L]{\headcolour\theleftfoot}
|
||||
\fancyfoot[C]{\headcolour\thepage}
|
||||
\fancyfoot[R]{\headcolour{}\therightfoot}
|
||||
\fancyhead[L]{\headcolour\thelefthead}
|
||||
\fancyhead[R]{\headcolour\therighthead}
|
||||
|
||||
%Make the headrule
|
||||
\renewcommand{\headrule}{\headrulecolour\vbox to 0pt{\hbox to\headwidth{\hrulefill}\vss}}
|
||||
\headheight=21pt
|
||||
|
||||
%Slightly modify line spacing
|
||||
\renewcommand{\baselinestretch}{1.1}
|
||||
|
||||
%Set the style and colour of the enumerate labels
|
||||
\renewcommand{\theenumi}{\alph{enumi}}
|
||||
\renewcommand{\theenumii}{\roman{enumii}}
|
||||
\renewcommand{\theenumiii}{\alph{enumiii}}
|
||||
\renewcommand{\labelenumi}{\enumcolour(\theenumi)}
|
||||
\renewcommand{\labelenumii}{\enumcolour(\theenumii)}
|
||||
\renewcommand{\labelenumiii}{\enumcolour(\theenumiii)}
|
||||
|
||||
%Make title info for maketitle
|
||||
\author{\titlecolour\theauthor}
|
||||
\title{\titlecolour\thetitle}
|
||||
\date{\titlecolour\today}
|
||||
|
||||
}{}
|
||||
|
||||
%---Notes specific layout and styling---
|
||||
\ifthenelse{\value{layout}=3}{%
|
||||
%Set the Notes specific headers and footers
|
||||
\rawlfoot{\authorname, \today}
|
||||
\rawrfoot{Notes}
|
||||
\rawlhead{\coursenumber -- \coursename}
|
||||
|
||||
%Assignment specific title and Author
|
||||
\rawauthor{\authorname}
|
||||
\rawtitle{\coursename -- Notes}
|
||||
}{}
|
||||
|
||||
%---Assignment specific layout and styling---
|
||||
\ifthenelse{\value{layout}=1}{%
|
||||
%Set the Assignment specific headers and footers
|
||||
\rawlfoot{\authorname}
|
||||
\rawrfoot{\today}
|
||||
\rawrhead{Assignment \assignmentnumber}
|
||||
\rawlhead{\coursenumber -- \coursename}
|
||||
|
||||
%Assignment specific title and Author
|
||||
\rawauthor{\authorname, Student \authornumber}
|
||||
\rawtitle{\coursenumber -- Assignment \assignmentnumber}
|
||||
}{}
|
||||
|
||||
|
||||
%Defining the Question-and-Answer environment for use in assignments.
|
||||
%%Formats Questions and Answers with incrementing counters
|
||||
%Base lengths
|
||||
%%Amount questions and answers are indented
|
||||
\newlength{\qmargin}
|
||||
\setlength{\qmargin}{5mm}
|
||||
|
||||
%%Amount of vertical space between question/answer title and question/answer.
|
||||
\newlength{\qgap}
|
||||
\setlength{\qgap}{5mm}
|
||||
|
||||
%%Amount of vertical space between end of question and answer.
|
||||
\newlength{\qagap}
|
||||
\setlength{\qagap}{0.5em}
|
||||
|
||||
|
||||
%Base colours, determined from colour option
|
||||
%%Default colour is gray anyway
|
||||
\newcommand{\qcolour}{%
|
||||
\ifthenelse{\boolean{fullcolour}}{\color{Gray}}{\colour}}
|
||||
|
||||
|
||||
\ifthenelse{\boolean{qthm}}{
|
||||
%Values
|
||||
\newcounter{question}
|
||||
\newcounter{nextanswer}
|
||||
|
||||
\newboolean{firstquest}
|
||||
\newboolean{questopen}
|
||||
\newboolean{answeropen}
|
||||
|
||||
\RequirePackage{amsthm}
|
||||
\theoremstyle{plain}
|
||||
\newtheorem{Question}[question]{Question}
|
||||
|
||||
|
||||
\makeatletter
|
||||
|
||||
\newcommand{\theanswer}{Answer \arabic{nextanswer}}
|
||||
|
||||
\newcommand{\qnum}[1]{%
|
||||
\setcounter{question}{#1}%
|
||||
\addtocounter{question}{-1}%
|
||||
\setcounter{nextanswer}{#1}%
|
||||
}
|
||||
|
||||
\newcommand{\startqanda}{%
|
||||
\setcounter{question}{0}%
|
||||
\setcounter{nextanswer}{0}%
|
||||
\setboolean{firstquest}{true}%
|
||||
\setboolean{questopen}{false}%
|
||||
\setboolean{answeropen}{false}%
|
||||
}
|
||||
|
||||
\newcommand{\closeqanda}{%
|
||||
\ifthenelse{\boolean{questopen}}{\closequestion}{}%
|
||||
\ifthenelse{\boolean{answeropen}}{\closeanswer}{}%
|
||||
}
|
||||
|
||||
\newcommand{\question}[1][0]{%
|
||||
\closeqanda%
|
||||
\ifthenelse{\boolean{sepquest} \and \not \boolean{firstquest} }{\newpage}{}%
|
||||
\setboolean{firstquest}{false}%
|
||||
\ifthenelse{#1=0}{}{\qnum{#1}}%
|
||||
\begin{Question}
|
||||
\@minipagetrue%
|
||||
\setboolean{questopen}{true}%
|
||||
\setcounter{nextanswer}{\value{question}}%
|
||||
}
|
||||
\newcommand{\closequestion}{%
|
||||
\end{Question}%
|
||||
\setboolean{questopen}{false}%
|
||||
}
|
||||
|
||||
\newcommand{\answer}[1][0]{%
|
||||
\closeqanda%
|
||||
\ifthenelse{#1=0}{}{\qnum{#1}\stepcounter{question}\stepcounter{question}}%
|
||||
\begin{proof}[\theanswer]
|
||||
\@minipagetrue%
|
||||
\setboolean{answeropen}{true}%
|
||||
\setcounter{question}{\value{nextanswer}}%
|
||||
\stepcounter{nextanswer}%
|
||||
\phantom{}%
|
||||
}
|
||||
\newcommand{\closeanswer}{%
|
||||
\ifvmode\vspace*{-\baselineskip}\fi%
|
||||
%\unskip%
|
||||
\end{proof}%
|
||||
\setboolean{answeropen}{false}%
|
||||
}
|
||||
|
||||
\newenvironment{qanda}{%
|
||||
\startqanda%
|
||||
}{%
|
||||
\closeqanda%
|
||||
}
|
||||
\makeatother
|
||||
}{
|
||||
|
||||
%Values
|
||||
\newcounter{question}
|
||||
\newcounter{lastquestion}
|
||||
\newboolean{firstquest}
|
||||
|
||||
|
||||
%Control Sequences
|
||||
\newcommand{\qnum}[1]{%
|
||||
\setcounter{question}{#1}%
|
||||
\addtocounter{question}{-1}%
|
||||
\setcounter{lastquestion}{\value{question}}}
|
||||
|
||||
%The environment
|
||||
\newenvironment{qanda}{%
|
||||
\setcounter{question}{0}%
|
||||
\setcounter{lastquestion}{0}%
|
||||
\setboolean{firstquest}{true}%
|
||||
\begin{list}{}{%
|
||||
\renewcommand{\makelabel}[1]{\qcolour\textbf{##1}\\}%
|
||||
\setlength{\leftmargin}{\qmargin}%
|
||||
}%
|
||||
}{%
|
||||
\end{list}%
|
||||
}
|
||||
|
||||
%The question and answer commands
|
||||
\makeatletter
|
||||
\newcommand{\question}[1][0]{%
|
||||
\ifthenelse{\boolean{sepquest} \and \not \boolean{firstquest} }{\newpage}{}%
|
||||
\setboolean{firstquest}{false}%
|
||||
\ifthenelse{#1=0}{%
|
||||
\stepcounter{question}%
|
||||
\setcounter{lastquestion}{\value{question}}%
|
||||
\item[Question \arabic{question}]%
|
||||
}{%
|
||||
\setcounter{lastquestion}{#1}%
|
||||
\item[Question #1]%
|
||||
}%
|
||||
\hfill\break\@minipagetrue%
|
||||
}
|
||||
|
||||
\newcommand{\answer}[1][0]{%
|
||||
\vspace{\the\qagap}%
|
||||
\ifthenelse{#1=0}{%
|
||||
\item[Answer \arabic{lastquestion}]%
|
||||
}{%
|
||||
\item[Answer #1]%
|
||||
}%
|
||||
\hfill\break%
|
||||
\@minipagetrue%
|
||||
}
|
||||
\makeatother
|
||||
}
|
||||
Reference in New Issue
Block a user