Fix even more \colon's

sed -r "s/(\\[ [^][ ()]+) : /\1 \\\\colon /g" -i (git ls-files '*.tex')

tex/H113/action.tex

@@ -182,7 +182,7 @@ \section{Conjugation of elements}
 \prototype{In $S_n$, conjugacy classes are ``cycle types''.}
 A particularly common type of action is the so-called \vocab{conjugation}.
 We let $G$ act on itself as follows:
-\[ g : h \mapsto ghg\inv. \]
+\[ g \colon h \mapsto ghg\inv. \]
 You might think this definition is a little artificial.
 Who cares about the element $ghg\inv$?
 Let me try to convince you this definition is not so unnatural.

tex/H113/grp-intro.tex

@@ -379,7 +379,7 @@ \section{Isomorphisms}
 Think about what this might mean formally for a moment.
 
 Specifically the map
-\[ \phi : \ZZ \to 10 \ZZ  \text{ by } x \mapsto 10 x \]
+\[ \phi \colon \ZZ \to 10 \ZZ  \text{ by } x \mapsto 10 x \]
 is a bijection of the underlying sets which respects the group operation.
 In symbols,
 \[ \phi(x + y) = \phi(x) + \phi(y). \]

tex/H113/quotient.tex

@@ -229,7 +229,7 @@ \section{Cosets and modding out}
 If this exposition doesn't work for you, try \cite{ref:gowers}.}
 
 Let $G$ and $Q$ be groups, and suppose there exists
-a \emph{surjective} homomorphism \[ \phi : G \surjto Q. \]
+a \emph{surjective} homomorphism \[ \phi \colon G \surjto Q. \]
 In other words, if $\phi$ is injective then $\phi \colon G \to Q$ is a bijection,
 and hence an isomorphism.
 But suppose we're not so lucky and $\ker\phi$ is bigger than just $\{1_G\}$.

tex/alg-NT/artin.tex

@@ -247,7 +247,7 @@ \section{Frobenius element and Artin symbol}
 	and prime factors of $m$.
 	Moreover, the Galois group $\Gal(L/K)$ is $(\ZZ/m\ZZ)^\times$:
 	the Galois group consists of elements of the form
-	\[ \sigma_n : \zeta \mapsto \zeta^n \]
+	\[ \sigma_n \colon \zeta \mapsto \zeta^n \]
 	and $\Gal(L/K) = \left\{ \sigma_n \mid n \in (\ZZ/m\ZZ)^\times \right\}$.
 
 	Then it follows just like before that

tex/alg-NT/classgrp.tex

@@ -259,7 +259,7 @@ \section{The signature of a number field}
 		The elements of $K$ are
 		\[ K = \left\{ a + b\cbrt2 + c\cbrt4 \mid a,b,c \in \QQ  \right\}. \]
 		Then the signature is $(1,1)$, because the three embeddings are
-		\[ \sigma_1 : \cbrt 2 \mapsto \cbrt 2,
+		\[ \sigma_1 \colon \cbrt 2 \mapsto \cbrt 2,
 			\quad
 			\sigma_2 : \cbrt 2 \mapsto \cbrt 2 \omega,
 			\quad
@@ -310,7 +310,7 @@ \section{The signature of a number field}
 \end{align*}
 \begin{example}[Example of canonical embedding]
 	As before let $K = \QQ(\sqrt[3]{2})$ and set
-	\[ \sigma_1 : \cbrt 2 \mapsto \cbrt 2,
+	\[ \sigma_1 \colon \cbrt 2 \mapsto \cbrt 2,
 		\quad
 		\sigma_2 : \cbrt 2 \mapsto \cbrt 2 \omega,
 		\quad

tex/alg-NT/finite-field.tex

@@ -12,7 +12,7 @@ \chapter{Finite fields}
 	so it's customary to use the notation $\FF_{p^n}$
 	for ``the'' finite field of order $p^n$ if we only care up to isomorphism.
 	\ii The extension $F/\FF_p$ is Galois, and $\Gal(F/\FF_p)$ is a cyclic group of order $n$.
-	The generator is the automorphism \[ \sigma : F \to F \quad\text{by}\quad x \mapsto x^p. \]
+	The generator is the automorphism \[ \sigma \colon F \to F \quad\text{by}\quad x \mapsto x^p. \]
 \end{itemize}
 If you're in a hurry you can just remember these results and skip to the next chapter.
 

tex/alg-NT/frobenius.tex

@@ -275,7 +275,7 @@ \section{Example: Frobenius elements of cyclotomic fields}
 \begin{itemize}
 	\ii $\Delta_L = \pm q^{q-2}$,
 	\ii $\OO_L = \ZZ[\zeta_q]$, and
-	\ii The map \[ \sigma_n : L \to L \quad\text{by}\quad \zeta_q \mapsto \zeta_q^n \]
+	\ii The map \[ \sigma_n \colon L \to L \quad\text{by}\quad \zeta_q \mapsto \zeta_q^n \]
 	is an automorphism of $L$ whenever $\gcd(n,q)=1$,
 	and depends only on $n \pmod q$.
 	In other words, the automorphisms of $L/\QQ$ just shuffle around the $q$th roots of unity.

tex/alg-geom/mor-scheme.tex

@@ -885,7 +885,7 @@ \section{Morphisms of sheaves}
 \begin{proposition}
 	[Morphisms determined by stalks]
 	A morphism of sheaves $\alpha \colon \SF \to \SG$ induces a morphism of stalks
-	\[ \alpha_p : \SF_p \to \SG_p \]
+	\[ \alpha_p \colon \SF_p \to \SG_p \]
 	for every point $p \in X$.
 	Moreover, the sequence $(\alpha_p)_{p \in X}$ determines $\alpha$ uniquely.
 \end{proposition}

tex/alg-geom/sheaves.tex

@@ -45,7 +45,7 @@ \subsection{Usual definition}
 \end{definition}
 \begin{definition}
 	A \vocab{pre-sheaf} of rings on a space $X$ is a function
-	\[ \SF : \Opens(X) \to \catname{Rings} \]
+	\[ \SF \colon \Opens(X) \to \catname{Rings} \]
 	meaning each open set gets associated with a ring $\SF(U)$.
 	Each individual element of $\SF(U)$ is called a \vocab{section}.
 
@@ -171,7 +171,7 @@ \subsection{Categorical definition}
 \end{abuse}
 \begin{definition}
 	A \vocab{pre-sheaf} of rings on $X$ is a contravariant functor
-	\[ \SF : \Opens(X)\op \to \catname{Rings}. \]
+	\[ \SF \colon \Opens(X)\op \to \catname{Rings}. \]
 \end{definition}
 \begin{exercise}
 	Check that these definitions are equivalent.

tex/calculus/integrate.tex

@@ -248,7 +248,7 @@ \section{Defining the Riemann integral}
 	+ f(t_2) \left( t_3 - t_2 \right)
 	+ \dots + f(t_n) \left( b - t_n \right).  \]
 We denote this function by
-\[ \Sigma : R([a,b]) \to \RR. \]
+\[ \Sigma \colon R([a,b]) \to \RR. \]
 
 \begin{theorem}
 	[The Riemann integral]

tex/diffgeo/forms.tex

@@ -75,15 +75,15 @@ \section{Pictures of differential forms}
 vector at that point, and spit out a real number.}
 So a $1$-form $\alpha$ is a smooth function on pairs $(p,v)$,
 where $v$ is a tangent vector at $p$, to $\RR$.  Hence
-\[ \alpha : U \times V \to \RR. \]
+\[ \alpha \colon U \times V \to \RR. \]
 
 Actually, for any point $p$, we will require that $\alpha(p,-)$
 is a linear function in terms of the vectors:
 i.e.\ we want for example that $\alpha(p,2v) = 2\alpha(p,v)$.
 So it is more customary to think of $\alpha$ as:
 \begin{definition}
 	A \vocab{$1$-form} $\alpha$ is a smooth function
-	\[ \alpha : U \to V^\vee. \]
+	\[ \alpha \colon U \to V^\vee. \]
 \end{definition}
 Like with $Df$, we'll use $\alpha_p$ instead of $\alpha(p)$.
 So, at every point $p$, $\alpha_p$ is some linear functional
@@ -94,7 +94,7 @@ \section{Pictures of differential forms}
 Next, we draw pictures of $2$-forms.
 This should, for example, let us integrate over a blob
 (a so-called $2$-cell) of the form
-\[ c : [0,1] \times [0,1] \to U \]
+\[ c \colon [0,1] \times [0,1] \to U \]
 i.e.\ for example, a square in $U$.
 In the previous example with $1$-forms,
 we looked at tangent vectors to the curve $c$.

tex/diffgeo/manifolds.tex

@@ -44,7 +44,7 @@ \section{Topological manifolds}
 	with an open cover $\{U_i\}$ of sets
 	homeomorphic to subsets of $\RR^n$,
 	say by homeomorphisms
-	\[ \phi_i : U_i \taking\cong E_i \subseteq \RR^n \]
+	\[ \phi_i \colon U_i \taking\cong E_i \subseteq \RR^n \]
 	where each $E_i$ is an open subset of $\RR^n$.
 	Each $\phi_i \colon U_i \to E_i$ is called a \vocab{chart},
 	and together they form a so-called \vocab{atlas}.

tex/diffgeo/multivar.tex

@@ -241,7 +241,7 @@ \section{Total and partial derivatives}
 \begin{example}[Partial derivatives of $f(x,y) = x^2+y^2$]
 	Let $f \colon \RR^2 \to \RR$ by $(x,y) \mapsto x^2+y^2$.
 	Then in our new language,
-	\[ Df : (x,y) \mapsto 2x \cdot \ee_1^\vee + 2y \cdot \ee_2^\vee. \]
+	\[ Df \colon (x,y) \mapsto 2x \cdot \ee_1^\vee + 2y \cdot \ee_2^\vee. \]
 	Thus the partials are
 	\[
 		\frac{\partial f}{\partial x} : (x,y) \mapsto 2x \in \RR
@@ -315,11 +315,11 @@ \section{(Optional) A word on higher derivatives}
 Thus it makes sense to write
 \[ D(Df) : U \to \Hom(V,\Hom(V,W)) \]
 which we abbreviate as $D^2 f$. Dropping all doubt and plunging on,
-\[ D^3f : U \to \Hom(V, \Hom(V,\Hom(V,W))). \]
+\[ D^3f \colon U \to \Hom(V, \Hom(V,\Hom(V,W))). \]
 I'm sorry.
 As consolation, we at least know that $\Hom(V,W) \cong V^\vee \otimes W$ in a natural way,
 so we can at least condense this to
-\[ D^kf : V \to (V^\vee)^{\otimes k} \otimes W \]
+\[ D^kf \colon V \to (V^\vee)^{\otimes k} \otimes W \]
 rather than writing a bunch of $\Hom$'s.
 \begin{remark}
 	If $k=2$, $W = \RR$, then $D^2f(v) \in (V^\vee)^{\otimes 2}$,

tex/diffgeo/stokes.tex

@@ -210,7 +210,7 @@ \section{Cells}
 		\ii As we saw, a $1$-cell is an arbitrary curve.
 		\ii A $2$-cell corresponds to a $2$-dimensional surface.
 		For example, the map $c \colon [0,R] \times [0,2\pi] \to V$ by
-		\[ c : (r,\theta) \mapsto (r\cos\theta, r\sin\theta) \]
+		\[ c \colon (r,\theta) \mapsto (r\cos\theta, r\sin\theta) \]
 		can be thought of as a disk of radius $R$.
 	\end{enumerate}
 \end{example}
@@ -384,7 +384,7 @@ \section{Boundaries}
 \begin{example}
 	[Boundary of a unit disk]
 	Consider the unit disk given by
-	\[ c : [0,1] \times [0,2\pi] \to \RR^2 \quad\text{by}\quad
+	\[ c \colon [0,1] \times [0,2\pi] \to \RR^2 \quad\text{by}\quad
 	(r,\theta) \mapsto s\cos(2\pi t)\ee_1 + s\sin(2\pi t)\ee_2. \]
 	The four parts of the boundary are shown in the picture below:
 	\begin{center}

tex/homology/cellular.tex

@@ -5,7 +5,7 @@ \chapter{Bonus: Cellular homology}
 \section{Degrees}
 \prototype{$z \mapsto z^d$ has degree $d$.}
 For any $n > 0$ and map $f \colon S^n \to S^n$, consider
-\[ f_\ast : \underbrace{H_n(S^n)}_{\cong \ZZ} \to \underbrace{H_n(S^n)}_{\cong \ZZ} \]
+\[ f_\ast \colon \underbrace{H_n(S^n)}_{\cong \ZZ} \to \underbrace{H_n(S^n)}_{\cong \ZZ} \]
 which must be multiplication by some constant $d$.
 This $d$ is called the \vocab{degree} of $f$, denoted $\deg f$.
 \begin{ques}
@@ -252,14 +252,14 @@ \section{The cellular boundary formula}
 Recalling that $H_k(X^k, X^{k-1})$ has a basis the $k$-cells of $X$, we obtain:
 \begin{theorem}
 	[Cellular boundary formula for $k=1$]
-	For $k=1$, \[ d_1 : \Cells_1(X) \to \Cells_0(X) \] is just the boundary map.
+	For $k=1$, \[ d_1 \colon \Cells_1(X) \to \Cells_0(X) \] is just the boundary map.
 \end{theorem}
 \begin{theorem}
 	[Cellular boundary for $k > 1$]
 	Let $k > 1$ be a positive integer.
 	Let $e^k$ be an $k$-cell, and let $\{e_\beta^{k-1}\}_\beta$
 	denote all $(k-1)$-cells of $X$.
-	Then \[ d_k : \Cells_k(X) \to \Cells_{k-1}(X) \]
+	Then \[ d_k \colon \Cells_k(X) \to \Cells_{k-1}(X) \]
 	is given on basis elements by
 	\[ d_k(e^k) = \sum_\beta d_\beta e_\beta^{k-1} \]
 	where $d_\beta$ is be the degree of the composed map

tex/homology/cohomology.tex

@@ -99,7 +99,7 @@ \section{Cohomology of spaces}
 	i.e.\ a homomorphism $\phi \colon C^0(X) \to G$,
 	what is $\delta\phi \colon C^1(X) \to G$?
 	Answer:
-	\[ \delta\phi : [v_0, v_1] \mapsto \phi([v_0]) - \phi([v_1]). \]
+	\[ \delta\phi \colon [v_0, v_1] \mapsto \phi([v_0]) - \phi([v_1]). \]
 	Hence, elements of
 	$\ker(C^0 \taking\delta C^1) \cong H^0(X;G)$
 	are those cochains
@@ -226,7 +226,7 @@ \section{Universal coefficient theorem}
 
 There are two things we need to explain, what the map $h$ is and the map $\Ext$ is.
 
-It's not too hard to guess how \[ h : H^n(A_\bullet; G) \to \Hom(H_n(A_\bullet), G) \] is defined.
+It's not too hard to guess how \[ h \colon H^n(A_\bullet; G) \to \Hom(H_n(A_\bullet), G) \] is defined.
 An element of $H^n(A_\bullet;G)$ is represented by a function which sends a cycle
 in $A_n$ to an element of $G$.
 The content of the theorem is to show that $h$ is surjective with kernel $\Ext(H_{n-1}(A_\bullet), G)$.

tex/homology/cup-product.tex

@@ -164,7 +164,7 @@ \section{Graded rings}
 \begin{lemma}
 	[Wedge product respects de Rham cohomology]
 	The wedge product induces a map
-	\[ \wedge : \HdR^k(M) \times \HdR^\ell(M) \to \HdR^{k+\ell}(M). \]
+	\[ \wedge \colon \HdR^k(M) \times \HdR^\ell(M) \to \HdR^{k+\ell}(M). \]
 \end{lemma}
 \begin{proof}
 	First, we recall that the operator $d$ satisfies
@@ -224,7 +224,7 @@ \section{Cup products}
 \end{proof}
 Thus, by the same routine we used for de Rham cohomology, we get
 an induced map
-\[ \smile : H^k(X;R) \times H^\ell(X;R) \to H^{k+\ell}(X;R).  \]
+\[ \smile \colon H^k(X;R) \times H^\ell(X;R) \to H^{k+\ell}(X;R).  \]
 We then define the \vocab{singular cohomology ring}
 whose elements are finite sums in
 \[ H^\bullet(X;R) = \bigoplus_{k \ge 0} H^k(X;R) \]

tex/homology/excision.tex

@@ -71,7 +71,7 @@ \section{The category of pairs}
 	Let $\varnothing \neq A \subseteq X$ and $\varnothing \neq B \subseteq X$
 	be subspaces, and consider a map $f \colon X \to Y$.
 	If $f\im(A) \subseteq B$ we write
-	\[ f : (X,A) \to (Y,B). \]
+	\[ f \colon (X,A) \to (Y,B). \]
 	We say $f$ is a \vocab{map of pairs},
 	between the pairs $(X,A)$ and $(Y,B)$.
 \end{definition}
@@ -103,10 +103,10 @@ \section{The category of pairs}
 We can do the same song and dance as before with the prism operator to obtain:
 \begin{lemma}[Induced maps of relative homology]
 	We have a functor
-	\[ H_n : \catname{hPairTop} \to \catname{Grp}. \]
+	\[ H_n \colon \catname{hPairTop} \to \catname{Grp}. \]
 \end{lemma}
 That is, if $f \colon (X,A) \to (Y,B)$ then we obtain an induced map
-\[ f_\ast : H_n(X,A) \to H_n(Y,B). \]
+\[ f_\ast \colon H_n(X,A) \to H_n(Y,B). \]
 and if two such $f$ and $g$ are pair-homotopic
 then $f_\ast = g_\ast$.
 
@@ -198,7 +198,7 @@ \section{Excision}
 	where the isomorphisms arise since $r$ is a pair-homotopy equivalence.
 	So $f$ is an isomorphism.
 	Similarly the map
-	\[ g : H_n(X/A, A/A) \to H_n(X/A, V/A) \]
+	\[ g \colon H_n(X/A, A/A) \to H_n(X/A, V/A) \]
 	is an isomorphism.
 
 	Now, consider the commutative diagram

tex/linalg/fourier.tex

@@ -205,7 +205,7 @@ \subsection{Fourier analysis on finite groups $Z$}
 For concreteness, suppose one wants to compute
 \[ \binom{1000}{0} + \binom{1000}{3} + \dots + \binom{1000}{999}. \]
 In that case, we can consider the function
-\[ w : \ZZ/3 \to \CC. \]
+\[ w \colon \ZZ/3 \to \CC. \]
 such that $w(0) = 1$ but $w(1) = w(2) = 0$.
 By abuse of notation we will also think of $w$
 as a function $w \colon \ZZ \surjto \ZZ/3 \to \CC$.

tex/rep-theory/applications.tex

@@ -195,7 +195,7 @@ \section{Frobenius determinant}
 	be the irreps of $G$.
 	Let's consider the map $T \colon \CC[G] \to \CC[G]$
 	which has matrix $M_G$ in the usual basis of $\CC[G]$, namely
-	\[ T : T(\{x_g\}_{g \in G}) = \sum_{g \in G} x_g \rho(g) \in \Mat(V). \]
+	\[ T \colon T(\{x_g\}_{g \in G}) = \sum_{g \in G} x_g \rho(g) \in \Mat(V). \]
 	Thus we want to examine $\det T$.
 
 	But we know that $V = \bigoplus_{i=1}^r V_i^{\oplus \dim V_i}$

tex/rep-theory/rep-alg.tex

@@ -612,7 +612,7 @@ \section{The representations of $\Mat_d(k)$}
 	\label{prob:regA_intertwine}
 	Fix an algebra $A$.
 	Find all intertwining operators
-	\[ T : \Reg(A) \to \Reg(A). \]
+	\[ T \colon \Reg(A) \to \Reg(A). \]
 	\begin{hint}
 		Right multiplication.
 	\end{hint}

tex/rep-theory/semisimple.tex

@@ -167,7 +167,7 @@ \section{Density theorem}
 	acts on $V = (V, \rho)$ by $\rho = \bigoplus_i \rho_i$.
 	Thus by \Cref{prob:reg_mat}, we can instead consider $\rho$
 	as an \emph{intertwining operator}
-	\[ \rho : \Reg(A) \to \bigoplus_{i=1}^r \Mat(V_i)
+	\[ \rho \colon \Reg(A) \to \bigoplus_{i=1}^r \Mat(V_i)
 		\cong \bigoplus_{i=1}^r V_i^{\oplus d_i}. \]
 	We will use this instead as it will be easier to work with.
 

tex/set-theory/zorn-lemma.tex

@@ -7,7 +7,7 @@ \chapter{Interlude: Cauchy's functional equation and Zorn's lemma}
 \medskip
 
 In the world of olympiad math, there's a famous functional equation that goes as follows:
-\[ f : \RR \to \RR \qquad f(x+y) = f(x) + f(y). \]
+\[ f \colon \RR \to \RR \qquad f(x+y) = f(x) + f(y). \]
 Everyone knows what its solutions are!
 There's an obvious family of solutions $f(x) = cx$.
 Then there's also this family of\dots\ uh\dots\ noncontinuous solutions (mumble grumble) pathological

tex/topology/cover-project.tex

@@ -299,7 +299,7 @@ \section{Lifting correspondence}
 \begin{proposition}
 	Let $p \colon (E,e_0) \to (B,b_0)$ be a covering projection.
 	Then we have a function of sets
-	\[ \Phi : \pi_1(B, b_0) \to p\pre(b_0) \]
+	\[ \Phi \colon \pi_1(B, b_0) \to p\pre(b_0) \]
 	by $[\gamma] \mapsto \tilde\gamma(1)$, where $\tilde\gamma$
 	is the unique lifting starting at $e_0$.
 	Furthermore,
@@ -345,7 +345,7 @@ \section{Lifting correspondence}
 	is a copy of the integers: naturally in bijection with loops in $S^1$.
 
 	You can show (and it's intuitively obvious) that the bijection
-	\[ \Phi : \pi_1(S^1) \leftrightarrow \ZZ \]
+	\[ \Phi \colon \pi_1(S^1) \leftrightarrow \ZZ \]
 	is in fact a group homomorphism if we equip $\ZZ$ with its
 	additive group structure $\ZZ$.
 	Since it's a bijection, this leads us to conclude $\pi_1(S^1) \cong \ZZ$.
@@ -474,7 +474,7 @@ \section{The algebra of fundamental groups}
 
 Let $X$ and $Y$ be topological spaces and $f \colon (X, x_0) \to (Y, y_0)$.
 Recall that we defined a group homomorphism
-\[ f_\sharp : \pi_1(X, x_0) \to \pi_1(Y_0, y_0)
+\[ f_\sharp \colon \pi_1(X, x_0) \to \pi_1(Y_0, y_0)
 	\quad\text{by}\quad
 	[\gamma] \mapsto [f \circ \gamma]. \]
 % which gave us a functor $\catname{Top}_\ast \to \catname{Grp}$.

tex/topology/top-more.tex

@@ -380,7 +380,7 @@ \section{Path-connected spaces}
 
 \begin{definition}
 	A \vocab{path} in the space $X$ is a continuous function
-	\[ \gamma : [0,1] \to X. \]
+	\[ \gamma \colon [0,1] \to X. \]
 	Its \vocab{endpoints} are the two points $\gamma(0)$ and $\gamma(1)$.
 \end{definition}