revert: d470fdf

Revert "fix: replace coloneqq with coloneq"

This reverts commit d470fdf.

Apparently, coloneq renders as :- on older versions, i.e. GitHub CI is
behind

tex/H113/grp-intro.tex

@@ -799,7 +799,7 @@ \section{\problemhead}
 
 		To give a construction from $n = 1009$,
 		note that $D_{2018}$ can be thought of the symmetries of a $1009$-gon.
-		If one labels the vertices of the $1009$-gon by $S \coloneq \{1,2,\dots,1009\}$,
+		If one labels the vertices of the $1009$-gon by $S \coloneqq \{1,2,\dots,1009\}$,
 		then elements of $D_{2018}$ induces permutations on $S$,
 		and the set of permutations achieved is the desired subgroup.
 	\end{sol}

tex/H113/ring-flavors.tex

@@ -863,7 +863,7 @@ \section{\problemhead}
 		and let $a + b \sqrt{2017}$ be a nonzero element of it.
 		Then $I$ also contains $(a^2-2017b^2)$.
 		That means when taking modulo $I$ we may take modulo the integer
-		$n \coloneq |a^2-2017b^2| \neq 0$.
+		$n \coloneqq |a^2-2017b^2| \neq 0$.
 
 		So every element in $R$ is equivalent modulo $I$
 		to an element of the form $x + y \sqrt{2017}$,

tex/diffgeo/stokes.tex

@@ -220,7 +220,7 @@ \section{Cells}
 	[How to integrate differential $k$-forms]
 	Take a differential $k$-form $\alpha$ and a $k$-cell $c \colon [0,1]^k \to V$.
 	Define the integral $\int_c \alpha$ using the pullback
-	\[ \int_c \alpha \coloneq \int_{[0,1]^k} c^\ast \alpha. \]
+	\[ \int_c \alpha \coloneqq \int_{[0,1]^k} c^\ast \alpha. \]
 \end{definition}
 Since $c^\ast \alpha$ is a $k$-form on the $k$-dimensional unit box,
 it can be written as $f(x_1, \dots, x_n) \; dx_1 \wedge \dots \wedge dx_n$,
@@ -716,7 +716,7 @@ \section{Back to Earth: A comparison to what you learned in vector calculus}
 	Specifically, if $\mathbf{v} = x \ee_1 + y \ee_2 + z \ee_3$
 	and $\mathbf{w} = x' \ee_1 + y' \ee_2 + z' \ee_3$,
 	the definition of cross products taught in school is
-	\[ \mathbf{v} \times \mathbf{w} \coloneq (yz'-y'z) \ee_1 + (zx'-xz') \ee_2 + (xy'-x'y) \ee_3. \]
+	\[ \mathbf{v} \times \mathbf{w} \coloneqq (yz'-y'z) \ee_1 + (zx'-xz') \ee_2 + (xy'-x'y) \ee_3. \]
 	Where does this come from?
 	The answer is that $\star(\mathbf{v} \vee \mathbf{w})$:
 	\begin{align*}
@@ -754,7 +754,7 @@ \section{Back to Earth: A comparison to what you learned in vector calculus}
 	On the 18.02 side, if we have
 	\[ \mathbf{F} = F_1 \ee_1 + F_2 \ee_2 + F_3 \ee_3 \]
 	then the 18.02 definition of curl is that
-	\[ \opname{curl}(\mathbf{F}) \coloneq \nabla \times \mathbf{F} \coloneq
+	\[ \opname{curl}(\mathbf{F}) \coloneqq \nabla \times \mathbf{F} \coloneqq
 		\left( \pdv{F_3}{y} - \pdv{F_2}{z} \right) \ee_1
 		+ \left( \pdv{F_1}{z} - \pdv{F_3}{x} \right) \ee_2
 		+ \left( \pdv{F_2}{x} - \pdv{F_1}{y} \right) \ee_3
@@ -783,7 +783,7 @@ \section{Back to Earth: A comparison to what you learned in vector calculus}
 	so this completes the correspondence between the 18.02 notation and the Napkin notation.
 
 	\ii In 18.02 terminology, the divergence $\opname{div}$ is defined by
-	\[ \opname{div}(\mathbf{F}) \coloneq \nabla \cdot \mathbf{F} \coloneq
+	\[ \opname{div}(\mathbf{F}) \coloneqq \nabla \cdot \mathbf{F} \coloneqq
 		\pdv{F_1}{x} + \pdv{F_2}{y} + \pdv{F_3}{z} \]
 	which is a scalar-valued function for input points $p \in \RR^3$.
 	We let you do this one in \Cref{prob:div1802}.
@@ -852,7 +852,7 @@ \section{Back to Earth: A comparison to what you learned in vector calculus}
 	That is, let $\mathbf{F} \colon \RR^3 \to \RR^3$ be a vector field,
 	and let $\alpha$ be the $2$-form corresponding to it in Napkin version.
 	Show that the scalar-valued function defined by
-	\[ \opname{div}(\mathbf{F}) \coloneq \nabla \cdot \mathbf{F} \coloneq
+	\[ \opname{div}(\mathbf{F}) \coloneqq \nabla \cdot \mathbf{F} \coloneqq
 		\pdv{F_1}{x} + \pdv{F_2}{y} + \pdv{F_3}{z} \]
 	coincides with evaluation at the $3$-form $d\alpha$.
 	\label{prob:div1802}

tex/macros.tex

@@ -67,7 +67,7 @@
 \DeclareMathOperator{\diam}{diam}
 \DeclareMathOperator{\ord}{ord}
 %\newcommand{\defeq}{\overset{\mathrm{def}}{=}}
-\newcommand{\defeq}{\coloneq}
+\newcommand{\defeq}{\coloneqq}
 
 %From the USAMO .tex files
 \newcommand{\st}{^{\text{st}}}

tex/rep-theory/characters.tex

@@ -82,7 +82,7 @@ \section{The dual space modulo the commutator}
 to be the $k$-vector subspace spanned by $xy-yx$ for $x,y \in A$.
 Thus $[A,A]$ is contained in the kernel of each $\chi_V$.
 \begin{definition}
-	The space $A\ab \coloneq A / [A,A]$ is called the \vocab{abelianization} of $A$.
+	The space $A\ab \coloneqq A / [A,A]$ is called the \vocab{abelianization} of $A$.
 	Each character of $A$ can be viewed as a map $A\ab \to k$, i.e.\ an element of $(A\ab)^\vee$.
 \end{definition}
 \begin{example}

tex/set-theory/zorn-lemma.tex

@@ -381,7 +381,7 @@ \section{\problemhead}
 	But then $f(2a) = f(a)+f(a) = 0$, so $f$ is not injective.
 
 	For the rest, fix a Hamel basis
-	\[ E = \{e_\alpha \mid \alpha \in S \coloneq \{ 0,1,2,\dots\} \dots \}. \]
+	\[ E = \{e_\alpha \mid \alpha \in S \coloneqq \{ 0,1,2,\dots\} \dots \}. \]
 	Here $S$ is an uncountable set of ordinals.
 	WLOG, $e_0 = 1$ and $e_\alpha > 0$ for all $\alpha \in S$.
 	Then $f$ is uniquely determined by the value of $f(e_\alpha)$ for each $\alpha \in S$.

tex/topology/compactness.tex

@@ -600,7 +600,7 @@ \section{\problemhead}
 		Assuming $M$ is not compact, construct an
 		unbounded continuous function $F \colon M \to \RR$.
 		Once such a function $F$ is defined, the metric
-		\[ d'(x, y) \coloneq d(x, y) + \abs{F(x) - F(y)} \]
+		\[ d'(x, y) \coloneqq d(x, y) + \abs{F(x) - F(y)} \]
 		will establish the contrapositive of the problem.
 	\end{hint}
 
@@ -612,7 +612,7 @@ \section{\problemhead}
 		The main step is to construction
 		an unbounded continuous function $F \colon M \to \RR$.
 		Once such a function $F$ is defined, the metric
-		\[ d'(x, y) \coloneq d(x, y) + \abs{F(x) - F(y)} \]
+		\[ d'(x, y) \coloneqq d(x, y) + \abs{F(x) - F(y)} \]
 		will solve the problem.
 
 		So, let $a_1$, $a_2$, \dots\ be a sequence in $M$

tex/topology/metric-prop.tex

@@ -337,7 +337,7 @@ \section{\problemhead}
 		Start with any point $x_0$.
 		Let $x_1 = T(x_0)$, $x_2 = T(x_1)$, $x_3 = T(x_2)$, \dots, and so on.
 		We contend that $(x_0, x_1, x_2, \dots)$ is a Cauchy sequence.
-		Indeed, if we let $r \coloneq 0.999 < 1$ and $c \coloneq d(x_0, x_1)$, then
+		Indeed, if we let $r \coloneqq 0.999 < 1$ and $c \coloneqq d(x_0, x_1)$, then
 		\begin{align*}
 			d(x_1, x_2) &< r \cdot c \\
 			d(x_2, x_3) &< r^2 \cdot c \\