From 32cde8595d5aeab72f5e23912e6ff638ff13654c Mon Sep 17 00:00:00 2001 From: Evan Chen Date: Sat, 18 Nov 2023 14:51:57 -0500 Subject: [PATCH] polish: "surjective head" style Closes #215 --- tex/H113/structure.tex | 2 +- tex/alg-NT/artin.tex | 6 +++--- tex/alg-geom/bezout.tex | 6 +++--- tex/cats/abelian.tex | 20 ++++++++++---------- tex/cats/categories.tex | 26 +++++++++++++------------- tex/homology/cellular.tex | 10 +++++----- tex/homology/long-exact.tex | 24 ++++++++++++------------ tex/preamble.tex | 6 +++++- tex/rep-theory/semisimple.tex | 2 +- 9 files changed, 53 insertions(+), 49 deletions(-) diff --git a/tex/H113/structure.tex b/tex/H113/structure.tex index 6328892b..f06e5af8 100644 --- a/tex/H113/structure.tex +++ b/tex/H113/structure.tex @@ -225,7 +225,7 @@ \section{Reduction to maps of free $R$-modules} \begin{center} \begin{tikzcd} & K \ar[rd, hook] \\ - R^{\oplus f} \ar[ru, two heads] \ar[rr, "T"'] && R^{\oplus d} \ar[r, two heads] & M + R^{\oplus f} \ar[ru, surjective head] \ar[rr, "T"'] && R^{\oplus d} \ar[r, surjective head] & M \end{tikzcd} \end{center} Observe that $M$ is the \emph{cokernel} of the linear map $T$, diff --git a/tex/alg-NT/artin.tex b/tex/alg-NT/artin.tex index 536d5f18..33eec561 100644 --- a/tex/alg-NT/artin.tex +++ b/tex/alg-NT/artin.tex @@ -374,8 +374,8 @@ \section{Artin reciprocity} the above theorem tells us we get a sequence of maps \begin{center} \begin{tikzcd} - I_K(\kf) \ar[r, two heads] & C_K(\kf) \ar[rd, two heads] - \ar[rr, "\left( \frac{L/K}{\bullet} \right)", two heads] + I_K(\kf) \ar[r, surjective head] & C_K(\kf) \ar[rd, surjective head] + \ar[rr, "\left( \frac{L/K}{\bullet} \right)", surjective head] && \Gal(L/K) \\ && I_K(\kf) / H(L/K, \kf) \ar[ru, "\cong", swap] & \end{tikzcd} @@ -472,7 +472,7 @@ \section{Artin reciprocity} I_K(\km) \ar[r, "\left( \frac{M/K}\bullet \right)"] \ar[rd, "\left( \frac{L/K}\bullet \right)", swap] - & \Gal(M/K) \ar[d, two heads] \\ + & \Gal(M/K) \ar[d, surjective head] \\ & \Gal(L/K) \end{tikzcd} \end{center} diff --git a/tex/alg-geom/bezout.tex b/tex/alg-geom/bezout.tex index 9791112c..1cfcfbd7 100644 --- a/tex/alg-geom/bezout.tex +++ b/tex/alg-geom/bezout.tex @@ -166,7 +166,7 @@ \section{Hilbert functions of finitely many points} \begin{tikzcd} 0 \ar[r] & \left[ S / (I \cap J) \right]^d \ar[r, hook] - & \left[ S / I \right]^d \oplus \left[ S / J \right]^d \ar[r, two heads] + & \left[ S / I \right]^d \oplus \left[ S / J \right]^d \ar[r, surjective head] & \left[ S / (I+J) \right]^d \ar[r] & 0 \\ & f \ar[r, mapsto] & (f,f) \\ && (f,g) \ar[r, mapsto] & f-g @@ -302,7 +302,7 @@ \section{Hilbert polynomials} \begin{tikzcd} 0 \ar[r] & \left[ S/I \right]^{d-1} \ar[r, hook] - & \left[ S / I \right]^d \ar[r, two heads] + & \left[ S / I \right]^d \ar[r, surjective head] & \left[ S / (I+(f)) \right]^d \ar[r] & 0 \\ & f \ar[r, mapsto] & f \cdot x_0 \\ && f \ar[r, mapsto] & f. @@ -365,7 +365,7 @@ \section{B\'ezout's theorem} \begin{tikzcd} 0 \ar[r] & \left[ S/I \right]^{d-k} \ar[r, hook] - & \left[ S / I \right]^d \ar[r, two heads] + & \left[ S / I \right]^d \ar[r, surjective head] & \left[ S / (I+(f)) \right]^d \ar[r] & 0. \end{tikzcd} \end{center} diff --git a/tex/cats/abelian.tex b/tex/cats/abelian.tex index 64e82118..ea77d710 100644 --- a/tex/cats/abelian.tex +++ b/tex/cats/abelian.tex @@ -49,7 +49,7 @@ \section{Zero objects, kernels, cokernels, and images} The \vocab{cokernel} of $f$ is a map $\coker f \colon B \surjto \Coker f$ such that \begin{center} \begin{tikzcd} - A \ar[r, "f"] \ar[rd, "0"', dashed] & B \ar[d, "\coker f", two heads] \\ + A \ar[r, "f"] \ar[rd, "0"', dashed] & B \ar[d, "\coker f", surjective head] \\ & \Coker f \end{tikzcd} \end{center} @@ -78,7 +78,7 @@ \section{Zero objects, kernels, cokernels, and images} \begin{center} \begin{tikzcd} A \ar[rd, "\exists!"'] \ar[rr, "f"] \ar[rrrd, "0"', near start, dashed] - && B \ar[rd, two heads, "\coker f"] \\ + && B \ar[rd, surjective head, "\coker f"] \\ & \Img f \ar[ur, hook] \ar[rr, "0", dashed] && \Coker f \end{tikzcd} \end{center} @@ -118,8 +118,8 @@ \section{Additive and abelian categories} \Img(f) \ar[rd, hook] && \Coker(f) \\ - & A \ar[ru, "\img(f)", two heads] \ar[rr, "f"', dashed] - && B \ar[ru, "\coker(f)"', two heads] + & A \ar[ru, "\img(f)", surjective head] \ar[rr, "f"', dashed] + && B \ar[ru, "\coker(f)"', surjective head] \end{tikzcd} \end{center} \end{definition} @@ -215,7 +215,7 @@ \section{Exact sequences} Adding in all the relevant objects, we get the commutative diagram below. \begin{center} \begin{tikzcd} - A \ar[rd, "f"] \ar[rr, dashed, "0"] \ar[dd, "\img f"', two heads] && C \\ + A \ar[rd, "f"] \ar[rr, dashed, "0"] \ar[dd, "\img f"', surjective head] && C \\ & B \ar[ru, "g"] \\ \Img f \ar[ru, hook, "\iota"] \ar[rr, dashed, "\exists!"] && \Ker g \ar["0"', dashed, uu] \ar[lu, hook'] @@ -284,10 +284,10 @@ \section{The Freyd-Mitchell embedding theorem} \begin{tikzcd} 0 \ar[r] & A \ar[r, hook, "p"] \ar[d, "\alpha", "\cong"'] - & B \ar[r, two heads, "q"] \ar[d, "\beta"] + & B \ar[r, surjective head, "q"] \ar[d, "\beta"] & C \ar[r] \ar[d, "\gamma", "\cong"'] & 0 \\ - 0 \ar[r] & A' \ar[r, hook, "p'"'] & B' \ar[r, two heads, "q'"'] & C' \ar[r] & 0 + 0 \ar[r] & A' \ar[r, hook, "p'"'] & B' \ar[r, surjective head, "q'"'] & C' \ar[r] & 0 \end{tikzcd} \end{center} and assume the top and bottom rows are exact. @@ -382,7 +382,7 @@ \section{Breaking long exact sequences} In an abelian category, consider the commutative diagram \begin{center} \begin{tikzcd} - A \ar[r, "p"] \ar[d, "\alpha"', two heads] + A \ar[r, "p"] \ar[d, "\alpha"', surjective head] & B \ar[r, "q"] \ar[d, "\beta"', hook] & C \ar[r, "r"] \ar[d, "\gamma"'] & D \ar[d, "\delta"', hook] \\ @@ -421,7 +421,7 @@ \section{Breaking long exact sequences} In an abelian category, consider the commutative diagram \begin{center} \begin{tikzcd} - A \ar[r, "p"] \ar[d, "\alpha"', two heads] + A \ar[r, "p"] \ar[d, "\alpha"', surjective head] & B \ar[r, "q"] \ar[d, "\beta"', "\cong"] & C \ar[r, "r"] \ar[d, "\gamma"'] & D \ar[r, "s"] \ar[d, "\delta"', "\cong"] @@ -441,7 +441,7 @@ \section{Breaking long exact sequences} In an abelian category, consider the diagram \begin{center} \begin{tikzcd} - & A \ar[r, "f"] \ar[d, "a"] & B \ar[r, "g", two heads] \ar[d, "b"] & C \ar[r] \ar[d, "c"] & 0 \\ + & A \ar[r, "f"] \ar[d, "a"] & B \ar[r, "g", surjective head] \ar[d, "b"] & C \ar[r] \ar[d, "c"] & 0 \\ 0 \ar[r] & A' \ar[r, hook, "f'"'] & B' \ar[r, "g'"'] & C' \end{tikzcd} \end{center} diff --git a/tex/cats/categories.tex b/tex/cats/categories.tex index 2ecfc868..bcc2f0e9 100644 --- a/tex/cats/categories.tex +++ b/tex/cats/categories.tex @@ -345,7 +345,7 @@ \section{Binary products} \begin{center} \begin{tikzcd} & X \\ - X \times Y \ar[ru, two heads, "\pi_X"] \ar[rd, two heads, "\pi_Y"'] & \\ + X \times Y \ar[ru, surjective head, "\pi_X"] \ar[rd, surjective head, "\pi_Y"'] & \\ & Y \end{tikzcd} \end{center} @@ -365,7 +365,7 @@ \section{Binary products} \begin{tikzcd} &&& X \\ A \ar[rrru, bend left, "g"'] \ar[rrrd, bend right, "h"] \ar[rr, dotted, "\exists! f"] && - X \times Y \ar[ru, two heads, "\pi_X"] \ar[rd, two heads, "\pi_Y"] & \\ + X \times Y \ar[ru, surjective head, "\pi_X"] \ar[rd, surjective head, "\pi_Y"] & \\ &&& Y \end{tikzcd} \end{center} @@ -404,9 +404,9 @@ \section{Binary products} \begin{tikzcd} & & X & & \\ \\ - P_1 \ar[rrdd, "\pi_Y^1"', two heads] \ar[rruu, "\pi_X^1", two heads] \ar[rr, "f", two heads] - && P_2 \ar[rr, "g", two heads] \ar[uu, "\pi_X^2"', two heads] \ar[dd, "\pi_Y^2", two heads] - && P_1 \ar[lluu, "\pi_X^1"', two heads] \ar[lldd, "\pi_Y^1"', two heads] \\ + P_1 \ar[rrdd, "\pi_Y^1"', surjective head] \ar[rruu, "\pi_X^1", surjective head] \ar[rr, "f", surjective head] + && P_2 \ar[rr, "g", surjective head] \ar[uu, "\pi_X^2"', surjective head] \ar[dd, "\pi_Y^2", surjective head] + && P_1 \ar[lluu, "\pi_X^1"', surjective head] \ar[lldd, "\pi_Y^1"', surjective head] \\ \\ && Y && \end{tikzcd} @@ -482,17 +482,17 @@ \section{Binary products} \begin{tikzcd} && A \ar[dd, "\exists! f"] - \ar[llddd, two heads, bend right] - \ar[lddd, two heads, bend right] - \ar[rddd, two heads, bend left] - \ar[rrddd, two heads, bend left] + \ar[llddd, surjective head, bend right] + \ar[lddd, surjective head, bend right] + \ar[rddd, surjective head, bend left] + \ar[rrddd, surjective head, bend left] && \\ &&&& \\ && P - \ar[lld, two heads] - \ar[ld, two heads] - \ar[rd, two heads] - \ar[rrd, two heads] + \ar[lld, surjective head] + \ar[ld, surjective head] + \ar[rd, surjective head] + \ar[rrd, surjective head] && \\ X_1 & X_2 && X_3 & X_4 \end{tikzcd} diff --git a/tex/homology/cellular.tex b/tex/homology/cellular.tex index f5b0b579..850862aa 100644 --- a/tex/homology/cellular.tex +++ b/tex/homology/cellular.tex @@ -123,22 +123,22 @@ \section{Cellular chain complex} \begin{tikzcd}[column sep=tiny] & \underbrace{H_3(X^2)}_{=0} \ar[d, "0"] \\ \CX{4} \ar[r, "\partial_4"] \ar[rd, "d_4", blue] - & H_3(X^3) \ar[r, two heads] \ar[d, "0"] + & H_3(X^3) \ar[r, surjective head] \ar[d, "0"] & \underbrace{H_3(X^4)}_{\cong H_3(X)} \ar[r, "0"] & \underbrace{H_3(X^4, X^3)}_{= 0} \\ & \CX{3} \ar[d, "\partial_3"] \ar[rd, "d_3", blue] && \underbrace{H_1(X^0)}_{=0} \ar[d, "0"] \\ \underbrace{H_2(X^1)}_{=0} \ar[r, "0"] - & H_2(X^2) \ar[r, hook] \ar[d, two heads] + & H_2(X^2) \ar[r, hook] \ar[d, surjective head] & \CX{2} \ar[r, "\partial_2"] \ar[rd, "d_2", blue] - & H_1(X^1) \ar[r, two heads] + & H_1(X^1) \ar[r, surjective head] & \underbrace{H_1(X^2)}_{\cong H_1(X)} \ar[r, "0"] & \underbrace{H_1(X^2, X^1)}_{=0} \\ & \underbrace{H_2(X^3)}_{\cong H_2(X)} \ar[d, "0"] && \CX{1} \ar[d, "\partial_1"] \ar[rd, "d_1", blue] \\ & \underbrace{H_2(X^3, X^2)}_{=0} & \underbrace{H_0(\varnothing)}_{=0} \ar[r, "0"] - & H_0(X^0) \ar[r, hook] \ar[d, two heads] + & H_0(X^0) \ar[r, hook] \ar[d, surjective head] & \CX{0} \ar[r, "\partial_0"] & \dots \\ &&& \underbrace{H_0(X^1)}_{\cong H_0(X)} \ar[d, "0"] \\ @@ -210,7 +210,7 @@ \section{Cellular chain complex} \begin{center} \begin{tikzcd} \underbrace{H_2(X^1)}_{=0} \ar[r, "0"] & H_2(X^2) \ar[r, hook] & - H_2(X^2, X^1) \ar[r, "\partial_2"] & H_1(X^1) \ar[r, two heads] & + H_2(X^2, X^1) \ar[r, "\partial_2"] & H_1(X^1) \ar[r, surjective head] & \underbrace{H_1(X^2)}_{\cong H_1(X)} \ar[r, "0"] & \underbrace{H_1(X^2, X^1)}_{=0} \end{tikzcd} diff --git a/tex/homology/long-exact.tex b/tex/homology/long-exact.tex index fee61cf4..5a9e3bba 100644 --- a/tex/homology/long-exact.tex +++ b/tex/homology/long-exact.tex @@ -27,17 +27,17 @@ \section{Short exact sequences and four examples} & \vdots \ar[d, "\partial_A"] & \vdots \ar[d, "\partial_B"] & \vdots \ar[d, "\partial_C"] & \\ 0 \ar[r] & A_{n+1} \ar[hook, r, "f_{n+1}"] \ar[d, "\partial_A"] - & B_{n+1} \ar[r, two heads, "g_{n+1}"] \ar[d, "\partial_B"] + & B_{n+1} \ar[r, surjective head, "g_{n+1}"] \ar[d, "\partial_B"] & C_{n+1} \ar[r] \ar[d, "\partial_C"] & 0 \\ 0 \ar[r] & A_n \ar[hook, r, "f_n"] \ar[d, "\partial_A"] - & B_n \ar[r, two heads, "g_n"] \ar[d, "\partial_B"] + & B_n \ar[r, surjective head, "g_n"] \ar[d, "\partial_B"] & C_n \ar[r] \ar[d, "\partial_C"] & 0 \\ 0 \ar[r] & A_{n-1} \ar[hook, r, "f_{n-1}"] \ar[d, "\partial_A"] - & B_{n-1} \ar[r, two heads, "g_{n-1}"] \ar[d, "\partial_B"] + & B_{n-1} \ar[r, surjective head, "g_{n-1}"] \ar[d, "\partial_B"] & C_{n-1} \ar[r] \ar[d, "\partial_C"] & 0 \\ & \vdots & \vdots & \vdots @@ -52,7 +52,7 @@ \section{Short exact sequences and four examples} For each $n$ consider \begin{center} \begin{tikzcd}[row sep=tiny] - C_n(U \cap V) \ar[r, hook] & C_n(U) \oplus C_n(V) \ar[r, two heads] & C_n(U + V) \\ + C_n(U \cap V) \ar[r, hook] & C_n(U) \oplus C_n(V) \ar[r, surjective head] & C_n(U + V) \\ c \ar[r, mapsto] & (c, -c) \\ & (c, d) \ar[r, mapsto] & c + d \end{tikzcd} @@ -71,8 +71,8 @@ \section{Short exact sequences and four examples} \begin{center} \begin{tikzcd}[row sep=large] 0 \ar[r] - & C_0(U \cap V) \ar[r, hook] \ar[d, "\eps"', two heads] - & C_0(U) \oplus C_0(V) \ar[r, two heads] \ar[d, "\eps \oplus \eps"', two heads] + & C_0(U \cap V) \ar[r, hook] \ar[d, "\eps"', surjective head] + & C_0(U) \oplus C_0(V) \ar[r, surjective head] \ar[d, "\eps \oplus \eps"', surjective head] & C_0(U+V) \ar[r] \ar[d, "\eps"'] & 0 \\ 0 \ar[r] @@ -105,8 +105,8 @@ \section{Short exact sequences and four examples} \begin{center} \begin{tikzcd} 0 \ar[r] - & C_0(A) \ar[r, hook] \ar[d, "\eps", two heads] - & C_0(X) \ar[r, two heads] \ar[d, "\eps", two heads] + & C_0(A) \ar[r, hook] \ar[d, "\eps", surjective head] + & C_0(X) \ar[r, surjective head] \ar[d, "\eps", surjective head] & C_0(X,A) \ar[r] & 0 \\ 0 \ar[r] & \ZZ \ar[r, "\id"'] & \ZZ \ar[r] & 0 \ar[r] & 0. @@ -162,8 +162,8 @@ \section{The long exact sequence of homology groups} Recall that $H_n$ is ``cycles modulo boundaries'', and consider the sub-diagram \begin{center} \begin{tikzcd} - & B_n \ar[r, "g_n", two heads] \ar[d, "\partial_B"'] & C_n \ar[d, "\partial_C"] \\ - A_{n-1} \ar[r, "f_{n-1}"', hook] & B_{n-1} \ar[r, "g_{n-1}"', two heads] & C_{n-1} + & B_n \ar[r, "g_n", surjective head] \ar[d, "\partial_B"'] & C_n \ar[d, "\partial_C"] \\ + A_{n-1} \ar[r, "f_{n-1}"', hook] & B_{n-1} \ar[r, "g_{n-1}"', surjective head] & C_{n-1} \end{tikzcd} \end{center} We need to take every cycle in $C_n$ to a cycle in $A_{n-1}$. @@ -560,9 +560,9 @@ \section{The Mayer-Vietoris sequence} \ii There is an isomorphism from $B$ to $A \oplus C$ such that the diagram \begin{center} \begin{tikzcd} - && B \ar[rd, "g", two heads] \ar[dd, leftrightarrow, "\cong"] \\ + && B \ar[rd, "g", surjective head] \ar[dd, leftrightarrow, "\cong"] \\ 0 \ar[r] & A \ar[ru, hook, "f"] \ar[rd, hook] && C \ar[r] & 0 \\ - && A \oplus C \ar[ru, two heads] + && A \oplus C \ar[ru, surjective head] \end{tikzcd} \end{center} commutes. (The maps attached to $A \oplus C$ are the obvious ones.) diff --git a/tex/preamble.tex b/tex/preamble.tex index fd0aa029..5f7252a3 100644 --- a/tex/preamble.tex +++ b/tex/preamble.tex @@ -71,7 +71,11 @@ arrow style=tikz, diagrams={>={Latex}}, tikzcd left hook/.tip={xGlyph[glyph math command=supset, swap, glyph axis = 5.7pt]}, - tikzcd right hook/.tip={xGlyph[glyph math command=supset, glyph axis = 5.7pt]} + tikzcd right hook/.tip={xGlyph[glyph math command=supset, glyph axis = 5.7pt]}, + surjective head arrow /.tip = {tikzcd to[sep=-1.5pt]tikzcd to}, + surjective head/.style={ + -surjective head arrow + } } %%fakesection Page layout diff --git a/tex/rep-theory/semisimple.tex b/tex/rep-theory/semisimple.tex index d6472557..e64a8c82 100644 --- a/tex/rep-theory/semisimple.tex +++ b/tex/rep-theory/semisimple.tex @@ -63,7 +63,7 @@ \section{Schur's lemma continued} by the $mn$ choices of compositions \begin{center} \begin{tikzcd} - V \ar[r, hook] & V^{\oplus m} \ar[r, "T"] & V^{\oplus n} \ar[r, two heads] & V + V \ar[r, hook] & V^{\oplus m} \ar[r, "T"] & V^{\oplus n} \ar[r, surjective head] & V \end{tikzcd} \end{center} where the first arrow is inclusion to the $i$th component of $V^{\oplus m}$