From 457f6882f63a7c92bbcce2ac9554f8df288a452f Mon Sep 17 00:00:00 2001 From: Marcello Seri Date: Sun, 31 Dec 2017 12:47:54 +0000 Subject: [PATCH] Replace hand-drawn commutative diagrams with TeX (#99) * preamble: add diagram package tikz-cd Signed-off-by: Marcello Seri * content/3.2: rewrite diagrams in tikz-cd Signed-off-by: Marcello Seri * content/3.9: rewrite diagrams in tikz-cd Signed-off-by: Marcello Seri * content/3.12: rewrite diagrams in tikz-cd Signed-off-by: Marcello Seri * 3.12: fix typo, missing \ on cat Signed-off-by: Marcello Seri * 3.12: adjust id.jpg size to improve pagination Signed-off-by: Marcello Seri * 2.4: fix typo, missing \ on cat Signed-off-by: Marcello Seri * content/3.12: use \id from category.tex instead of \rm{id} Signed-off-by: Marcello Seri * content/3.14: rewrite diagrams in tikz-cd Signed-off-by: Marcello Seri * content/3.12: correct \rho -> \lambda on commutative diagram Signed-off-by: Marcello Seri * README: add myself among the contributors Signed-off-by: Marcello Seri --- README.md | 1 + src/content/2.4/Representable Functors.tex | 2 +- src/content/3.12/Enriched Categories.tex | 90 +++++++++++++++++---- src/content/3.12/images/assoc.jpg | Bin 122098 -> 0 bytes src/content/3.12/images/compcoherence.jpg | Bin 73545 -> 0 bytes src/content/3.12/images/functorcomp.jpg | Bin 107554 -> 0 bytes src/content/3.12/images/functorid.jpg | Bin 74704 -> 0 bytes src/content/3.12/images/idcoherence.jpg | Bin 103630 -> 0 bytes src/content/3.12/images/leftid.jpg | Bin 80529 -> 0 bytes src/content/3.12/images/rightid.jpg | Bin 82839 -> 0 bytes src/content/3.14/Lawvere Theories.tex | 27 ++++++- src/content/3.14/images/equalize1.png | Bin 16528 -> 0 bytes src/content/3.14/images/equalize2.png | Bin 18092 -> 0 bytes src/content/3.14/images/liftl.png | Bin 14131 -> 0 bytes src/content/3.2/Adjunctions.tex | 13 ++- src/content/3.2/images/triangles-2.png | Bin 33177 -> 0 bytes src/content/3.2/images/triangles.png | Bin 26069 -> 0 bytes src/content/3.9/Algebras for Monads.tex | 74 ++++++++++++++--- src/content/3.9/images/talg1.png | Bin 32016 -> 0 bytes src/content/3.9/images/talg2.png | Bin 54506 -> 0 bytes src/content/3.9/images/talg31.png | Bin 44964 -> 0 bytes src/content/3.9/images/talg4.png | Bin 68914 -> 0 bytes src/content/3.9/images/talg5.png | Bin 80828 -> 0 bytes src/preamble.tex | 3 + 24 files changed, 177 insertions(+), 33 deletions(-) delete mode 100644 src/content/3.12/images/assoc.jpg delete mode 100644 src/content/3.12/images/compcoherence.jpg delete mode 100644 src/content/3.12/images/functorcomp.jpg delete mode 100644 src/content/3.12/images/functorid.jpg delete mode 100644 src/content/3.12/images/idcoherence.jpg delete mode 100644 src/content/3.12/images/leftid.jpg delete mode 100644 src/content/3.12/images/rightid.jpg delete mode 100644 src/content/3.14/images/equalize1.png delete mode 100644 src/content/3.14/images/equalize2.png delete mode 100644 src/content/3.14/images/liftl.png delete mode 100644 src/content/3.2/images/triangles-2.png delete mode 100644 src/content/3.2/images/triangles.png delete mode 100644 src/content/3.9/images/talg1.png delete mode 100644 src/content/3.9/images/talg2.png delete mode 100644 src/content/3.9/images/talg31.png delete mode 100644 src/content/3.9/images/talg4.png delete mode 100644 src/content/3.9/images/talg5.png diff --git a/README.md b/README.md index 154759c..dbde6ae 100644 --- a/README.md +++ b/README.md @@ -57,6 +57,7 @@ Thanks to the following people for contributing corrections/conversions: * Paolo G. Giarrusso * Adi Shavit * Mico from the TeX.StackExchange community +* Marcello Seri * ...and many others! License diff --git a/src/content/2.4/Representable Functors.tex b/src/content/2.4/Representable Functors.tex index 06e33aa..a2f2000 100644 --- a/src/content/2.4/Representable Functors.tex +++ b/src/content/2.4/Representable Functors.tex @@ -145,7 +145,7 @@ We've seen that, for every choice of an object $a$ in $\cat{C}$, we get a functor from $\cat{C}$ to $\Set$. This kind of structure-preserving mapping to $\Set$ is often called a \newterm{representation}. We are representing objects and morphisms of -$cat{C}$ as sets and functions in $\Set$. +$\cat{C}$ as sets and functions in $\Set$. The functor $\cat{C}(a, -)$ itself is sometimes called representable. More generally, any functor $F$ that is naturally isomorphic to diff --git a/src/content/3.12/Enriched Categories.tex b/src/content/3.12/Enriched Categories.tex index 99e6688..99650d3 100644 --- a/src/content/3.12/Enriched Categories.tex +++ b/src/content/3.12/Enriched Categories.tex @@ -89,12 +89,30 @@ The associator and the unitors must satisfy coherence conditions: \begin{figure}[H] \centering -\includegraphics[width=80mm]{images/assoc.jpg} +\begin{tikzcd}[row sep=large] +((a \otimes b) \otimes c) \otimes d +\arrow[d, "\alpha_{(a \otimes b)cd}"] +\arrow[rr, "\alpha_{abc} \otimes \id_d"] + & & (a \otimes (b \otimes c)) \otimes d + \arrow[d, "\alpha_{a(b \otimes c)d}"] \\ +(a \otimes b) \otimes (c \otimes d) +\arrow[rd, "\alpha_{ab(c \otimes d)}"] + & & a \otimes ((b \otimes c) \otimes d) + \arrow[ld, "\id_a \otimes \alpha_{bcd}"] \\ + & a \otimes (b \otimes (c \otimes d)) +\end{tikzcd} \end{figure} \begin{figure}[H] \centering -\includegraphics[width=60mm]{images/idcoherence.jpg} +\begin{tikzcd}[row sep=large] +(a \otimes i) \otimes b +\arrow[dr, "\rho_{a} \otimes \id_b"'] +\arrow[rr, "\alpha_{aib}"] + & & a \otimes (i \otimes b) + \arrow[dl, "\id_a \otimes \lambda_b"] \\ + & a \otimes b +\end{tikzcd} \end{figure} \noindent @@ -156,7 +174,7 @@ where $i$ is the tensor unit in $\cat{V}$. \begin{figure}[H] \centering -\includegraphics[width=60mm]{images/id.jpg} +\includegraphics[width=40mm]{images/id.jpg} \end{figure} \noindent @@ -165,20 +183,48 @@ $\cat{V}$: \begin{figure}[H] \centering -\includegraphics[width=80mm]{images/compcoherence.jpg} +\begin{tikzcd}[column sep=large] +(\cat{C}(c,d) \otimes \cat{C}(b,c)) \otimes \cat{C}(a,b) +\arrow[r, "\circ\otimes\id"] +\arrow[dd, "\alpha"] + & \cat{C}(b,d) \otimes \cat{C}(a,b) + \arrow[dr, "\circ"] \\ + & & \cat{C}(a,d) \\ +\cat{C}(c,d) \otimes (\cat{C}(b,c) \otimes \cat{C}(a,b)) +\arrow[r, "\id\otimes\circ"] + & \cat{C}(c,d) \otimes \cat{C}(a,c) + \arrow[ur, "\circ"] +\end{tikzcd} \end{figure} \noindent Unit laws are likewise expressed in terms of unitors: \begin{figure}[H] -\centering -\includegraphics[width=70mm]{images/rightid.jpg} -\end{figure} - -\begin{figure}[H] -\centering -\includegraphics[width=70mm]{images/leftid.jpg} + \centering + \begin{subfigure} + \centering + \begin{tikzcd}[row sep=large] + \cat{C}(a,b) \otimes i + \arrow[rr, "\id \otimes j_a"] + \arrow[dr, "\rho"] + & & \cat{C}(a,b) \otimes \cat{C}(a,a) + \arrow[dl, "\circ"] \\ + & \cat{C}(a,b) + \end{tikzcd} + \end{subfigure} + \hspace{1cm} + \begin{subfigure} + \centering + \begin{tikzcd}[row sep=large] + i \otimes \cat{C}(a,b) + \arrow[rr, "j_b \otimes \id"] + \arrow[dr, "\lambda"] + & & \cat{C}(b,b) \otimes \cat{C}(a,b) + \arrow[dl, "\circ"] \\ + & \cat{C}(a,b) + \end{tikzcd} + \end{subfigure} \end{figure} \section{Preorders} @@ -265,9 +311,9 @@ us that the distance from $a$ to $a$ is always zero. Check! Now let's talk about composition. We start with the tensor product of -two abutting hom-objects, $cat{C}(b, c) \otimes \cat{C}(a, b)$. We have defined +two abutting hom-objects, $\cat{C}(b, c) \otimes \cat{C}(a, b)$. We have defined the tensor product as the sum of the two distances. Composition is a -morphism in $\cat{V}$ from this product to $cat{C}(a, c)$. A morphism +morphism in $\cat{V}$ from this product to $\cat{C}(a, c)$. A morphism in $\cat{V}$ is defined as the greater-or-equal relation. In other words, the sum of distances from $a$ to $b$ and from $b$ to $c$ is greater than or equal to the distance from $a$ @@ -297,7 +343,16 @@ preservation of composition means that the following diagram commute: \begin{figure}[H] \centering -\includegraphics[width=80mm]{images/functorcomp.jpg} +\begin{tikzcd}[column sep=large, row sep=large] + \cat{C}(b,c) \otimes \cat{C}(a,b) + \arrow[r, "\circ"] + \arrow[d, "F_{bc} \otimes F_{ab}"] + & \cat{C}(a,c) + \arrow[d, "F_{ac}"] \\ + \cat{D}(F\ b, F\ c) \otimes \cat{D}(F\ a, F\ b) + \arrow[r, "\circ"] + & \cat{D}(F\ a, F\ c) +\end{tikzcd} \end{figure} \noindent @@ -306,7 +361,12 @@ morphisms in $\cat{V}$ that ``select'' the identity: \begin{figure}[H] \centering -\includegraphics[width=70mm]{images/functorid.jpg} +\begin{tikzcd}[row sep=large] + & i \arrow[dl, "j_a"'] \arrow[dr, "j_{F\ a}"] & \\ + \cat{C}(a,a) + \arrow[rr, "F_{aa}"] + & & \cat{D}(F\ a, F\ a) +\end{tikzcd} \end{figure} \section{Self Enrichment} diff --git 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