minor

exam/main.tex

@@ -33,7 +33,7 @@
   \begin{parts}
     \part Was sind F-Algebren?
     \part Was sind initiale Algebren?
-    \part Hat jede Algebra eine initiale Algebra? Wenn nein Gegenbeispiel
+    \part Hat jeder Funktor eine initiale Algebra? Wenn nein Gegenbeispiel
     \part Wie beweist man Lambeks Lemma?
     \part Wie konstruiert man F-Algebren?
   \end{parts}

tex/.vscode/ltex.dictionary.en-US.txt

@@ -68,3 +68,5 @@ Cocones
 coequalizer
 coequalizers
 pushouts
+pushout
+monic

tex/main.tex

@@ -25,6 +25,7 @@
 \pagestyle{scrheadings}
 \newcounter{resumeenum} % for resuming enumerated lists, https://tex.stackexchange.com/a/1702
 \usepackage{catprog}
+\usepackage{multicol}
 %%%%
 
 %%%% Metadata

tex/sections/02_categories.tex

@@ -940,11 +940,12 @@ Dual to F-algebras the \emph{initial F-coalgebra} is trivial:
 \end{remark}
 \begin{example}
   As a consequence of the previous remark, we can follow that for $FX = 2 \times X^\Sigma$ the following holds:
-  \[x \sim y \iff x,y accept the same formal language.\]
+  \[x \sim y \iff x,y \text{ accept the same formal language.}\]
 \end{example}
 
 \begin{definition}[Bisimulation]
-  Let $F : \Set \to \Set$ and let $(C,c), (D,d)$ be F-coalgebras. A \emph{bisimulation} is a relation $R \subseteq C \times C$ such that a coalgebra $(R, r)$ exists where
+  Let $F : \Set \to \Set$ and let $(C,c), (D,d)$ be F-coalgebras. 
+  A \emph{bisimulation} is a relation $R \subseteq C \times D$ such that a coalgebra $(R, r)$ exists where
   % https://q.uiver.app/#q=WzAsOCxbMCwwLCJDIl0sWzAsMiwiRkMiXSxbMiwwLCJSIl0sWzIsMiwiRlIiXSxbNCwwLCJEIl0sWzQsMiwiRkQiXSxbMSwxLCJcXGNvbW0iXSxbMywxLCJcXGNvbW0iXSxbMCwxLCJjIl0sWzIsMywiciJdLFs0LDUsImQiXSxbMiwwLCJcXHBpXzEiLDJdLFsyLDQsIlxccGlfMiJdLFszLDEsIkZcXHBpXzEiLDJdLFszLDUsIkZcXHBpXzIiXV0=
   \[\begin{tikzcd}
       C && R && D \\
@@ -1107,14 +1108,13 @@ The notion of limit can be instantiated to many interesting notions:
 
 \begin{definition}[Pullback]
   Let $\CD$ be a poset category illustrated by the following diagram:
-  % https://q.uiver.app/#q=WzAsMyxbMSwwLCJcXGJ1bGxldCJdLFswLDEsIlxcYnVsbGV0Il0sWzIsMSwiXFxidWxsZXQiXSxbMCwxXSxbMCwyXV0=
+  % https://q.uiver.app/#q=WzAsMyxbMCwwLCJcXGJ1bGxldCJdLFsyLDAsIlxcYnVsbGV0Il0sWzEsMSwiXFxidWxsZXQiXSxbMCwyXSxbMSwyXV0=
   \[\begin{tikzcd}
-      & \bullet \\
+      \bullet && \bullet \\
-      \bullet && \bullet
+      & \bullet
-      \arrow[from=1-2, to=2-1]
+      \arrow[from=1-1, to=2-2]
-      \arrow[from=1-2, to=2-3]
+      \arrow[from=1-3, to=2-2]
     \end{tikzcd}\]
-
   Diagrams consist of two arrows with the same codomain.
   Cones are of the form:
   % https://q.uiver.app/#q=WzAsNixbMiwwLCJDIl0sWzAsMiwiQSJdLFs0LDIsIkIiXSxbMiw0LCJEIl0sWzEsMiwiXFxjb21tIl0sWzMsMiwiXFxjb21tIl0sWzAsMSwiZiIsMl0sWzAsMiwiZyJdLFsxLDMsIm0iLDJdLFsyLDMsIm4iXSxbMCwzLCJoIiwyXV0=
@@ -1160,7 +1160,35 @@ The notion of limit can be instantiated to many interesting notions:
   Limits are unique up to isomorphism.
 \end{proposition}
 \begin{proof}
-  % TODO proof that limits are up to iso
+  Let $\CD$ be a small category and $D : \CD \to \CC$ a diagram.
+  Furthermore, let $(L, {out}_d)$ and $(L', {out'}_d)$ be two limits of this diagram.
+  Note that both $L$ and $L'$ are apexes of cones, thus they induce morphisms:
+  \begin{multicols}{2}
+    % https://q.uiver.app/#q=WzAsMyxbMCwwLCJMIl0sWzIsMCwiTCciXSxbMSwyLCJEZCJdLFswLDIsIntvdXR9X2QiLDJdLFsxLDIsIntvdXQnfV9kIl0sWzAsMSwiaSIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==
+    \[\begin{tikzcd}
+        L && {L'} \\
+        \\
+        & Dd
+        \arrow["{{out}_d}"', from=1-1, to=3-2]
+        \arrow["{{out'}_d}", from=1-3, to=3-2]
+        \arrow["i", dashed, from=1-1, to=1-3]
+      \end{tikzcd}\]
+    \columnbreak
+    % https://q.uiver.app/#q=WzAsMyxbMiwwLCJMIl0sWzAsMCwiTCciXSxbMSwyLCJEZCJdLFswLDIsIntvdXR9X2QiXSxbMSwyLCJ7b3V0J31fZCIsMl0sWzEsMCwiaV5cXG1vbmUiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=
+    \[\begin{tikzcd}
+        {L'} && L \\
+        \\
+        & Dd
+        \arrow["{{out}_d}", from=1-3, to=3-2]
+        \arrow["{{out'}_d}"', from=1-1, to=3-2]
+        \arrow["{i^\mone}", dashed, from=1-1, to=1-3]
+      \end{tikzcd}\]
+  \end{multicols}
+  Now, using the fact that the ${out}_d$ and ${out'}_d$ are jointly monic, we are done by:
+  \[{out}_d \comp i^\mone \comp i = {out'}_d \comp i = {out}_d = {out}_d \comp id_L \Rightarrow i^\mone \comp i = id_L\]
+  and
+  \[{out'}_d \comp i \comp i^\mone = {out}_d \comp i^\mone = {out'}_d = {out'}_d \comp id_{L'} \Rightarrow i \comp i^\mone = id_{L'}\]
+  Thus $L \iso L'$.
 \end{proof}
 
 \begin{definition}[Complete Category]
@@ -1311,7 +1339,49 @@ Now we can instantiate the notion of colimit to the dual notions of \autoref{sec
 \end{proof}
 
 \begin{definition}[Pushout]
-  % TODO
+  Let $\CD$ be the poset:
+  % https://q.uiver.app/#q=WzAsMyxbMSwwLCJcXGJ1bGxldCJdLFswLDEsIlxcYnVsbGV0Il0sWzIsMSwiXFxidWxsZXQiXSxbMCwxXSxbMCwyXV0=
+  \[\begin{tikzcd}
+      & \bullet \\
+      \bullet && \bullet
+      \arrow[from=1-2, to=2-1]
+      \arrow[from=1-2, to=2-3]
+    \end{tikzcd}\]
+
+  Diagrams consist of two arrows with the same domain and cocones are of the form:
+  % https://q.uiver.app/#q=WzAsNixbMiwwLCJIIl0sWzAsMiwiQSJdLFs0LDIsIkIiXSxbMiw0LCJDIl0sWzEsMiwiXFxjb21tIl0sWzMsMiwiXFxjb21tIl0sWzAsMSwibSIsMl0sWzAsMiwibiJdLFsxLDMsImYiLDJdLFsyLDMsImciXSxbMCwzLCJoIiwyXV0=
+  \[\begin{tikzcd}
+      && H \\
+      \\
+      A & \comm && \comm & B \\
+      \\
+      && C
+      \arrow["m"', from=1-3, to=3-1]
+      \arrow["n", from=1-3, to=3-5]
+      \arrow["f"', from=3-1, to=5-3]
+      \arrow["g", from=3-5, to=5-3]
+      \arrow["h"', from=1-3, to=5-3]
+    \end{tikzcd}\]
+  where $h$ is usually omitted, and instead the condition is expressed as $f \comp m = g \comp n$.
+  A colimit of such a diagram is called a \emph{pushout} and usually denoted:
+  % https://q.uiver.app/#q=WzAsNSxbMiwwLCJIIl0sWzAsMiwiQSJdLFs0LDIsIkIiXSxbMiw0LCJQIl0sWzIsNiwiQyJdLFsxLDMsInBfMSIsMl0sWzIsMywicF8yIl0sWzAsMSwiZiIsMl0sWzAsMiwiZyJdLFszLDQsIiIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFsxLDQsImEiLDIseyJjdXJ2ZSI6M31dLFsyLDQsImIiLDAseyJjdXJ2ZSI6LTN9XSxbMywwLCIiLDAseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0=
+  \[\begin{tikzcd}
+      && H \\
+      \\
+      A &&&& B \\
+      \\
+      && P \\
+      \\
+      && C
+      \arrow["{p_1}"', from=3-1, to=5-3]
+      \arrow["{p_2}", from=3-5, to=5-3]
+      \arrow["f"', from=1-3, to=3-1]
+      \arrow["g", from=1-3, to=3-5]
+      \arrow[dashed, from=5-3, to=7-3]
+      \arrow["a"', curve={height=18pt}, from=3-1, to=7-3]
+      \arrow["b", curve={height=-18pt}, from=3-5, to=7-3]
+      \arrow["\lrcorner"{anchor=center, pos=0.125, rotate=135}, draw=none, from=5-3, to=1-3]
+    \end{tikzcd}\]
 \end{definition}
 
 \begin{lemma}

tex/sections/03_constructions.tex

@@ -120,7 +120,7 @@ A consequence of \autoref{lem:finclosed} is that for any signature $\Sigma$, the
 is finitary.
 
 \begin{theorem}
-  Let $\CC$ be cocomplete and $F : \CC \to \CC$ $\sigma$-cocontinuous. Then $F$ has the initial algebra $I = \oname{colim}_{n\in\mathbb{N}} F^n 0$.
+  Let $\CC$ be cocomplete and $F : \CC \to \CC$ $\omega$-cocontinuous. Then $F$ has the initial algebra $I = \oname{colim}_{n\in\mathbb{N}} F^n 0$.
   More concretely $I$ is the colimit of the $\omega$-chain:
   % https://q.uiver.app/#q=WzAsNSxbMCwwLCIwIl0sWzIsMCwiRjAiXSxbNCwwLCJGRjAiXSxbNiwwLCJcXGxkb3RzIl0sWzIsMiwiXFxtdSBGIl0sWzAsMSwiaSJdLFsxLDIsIkZpIl0sWzIsMywiXFxsZG90cyJdLFswLDQsIlxcaW90YV8wIiwyLHsiY3VydmUiOjJ9XSxbMSw0LCJcXGlvdGFfMSIsMl0sWzIsNCwiXFxpb3RhXzIiLDIseyJjdXJ2ZSI6LTN9XSxbMyw0LCJcXGxkb3RzIiwwLHsiY3VydmUiOi0zfV1d
   \[\begin{tikzcd}