diff --git a/tex/main.pdf b/tex/main.pdf index a550413..14344e8 100644 Binary files a/tex/main.pdf and b/tex/main.pdf differ diff --git a/tex/sections/02_categories.tex b/tex/sections/02_categories.tex index 4c4d19c..a4aa6ff 100644 --- a/tex/sections/02_categories.tex +++ b/tex/sections/02_categories.tex @@ -980,4 +980,138 @@ Dual to F-algebras the \emph{initial F-coalgebra} is trivial: \end{example} \section{Limits} -\section{Colimits} \ No newline at end of file +Limits are an abstraction of products and many other categorical concepts. +\begin{definition}[Limit] We will need to introduce some related notions first. + \begin{enumerate} + \item A \emph{diagram} in $\CC$ is a functor $D : \CD \to \CC$, where $\CD$ is small. + \item A \emph{cone} of a diagram $D : \CD \to \CC$ consists of + \begin{itemize} + \item an object $C \in \obj{\CC}$ called the \emph{apex} and + \item a family of morphisms ${(f_d : C \to Dd)}_{d\in\obj{\CD}}$ such that + % https://q.uiver.app/#q=WzAsMyxbMCwwLCJDIl0sWzAsMiwiRGQiXSxbMiwyLCJEZCciXSxbMCwxLCJmX2QiLDJdLFswLDIsImZfe2QnfSJdLFsxLDIsInUiXV0= + \[\begin{tikzcd} + C \\ + \\ + Dd && {Dd'} + \arrow["{f_d}"', from=1-1, to=3-1] + \arrow["{f_{d'}}", from=1-1, to=3-3] + \arrow["u", from=3-1, to=3-3] + \end{tikzcd}\] + commutes for every $u : d \to d'$. + \end{itemize} + \item A \emph{limit} of a diagram $D$ is a universal cone, i.e.\ a cone $(L, out_d)$ such that for every cone $(C, f_d)$ there exists a unique morphism $h : C \to L$ such that $out_d \comp h = f_d$ for all $d \in \obj{\CD}$: + % https://q.uiver.app/#q=WzAsMyxbMiwwLCJMIl0sWzIsMiwiRGQiXSxbMCwwLCJDIl0sWzAsMSwib3V0X2QiXSxbMiwwLCJoIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzIsMSwiZl9kIiwyXV0= + \[\begin{tikzcd} + C && L \\ + \\ + && Dd + \arrow["{out_d}", from=1-3, to=3-3] + \arrow["h", dashed, from=1-1, to=1-3] + \arrow["{f_d}"', from=1-1, to=3-3] + \end{tikzcd}\] + \end{enumerate} +\end{definition} + +The notion of limit can be instantiated to many interesting notions: + +\begin{definition}[Products (as limits)] + Let $\CD$ be the discrete category with 2 elements. Diagrams $D$ are pairs $(A,B)$ of objects of $\CC$, cones are pairs of morphisms + \[A \overset{f}{\longleftarrow} C \overset{g}{\longrightarrow} B\] + and limits of such diagrams are exactly products: + \[A \overset{\fst}{\longleftarrow} A\times B \overset{\snd}{\longrightarrow} B.\] +\end{definition} + +\begin{definition}[Equalizer] + Let $\CD$ be a category with two non-trivial and parallel morphisms $u,v : 1 \to 2$. Diagrams are parallel morphisms + % https://q.uiver.app/#q=WzAsMixbMCwwLCJBXzEiXSxbMiwwLCJBXzIiXSxbMCwxLCJmIiwwLHsib2Zmc2V0IjotMX1dLFswLDEsImciLDIseyJvZmZzZXQiOjF9XV0= + \[\begin{tikzcd} + {A_1} && {A_2} + \arrow["f", shift left, from=1-1, to=1-3] + \arrow["g"', shift right, from=1-1, to=1-3] + \end{tikzcd}\] + and cones are pairs of morphisms $c : C \to A_1, d C \to A_2$, such that $f \comp c = d = g \comp c$: + % https://q.uiver.app/#q=WzAsMyxbMCwyLCJBXzEiXSxbMiwyLCJBXzIiXSxbMCwwLCJDIl0sWzAsMSwiZiIsMCx7Im9mZnNldCI6LTF9XSxbMCwxLCJnIiwyLHsib2Zmc2V0IjoxfV0sWzIsMCwiYyIsMl0sWzIsMSwiZCJdXQ== + \[\begin{tikzcd} + C \\ + \\ + {A_1} && {A_2} + \arrow["f", shift left, from=3-1, to=3-3] + \arrow["g"', shift right, from=3-1, to=3-3] + \arrow["c"', from=1-1, to=3-1] + \arrow["d", from=1-1, to=3-3] + \end{tikzcd}\] + A limit of such a diagram is called an \emph{equalizer} of $f$ and $g$: + % https://q.uiver.app/#q=WzAsNCxbMiwyLCJBXzEiXSxbNCwyLCJBXzIiXSxbMiwwLCJDIl0sWzAsMiwiRSJdLFswLDEsImYiLDAseyJvZmZzZXQiOi0xfV0sWzAsMSwiZyIsMix7Im9mZnNldCI6MX1dLFsyLDAsIlxcZm9yYWxsIGMiLDJdLFsyLDEsImQiXSxbMywwLCJlIl0sWzIsMywiXFxleGlzdHMhaCIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ== + \[\begin{tikzcd} + && C \\ + \\ + E && {A_1} && {A_2} + \arrow["f", shift left, from=3-3, to=3-5] + \arrow["g"', shift right, from=3-3, to=3-5] + \arrow["{\forall c}"', from=1-3, to=3-3] + \arrow["d", from=1-3, to=3-5] + \arrow["e", from=3-1, to=3-3] + \arrow["{\exists!h}"', dashed, from=1-3, to=3-1] + \end{tikzcd}\] +\end{definition} + +% TODO equalizer in Set +% \begin{example} +% In $\Set$... +% \end{example} + +\begin{definition}[Regular Monomorphism] + A monomorphism is called \emph{regular} if it is also an equalizer. +\end{definition} + +\begin{proposition} + Every equalizer is a monomorphism and thus a regular monomorphism. +\end{proposition} +\begin{proof} + % TODO prove that eq are mono. +\end{proof} + +\begin{proposition} + $e$ is a regular monomorphism and an epimorphism $\iff$ e is an isomorphism. +\end{proposition} +\begin{proof} + % TODO prove that regular mono + epi == iso +\end{proof} + +\begin{definition}[Pullback] + % TODO pullback +\end{definition} +% TODO pullback in Set +% \begin{example} +% In $\Set$... +% \end{example} + +\begin{proposition} + Limits are unique up to isomorphism. +\end{proposition} + +\begin{definition}[Complete Category] + A category $\CC$ is called \emph{complete} if every diagram in $\CC$ has a limit. +\end{definition} + +\begin{proposition} + $\CC$ is complete iff $\CC$ has all products and equalizers, i.e.\ using products and equalizer one can construct arbitrary limits. +\end{proposition} + +\begin{definition}[Finitely Complete Category] + A category $\CC$ is called \emph{finitely complete} if every finite diagram in $\CC$ has a limit. +\end{definition} + +\begin{proposition} + The following are equivalent: + \begin{enumerate} + \item $\CC$ is finitely complete + \item $\CC$ has finite products and equalizers + \item $\CC$ has finite products and pullbacks + \item $\CC$ has a terminal object and pullbacks + \end{enumerate} +\end{proposition} + + +\section{Colimits} +% TODO colimits \ No newline at end of file