From 16e70f6bb24032a8f4e823fb957736565f27e7d6 Mon Sep 17 00:00:00 2001 From: Leon Vatthauer Date: Thu, 28 Mar 2024 19:46:30 +0100 Subject: [PATCH] somewhat finish coalgebras --- tex/.vscode/ltex.dictionary.en-US.txt | 3 + tex/.vscode/settings.json | 3 + tex/catprog.sty | 8 +-- tex/main.pdf | Bin 163457 -> 168091 bytes tex/sections/02_categories.tex | 98 ++++++++++++++++++++------ 5 files changed, 87 insertions(+), 25 deletions(-) diff --git a/tex/.vscode/ltex.dictionary.en-US.txt b/tex/.vscode/ltex.dictionary.en-US.txt index e440aa4..785c4eb 100644 --- a/tex/.vscode/ltex.dictionary.en-US.txt +++ b/tex/.vscode/ltex.dictionary.en-US.txt @@ -48,3 +48,6 @@ coalgebras F-coalgebras coalgebra corecursion +bisimulation +bisimilar +bisimilarity diff --git a/tex/.vscode/settings.json b/tex/.vscode/settings.json index 7116be1..d918a3e 100644 --- a/tex/.vscode/settings.json +++ b/tex/.vscode/settings.json @@ -30,5 +30,8 @@ "\\setmintedinline[]{}": "ignore", "\\setmathfont[]{}": "ignore", "\\setmathfont{}": "ignore" + }, + "ltex.latex.environments": { + "cases": "ignore" } } \ No newline at end of file diff --git a/tex/catprog.sty b/tex/catprog.sty index 5e495df..8be2316 100644 --- a/tex/catprog.sty +++ b/tex/catprog.sty @@ -146,16 +146,16 @@ \providecommand{\bigmeet}{\bigsqcap} %% Products -\providecommand{\fst}{\oname{fst}} -\providecommand{\snd}{\oname{snd}} +\providecommand{\fst}{\pi_1} +\providecommand{\snd}{\pi_2} \providecommand{\pr}{\oname{pr}} \providecommand{\brks}[1]{\langle #1\rangle} \providecommand{\Brks}[1]{\bigl\langle #1\bigr\rangle} %% Coproducts -\providecommand{\inl}{\oname{inl}} -\providecommand{\inr}{\oname{inr}} +\providecommand{\inl}{\oname{i}_1} +\providecommand{\inr}{\oname{i}_2} \providecommand{\inj}{\oname{in}} \DeclareSymbolFont{Symbols}{OMS}{cmsy}{m}{n} diff --git a/tex/main.pdf b/tex/main.pdf index bd02ec1a9b211f170445b68b50c36cd055347fb7..a5504139bff3d98b9dcddb427baab79f0fd5e5ea 100644 GIT binary patch delta 61837 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yields another \section{Products and Coproducts} The categorical abstraction of Cartesian products is: \begin{definition}[Product] - The \emph{product} of two objects $A, B \in \obj{\CC}$ is an object that we call $A \times B$ together with morphisms $\pi_1 : A \times B \to A$ and $\pi_2 : A \times B \to B$ (the projections), where the following property holds: + The \emph{product} of two objects $A, B \in \obj{\CC}$ is an object that we call $A \times B$ together with morphisms $\fst : A \times B \to A$ and $\snd : A \times B \to B$ (the projections), where the following property holds: % https://q.uiver.app/#q=WzAsNCxbMiwyLCJBIFxcdGltZXMgQiJdLFs0LDIsIkIiXSxbMCwyLCJBIl0sWzIsMCwiQyJdLFswLDIsIlxccGlfMSJdLFswLDEsIlxccGlfMiIsMl0sWzMsMiwiZiIsMl0sWzMsMSwiZyJdLFszLDAsIlxcZXhpc3RzIVxcbGFuZ2xlIGYsIGcgXFxyYW5nbGUiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0= \[ \begin{tikzcd} && C \\ \\ A && {A \times B} && B - \arrow["{\pi_1}", from=3-3, to=3-1] - \arrow["{\pi_2}"', from=3-3, to=3-5] + \arrow["{\fst}", from=3-3, to=3-1] + \arrow["{\snd}"', from=3-3, to=3-5] \arrow["f"', from=1-3, to=3-1] \arrow["g", from=1-3, to=3-5] \arrow["{\exists!\langle f, g \rangle}", dashed, from=1-3, to=3-3] @@ -189,15 +189,15 @@ The categorical abstraction of Cartesian products is: \end{example} Dual to products are: \begin{definition}[Coproduct] - The \emph{coproduct} of two objects $A, B \in \obj{\CC}$ is an object that we call $A + B$ together with morphisms $i_1 : A \to A + B$ and $i_2 : B \to A + B$ (the injections), where the following property holds: + The \emph{coproduct} of two objects $A, B \in \obj{\CC}$ is an object that we call $A + B$ together with morphisms $\inl : A \to A + B$ and $\inr : B \to A + B$ (the injections), where the following property holds: % https://q.uiver.app/#q=WzAsNCxbMiwwLCJBK0IiXSxbMCwwLCJBIl0sWzQsMCwiQiJdLFsyLDIsIkMiXSxbMSwwLCJpXzEiXSxbMiwwLCJpXzIiLDJdLFsxLDMsImYiLDJdLFsyLDMsImciXSxbMCwzLCJcXGV4aXN0cyFbZixnXSIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ== \[ \begin{tikzcd} A && {A+B} && B \\ \\ && C - \arrow["{i_1}", from=1-1, to=1-3] - \arrow["{i_2}"', from=1-5, to=1-3] + \arrow["{\inl}", from=1-1, to=1-3] + \arrow["{\inr}"', from=1-5, to=1-3] \arrow["f"', from=1-1, to=3-3] \arrow["g", from=1-5, to=3-3] \arrow["{\exists![f,g]}", dashed, from=1-3, to=3-3] @@ -249,27 +249,27 @@ We can now characterize products (and later dually coproducts) as a commutative $1$ is a unit of $\times$, i.e.\ $A \iso A \times 1$ for any $A \in \obj{\CC}$. \end{proposition} \begin{proof} - Take $\langle id_A , !_A \rangle : A \to A \times 1$ and $\pi_1 : A \times 1 \to A$, this indeed constitutes an isomorphism, since - \[\pi_1 \comp \langle id_A , !_A \rangle = id_A \] + Take $\langle id_A , !_A \rangle : A \to A \times 1$ and $\fst : A \times 1 \to A$, this indeed constitutes an isomorphism, since + \[\fst \comp \langle id_A , !_A \rangle = id_A \] by definition and - \[\langle id_A , !_A \rangle \comp \pi_1 = \langle \pi_1 , !_A \rangle = \langle \pi_1 , \pi_2 \rangle = id_{A \times 1},\] - because $\pi_2 = !_A : A \times 1 \to 1$ by uniqueness of $!_A$. + \[\langle id_A , !_A \rangle \comp \fst = \langle \fst , !_A \rangle = \langle \fst , \snd \rangle = id_{A \times 1},\] + because $\snd = !_A : A \times 1 \to 1$ by uniqueness of $!_A$. \end{proof} \begin{proposition} $\times$ is associative, i.e.\ $A \times (B \times C) \iso (A \times B) \times C$ for any $A,B,C \in \obj{\CC}$. \end{proposition} \begin{proof} - Take \[\alpha = \langle \langle \pi_1 , \pi_1 \comp \pi_2 \rangle , \pi_2 \comp \pi_2 \rangle : A \times (B \times C) \to (A \times B) \times C\] - and \[\alpha^{-1} = \langle \pi_1 \comp \pi_1 , \langle \pi_2 \comp \pi_1 , \pi_2 \rangle \rangle : (A \times B) \times C \to A \times (B \times C).\] + Take \[\alpha = \langle \langle \fst , \fst \comp \snd \rangle , \snd \comp \snd \rangle : A \times (B \times C) \to (A \times B) \times C\] + and \[\alpha^{-1} = \langle \fst \comp \fst , \langle \snd \comp \fst , \snd \rangle \rangle : (A \times B) \times C \to A \times (B \times C).\] The rest of the proof then amounts to simply rewriting. \end{proof} \begin{proposition} $\times$ is commutative, i.e.\ $A \times B \iso B \times A$ for any $A, B \in \obj{\CC}$. \end{proposition} \begin{proof} - Take \[\langle \pi_2 , \pi_1 \rangle : A \times B \to B \times A\] - and \[\langle \pi_2 , \pi_1 \rangle : B \times A \to A \times B.\] - Indeed, $\langle \pi_2 , \pi_1 \rangle \comp \langle \pi_2 , \pi_1 \rangle = \langle \pi_2 \comp \langle \pi_2 , \pi_1 \rangle , \pi_1 \comp \langle \pi_2 , \pi_1 \rangle \rangle = \langle \pi_1 , \pi_2 \rangle = id$. + Take \[\langle \snd , \fst \rangle : A \times B \to B \times A\] + and \[\langle \snd , \fst \rangle : B \times A \to A \times B.\] + Indeed, $\langle \snd , \fst \rangle \comp \langle \snd , \fst \rangle = \langle \snd \comp \langle \snd , \fst \rangle , \fst \comp \langle \snd , \fst \rangle \rangle = \langle \fst , \snd \rangle = id$. \end{proof} Duality instantly yields the commutative monoid structure of coproducts: @@ -334,7 +334,7 @@ We can however consider structures like products and isomorphisms in the quasi-c \item $\obj{\CC \times \CD} = \obj{\CC} \times \obj{\CD}$, \item $(\CC \times \CD)((A_1, A_2),(B_1,B_2)) = \CC(A_1, B_1) \times D(A_2, B_2)$, \end{itemize} - with projection functors $\pi_1 : \CC \times \CD \to \CC, \pi_2 : \CC \times \CD \to \CD$. + with projection functors $\fst : \CC \times \CD \to \CC, \snd : \CC \times \CD \to \CD$. \end{definition} \begin{example} More examples of functors include: \begin{enumerate} \setcounter{enumi}{\value{resumeenum}} @@ -895,7 +895,7 @@ Dual to F-algebras the \emph{initial F-coalgebra} is trivial: \arrow["zip"', from=1-1, to=3-1] \arrow["{\langle hd , tl \rangle}"', from=3-1, to=3-4] \arrow["{id \times zip}", from=1-4, to=3-4] - \arrow["{\langle hd \comp \pi_1 , \langle \pi_2 , tl \comp \pi_1 \rangle \rangle}", from=1-1, to=1-4] + \arrow["{\langle hd \comp \fst , \langle \snd , tl \comp \fst \rangle \rangle}", from=1-1, to=1-4] \end{tikzcd}\] \item Similarly, for $f : A \rightarrow B$ we can define $map\;f : A^\omega \to B^\omega$ by \[hd(map\;f\;as) = f(hd\;as); \qquad tl(map\;f\;as) = map\;f\;(tl\;as),\] @@ -919,9 +919,65 @@ Dual to F-algebras the \emph{initial F-coalgebra} is trivial: \[(C,c) \overset{h}{\longrightarrow} (E,e) \overset{k}{\longleftarrow} (D,d),\] such that $h\;x = k\;y$. \end{definition} -% TODO if \nu F exists, ... -% TODO bisimilarity -% TODO proof bisimilar \to \sim, etc -% TODO examples of bisimilarity + +\begin{remark} + If $\nu F$ exists then + \[x \sim y \iff h_c\;x = h_d\;y,\] + where $h_c : (C,c) \to (\nu F, t), h_d : (D,d) \to (\nu F, t)$ are the unique morphisms into the terminal coalgebra. + The proof of ``$\Leftarrow$'' is clear, for ``$\Rightarrow$'' consider the diagram: + % https://q.uiver.app/#q=WzAsNixbMCwwLCIoQywgYykiXSxbMywwLCIoRSxlKSJdLFs2LDAsIihELGQpIl0sWzMsMywiKFxcbnUgRiwgdCkiXSxbMiwxLCJcXGNvbW0iXSxbNCwxLCJcXGNvbW0iXSxbMCwxLCJoIl0sWzIsMSwiayIsMl0sWzAsMywiaF9jIiwyXSxbMiwzLCJoX2QiXSxbMSwzLCJoX2UiLDJdXQ== + \[\begin{tikzcd} + {(C, c)} &&& {(E,e)} &&& {(D,d)} \\ + && \comm && \comm \\ + \\ + &&& {(\nu F, t)} + \arrow["h", from=1-1, to=1-4] + \arrow["k"', from=1-7, to=1-4] + \arrow["{h_c}"', from=1-1, to=4-4] + \arrow["{h_d}", from=1-7, to=4-4] + \arrow["{h_e}"', from=1-4, to=4-4] + \end{tikzcd}\] +\end{remark} +\begin{example} + As a consequence of the previous remark, we can follow that for $FX = 2 \times X^\Sigma$ the following holds: + \[x \sim y \iff x,y accept the same formal language.\] +\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 + % https://q.uiver.app/#q=WzAsOCxbMCwwLCJDIl0sWzAsMiwiRkMiXSxbMiwwLCJSIl0sWzIsMiwiRlIiXSxbNCwwLCJEIl0sWzQsMiwiRkQiXSxbMSwxLCJcXGNvbW0iXSxbMywxLCJcXGNvbW0iXSxbMCwxLCJjIl0sWzIsMywiciJdLFs0LDUsImQiXSxbMiwwLCJcXHBpXzEiLDJdLFsyLDQsIlxccGlfMiJdLFszLDEsIkZcXHBpXzEiLDJdLFszLDUsIkZcXHBpXzIiXV0= + \[\begin{tikzcd} + C && R && D \\ + & \comm && \comm \\ + FC && FR && FD + \arrow["c", from=1-1, to=3-1] + \arrow["r", from=1-3, to=3-3] + \arrow["d", from=1-5, to=3-5] + \arrow["{\fst}"', from=1-3, to=1-1] + \arrow["{\snd}", from=1-3, to=1-5] + \arrow["{F\fst}"', from=3-3, to=3-1] + \arrow["{F\snd}", from=3-3, to=3-5] + \end{tikzcd}\] + commutes. + Two elements $c \in C, d \in D$ are called \emph{bisimilar} if there exists a bisimulation $R$ such that $xRy$ holds. +\end{definition} + +\begin{proposition} + \[x,y \text{ bisimilar} \Rightarrow x \sim y\] +\end{proposition} +% TODO proof and backwards direction. + +\begin{example} + Let us examine bisimilarity of streams, i.e.\ consider the functor $FX = A \times X$ and coalgebras + \[C \overset{\langle h , t \rangle}{\longrightarrow} A \times C; \qquad D \overset{\langle h', t'\rangle}{\longrightarrow} A \times D.\] + A relation $R \subseteq C \times D$ is a bisimulation if + \begin{itemize} + \item $h\;x = h'\;y$, + \item $xRy \Rightarrow (t\;x)R(t'\;y)$. + \end{itemize} + + % TODO example proof, needs some interesting functions introduced earlier. +\end{example} + \section{Limits} \section{Colimits} \ No newline at end of file