Work on slides

slides/bibliography.bib

@@ -1,3 +1,17 @@
+@article{eqlm,
+  title    = {Equational Lifting Monads},
+  journal  = {Electronic Notes in Theoretical Computer Science},
+  volume   = {29},
+  pages    = {22},
+  year     = {1999},
+  note     = {CTCS '99, Conference on Category Theory and Computer Science},
+  issn     = {1571-0661},
+  doi      = {https://doi.org/10.1016/S1571-0661(05)80303-2},
+  url      = {https://www.sciencedirect.com/science/article/pii/S1571066105803032},
+  author   = {Anna Bucalo and Carsten Führmann and Alex Simpson},
+  abstract = {We introduce the notion of an “equational lifting monad” : a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category of the former fully embeds in the partial category of the latter. Thus equational lifting monads precisely capture the (partial) equational properties of partial map classifiers. The representation theorem also provides a tool for transferring non-equational properties of partial map classifiers to equational lifting monads.}
+}
+
 @article{moggi,
   author     = {Moggi, Eugenio},
   title      = {Notions of Computation and Monads},
@@ -11,7 +25,7 @@ issn = {0890-5401},
   url        = {https://doi.org/10.1016/0890-5401(91)90052-4},
   doi        = {10.1016/0890-5401(91)90052-4},
   journal    = {Inf. Comput.},
-month = {jul},
+  month      = {7},
   pages      = {55–92},
   numpages   = {38}
 }
@@ -34,17 +48,3 @@ month = {1},
   pages      = {223–259},
   numpages   = {37}
 }
-
-@article{eqlm,
-title = {Equational Lifting Monads},
-journal = {Electronic Notes in Theoretical Computer Science},
-volume = {29},
-pages = {22},
-year = {1999},
-note = {CTCS '99, Conference on Category Theory and Computer Science},
-issn = {1571-0661},
-doi = {https://doi.org/10.1016/S1571-0661(05)80303-2},
-url = {https://www.sciencedirect.com/science/article/pii/S1571066105803032},
-author = {Anna Bucalo and Carsten Führmann and Alex Simpson},
-abstract = {We introduce the notion of an “equational lifting monad” : a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category of the former fully embeds in the partial category of the latter. Thus equational lifting monads precisely capture the (partial) equational properties of partial map classifiers. The representation theorem also provides a tool for transferring non-equational properties of partial map classifiers to equational lifting monads.}
-}

slides/main.tex

@@ -160,6 +160,7 @@ Leon Vatthauer%\inst{1}
 \usepackage{tikz}
 \usepackage{tikz-cd}
 \usetikzlibrary{shapes.callouts}
+\usepackage{mathpartir}
 
 \usepackage{xparse}
 

slides/sections/01_abstracting.tex

@@ -4,7 +4,7 @@
 Goal: interpret an effectul programming language in a category $\mathcal{C}$
 
 \begin{itemize}
-  \item<2-> Take a Monad $T$ on $\mathcal{C}$, then values are denoted by objects $A$ and programs by $TA$
+  \item<2-> Take a Monad $T$ on $\mathcal{C}$, then values are denoted by objects $A$ and computations by $TA$
   \item<3-> Programs form a category $\mathcal{C}_T$ with $\mathcal{C}_T(X,Y) := \mathcal{C}(X, TY)$
 \end{itemize}
 
@@ -32,7 +32,7 @@ What properties should a partiality monad $T$ have?
 
 \end{frame}
 
-\newcommand{\tdom}{\text{dom}}
+\newcommand{\tdom}{\text{dom}\;}
 
 \begin{frame}[t, fragile]{Capturing Partiality Categorically}{Restriction Categories~\cite{restriction}}
 \begin{definition}<1->
@@ -46,15 +46,17 @@ What properties should a partiality monad $T$ have?
   for any $X, Y, Z \in \vert\mathcal{C}\vert, f : X \rightarrow KY, g : X \rightarrow KZ, h: Y \rightarrow KZ$.
 \end{definition}
 
+\onslide<2->
 Intuitively $\tdom f$ captures the domain of definiteness of $f$.
 
-\begin{block}{Remark}<2->
+\begin{block}{Remark}<3->
   Every category has a trivial restriction structure $\tdom f = id$, we call categories with a non-trivial restriction structure \textit{restriction categories}.
 \end{block}
 \end{frame}
 
 \begin{frame}[t, fragile]{Capturing Partiality Categorically}{Equational Lifting Monads~\cite{eqlm}}
   The following criterion is sufficient for guaranteeing that the kleisli category is a restriction category:
+  \pause
   \begin{definition}
     A commutative monad $T$ is called an \textit{equational lifting monad} if the following diagram commutes:
 
@@ -77,9 +79,24 @@ Intuitively $\tdom f$ captures the domain of definiteness of $f$.
 % TODO is the maybe monad an equational lifting monad? Ask sergey
 
 \begin{frame}[t, fragile]{The Maybe Monad}
-\begin{itemize}
-  \item Short definition
-  \item is equational lifting monad
+\begin{itemize}[<+->]
+  \item $MX = X + 1$
+  \item on a distributive category the maybe monad is strong and commutative:
+    \[ \tau_{X,Y} := X \times (Y + 1) \overset{dstr}{\longrightarrow} (X \times Y) + (X \times 1) \overset{id+1}{\longrightarrow} (X \times Y) + 1 \]
+  \item and the following diagram commutes (i.e. it's an equational lifting monad):
+    % https://q.uiver.app/#q=WzAsNCxbMCwwLCJYKzEiXSxbMywwLCIoWCsxKVxcdGltZXMoWCsxKSJdLFszLDIsIigoWCsxKVxcdGltZXMgWCkgKygoWCsxKVxcdGltZXMgMSkiXSxbMyw0LCIoKFgrMSlcXHRpbWVzIFgpKzEiXSxbMCwxLCJcXERlbHRhIl0sWzEsMiwiZHN0ciJdLFsyLDMsImlkKyEiXSxbMCwzLCJcXGxhbmdsZSBpbmwsaWRcXHJhbmdsZSArICEiLDJdXQ==
+    \[\begin{tikzcd}
+      {X+1} &&& {(X+1)\times(X+1)} \\
+      \\
+      &&& {((X+1)\times X) +((X+1)\times 1)} \\
+      \\
+      &&& {((X+1)\times X)+1}
+      \arrow["\Delta", from=1-1, to=1-4]
+      \arrow["dstr", from=1-4, to=3-4]
+      \arrow["{id+!}", from=3-4, to=5-4]
+      \arrow["{\langle inl,id\rangle + !}"', from=1-1, to=5-4]
+    \end{tikzcd}\]
+  
 \end{itemize}
 \end{frame}
 
@@ -87,13 +104,18 @@ Intuitively $\tdom f$ captures the domain of definiteness of $f$.
 
 \begin{frame}[t, fragile]{The Delay Monad}
 \begin{itemize}
-  \item Definition
-  \item Strong-Bisimilarity
-  \item Weak-Bisimilarity (Monad?)
+  \item Recall the delay codatatype:
+    
+    \[\mprset{fraction={===}}
+    \inferrule {\;} {now\; x : DX}\hskip 2cm
+    \inferrule {c : DX} {later\; c : DX}\]
+
+  \item Categorically: $DX = \nu \gamma. X + \gamma$
+  \item $D$ (if it exists) is a strong and commutative monad (on a cartesian, cocartesian, distributive category)
 \end{itemize}
 \end{frame}
 
-\begin{frame}[t, fragile]{Iteration}
+\begin{frame}[t, fragile]{Partiality from iteration}
 \begin{itemize}
   \item Elgot-Algebras
   \item Free Elgot-Algebras yield monad K