From e6aec55e066cba294af9ef7fe264bb21087295be Mon Sep 17 00:00:00 2001
From: ice1000 <ice1000kotlin@foxmail.com>
Date: Mon, 25 Mar 2024 16:42:19 -0400
Subject: [PATCH] WIP

---
 week8.tex | 30 ++++++++++++++++++++++++++++++
 1 file changed, 30 insertions(+)

diff --git a/week8.tex b/week8.tex
index 2b34d2a..a4bdfc6 100644
--- a/week8.tex
+++ b/week8.tex
@@ -77,6 +77,11 @@ \subsection{Statics}
   &Γ \entails{\isOfTp C X} && \quad \text{Computation typing}
 \end{align*}
 
+Similar to the lax framework, the judgment form is stratified into two categories,
+but the context used is always a context of value types.
+Therefore, we only need to (and only intend to) consider value substitutions
+into both of these judgment forms.
+
 The rules for values are given by the typing rules in~\cref{fig:cbpv-values},
 which only includes identity and the introduction of value types.
 
@@ -187,6 +192,31 @@ \subsection{Statics}
     to the CBPV framework, which possibly includes dependent function types and dependent pairs.
 \end{description}
 
+\begin{remark}
+We refrain from discussing the dynamics of CBPV in this note,
+because the standard presentation of the dynamics of CBPV uses the CK-Machine,
+which is different from our usual presentation of the semantics of type theories
+based on reductions.
+\end{remark}
+
+\subsection{Categorical semantics}
+\newcommand{\Ctx}{\mathsf{Ctx}}
+\newcommand{\CatComp}{\mathsf{Comp}}
+
+Under the standard categorical interpretation of type theories,
+we may interpret the CBPV language as follows:
+
+\begin{itemize}
+\item We interpret the category of context of value types, $\Ctx$,
+  as a bicartesian monoidal category, i.e. with finite products and coproducts.
+  Morphisms in this category correspond to value substitutions,
+  and we identify value types with contexts of length $1$.
+\item The judgment for computations uses value contexts, therefore they have to
+  be \emph{indexed} by $\Ctx$, we denote it as $\CatComp$.
+  Formally speaking, $\CatComp$ are (strict) $\Ctx$-indexed categories,
+  with all the base change functors being identity on objects.
+\end{itemize}
+
 \subsection{Compiling lax to CBPV}
 
 \end{document}
\ No newline at end of file