Skip to content

Commit

Permalink
Small remark on cubical canonicity
Browse files Browse the repository at this point in the history
  • Loading branch information
ice1000 committed Mar 4, 2023
1 parent 8a5f125 commit f5dac11
Show file tree
Hide file tree
Showing 2 changed files with 14 additions and 0 deletions.
7 changes: 7 additions & 0 deletions chapters/hott.tex
Original file line number Diff line number Diff line change
Expand Up @@ -344,6 +344,8 @@ \section{立方类型论}
生成的立方体作为基础, 得到了ABCFHL模型
\textbf{积立方类型论}.

立方类型论有自己的闭典范性定义, 在 2018 年由
Huber~\cite{huber:2018:canonicity} 提出并证明.
2021年, Sterling等人发展了范畴论工具, 证明了立方类型论的
典范性. 在他的学位论文~\cite{sterling:2021:thesis}中,
Sterling进一步描述了这些范畴论工具, 并且指出语法与语义研究
Expand Down Expand Up @@ -394,5 +396,10 @@ \subsection{立方类型论简介}

可以看到, 将相等类型使用道路空间的方式定义, 使得证明更加简洁直观.
同时函数外延性在用泛等公理证明时十分复杂, 而使用立方类型论则是最基础的结论.

立方类型论版本的闭典范性, 又叫立方闭典范性, 是指将传统意义上的闭典范性中的
空语境换成完全由类型 \(\mathbb I\) 拼成的语境
\(x_1:\mathbb I, x_2:\mathbb I, \cdots, x_n : \mathbb I\).

希望继续了解立方类型论的读者可以
参考\href{https://1lab.dev/1Lab.intro.html}{这篇文章}~\cite{amelia:2023:1lab}.
7 changes: 7 additions & 0 deletions history.bib
Original file line number Diff line number Diff line change
Expand Up @@ -458,6 +458,13 @@ @misc{henry:2019:constructive
publisher = {arXiv},
year = {2019}
}
@article{huber:2018:canonicity,
doi = {10.1007/s10817-018-9469-1},
journal = {J Autom Reasoning},
author = {Simon Huber},
title = {Canonicity for Cubical Type Theory},
year = {2018},
}
@article{cartmell:1986:contextualcat,
title = {Generalised algebraic theories and contextual categories},
journal = {Annals of Pure and Applied Logic},
Expand Down

0 comments on commit f5dac11

Please sign in to comment.