On Hofmann–Streicher universes

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS Mathematical Structures in Computer Science Pub Date : 2024-09-19 DOI:10.1017/s0960129524000203
Steve Awodey
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Abstract

We take another look at the construction by Hofmann and Streicher of a universe $(U,{\mathcal{E}l})$ for the interpretation of Martin-Löf type theory in a presheaf category $[{{{\mathbb{C}}}^{\textrm{op}}},\textsf{Set}]$ . It turns out that $(U,{\mathcal{E}l})$ can be described as the nerve of the classifier $\dot{{\textsf{Set}}}^{\textsf{op}} \rightarrow{{\textsf{Set}}}^{\textsf{op}}$ for discrete fibrations in $\textsf{Cat}$ , where the nerve functor is right adjoint to the so-called “Grothendieck construction” taking a presheaf $P :{{{\mathbb{C}}}^{\textrm{op}}}\rightarrow{\textsf{Set}}$ to its category of elements $\int _{\mathbb{C}} P$ . We also consider change of base for such universes, as well as universes of structured families, such as fibrations.
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关于霍夫曼-斯特赖歇尔宇宙
我们再来看一下霍夫曼和施特莱歇尔构建的宇宙 $(U,{\mathcal{E}l})$ 对于马丁-洛夫类型理论在预设范畴 $[{{{\mathbb{C}}^{\textrm{op}}},\textsf{Set}]$ 中的解释。事实证明,$(U,{{mathcal{E}l}})$ 可以被描述为分类器 $\dot{{\textsf{Set}}}^{textsf{op}} 的神经。\其中神经函子与所谓的 "格罗thendieck 构造 "是右邻接的,所谓的 "格罗thendieck 构造 "是将预叶 $P :{{{textrm{op}}}^{textsf{Set}}$ 取为其元素类别 $int _{{mathbb{C}} 。P$ .我们还考虑了这类宇宙的基底变化,以及结构族的宇宙,如纤维。
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来源期刊
Mathematical Structures in Computer Science
Mathematical Structures in Computer Science 工程技术-计算机:理论方法
CiteScore
1.50
自引率
0.00%
发文量
30
审稿时长
12 months
期刊介绍: Mathematical Structures in Computer Science is a journal of theoretical computer science which focuses on the application of ideas from the structural side of mathematics and mathematical logic to computer science. The journal aims to bridge the gap between theoretical contributions and software design, publishing original papers of a high standard and broad surveys with original perspectives in all areas of computing, provided that ideas or results from logic, algebra, geometry, category theory or other areas of logic and mathematics form a basis for the work. The journal welcomes applications to computing based on the use of specific mathematical structures (e.g. topological and order-theoretic structures) as well as on proof-theoretic notions or results.
期刊最新文献
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