← 研踪 PaperTrailarXiv:2501.13229 · v4
The Yoneda embedding in simplicial type theory
首次提交 2025-01-23 05:28
版本更新 2026-11-06 17:21
雷达观测 2026-11-09 11:19
摘要
Riehl and Shulman introduced simplicial type theory (STT), a variant of homotopy type theory which aimed to study not just homotopy theory, but its fusion with category theory: $(\infty,1)$-category theory. While notoriously technical, manipulating $\infty$-categories in simplicial type theory is often easier than working with ordinary categories, with the type theory handling infinite stacks of coherences in the background. We capitalize on recent work by Gratzer et al. defining the $(\infty,1)$-category of $\infty$-groupoids in STT to define presheaf categories within STT and systematically develop their theory. In particular, we construct the Yoneda embedding, prove the universal property of presheaf categories, refine the theory of adjunctions in STT, introduce the theory of Kan extensions, and prove Quillen's Theorem A.
作者备注：Submitted to LMCS. Revised according to first round of feedback
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v4 · 2026-11-06 17:21 · 最新观测版本
The Yoneda embedding in simplicial type theory
Riehl and Shulman introduced simplicial type theory (STT), a variant of homotopy type theory which aimed to study not just homotopy theory, but its fusion with category theory: $(\infty,1)$-category theory. While notoriously technical, manipulating $\infty$-categories in simplicial type theory is often easier than working with ordinary categories, with the type theory handling infinite stacks of coherences in the background. We capitalize on recent work by Gratzer et al. defining the $(\infty,1)$-category of $\infty$-groupoids in STT to define presheaf categories within STT and systematically develop their theory. In particular, we construct the Yoneda embedding, prove the universal property of presheaf categories, refine the theory of adjunctions in STT, introduce the theory of Kan extensions, and prove Quillen's Theorem A.
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