This page collects a selection of my mathematical notes: seminar notes, lecture notes, and expository material in algebra, topology, and higher category theory.

Longer write-ups are maintained as PDFs; shorter or in-progress material is kept in Markdown and indexed below.

Other Notes

Shorter or less polished notes maintained in Markdown form on this site. These include ongoing working notes and thematic collections.

Topics

    Notes

    • Sheaf Cohomology as Sheafification— 2025-12-27
      Abstract
      Sheaf cohomology is evaluation of the sheafification: \(\Gamma(X, \mathcal{F}) \coloneqq (L\mathcal{F})(X)\). Reading Mayer–Vietoris, derived Čech, pushforward, base change and the vanishing \(H^i(\mathrm{Spec}\,R, \widetilde{M}) = 0\) directly off the sheaf condition, without spectral sequences or injective resolutions.
    • Synthetic category theory and type theory— 2025-12-24
      Abstract

      This page aims to explain how type theory can be understood within the framework of synthetic category theory.

      The content of this page is derived from my questions to Tashi during the second exercise class and Tashi’s responses. I would like to express my gratitude to Tashi here.

      We focus on the following two questions:

      • Question I. How should we understand the notion of isofibration (hereafter referred to as a fibration) in synthetic category theory?
      • Question II. Do we still have a weak factorization system in this context?

      Next, we will answer these questions through the lens of type theory and the categorical perspective of synthetic category theory.

    • Stable Dold–Kan and Descent— 2025-11-27
      Abstract
      Unified exposition of the stable Dold–Kan correspondence and its application to descent theory in stable categories.
    • Basic Concepts on Higher Algebra— 2025-09-22
      Abstract
      This note introduces algebraic patterns and Segal objects, develops operads over algebraic patterns, and studies $\mathcal{O}$-monoidal categories together with $\mathcal{O}$-algebras in the Cartesian setting.