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

  • Continuous K-theory
    Overview
    Notes on continuous (Efimov) K-theory: from classical connective and non-connective algebraic K-theory, through compactly assembled categories, to Efimov’s universality theorem and the sheaf formula.
  • Gestalten
    Overview
    Notes on Gestalten and higher-categorical geometry following Scholze’s Geometry and Higher Category Theory.

Notes

  • A brief introduction to 6-functor formalisms— 2026-04-24
    Abstract
    A coherent pass through six-functor formalisms. From the structural properties one wants of cohomology, through the Liu–Zheng / Mann / Scholze span-category packaging and the Cnossen–Lenz–Linskens universal property, to Scholze’s organising observation: Poincaré duality for a morphism is precisely a dualizability statement in the $2$-category of kernels. We close with Heyer–Mann’s suave/prim weakening, Aoki’s one-step étale/proper picture, and transmutation to Gestalten.
  • 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.