https://ou-liu-red-sugar.github.io/notes/Recent content in Notes on Ou LiuHugo -- 0.146.0en-usFri, 24 Apr 2026 00:00:00 +0000https://ou-liu-red-sugar.github.io/notes/notes/basic-concepts-on-higher-algebra/Mon, 22 Sep 2025 00:00:00 +0000https://ou-liu-red-sugar.github.io/notes/notes/basic-concepts-on-higher-algebra/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.https://ou-liu-red-sugar.github.io/notes/notes/a-brief-introduction-to-6-functor-formalisms/Fri, 24 Apr 2026 00:00:00 +0000https://ou-liu-red-sugar.github.io/notes/notes/a-brief-introduction-to-6-functor-formalisms/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&#39;s organising observation: Poincaré duality for a morphism is *precisely* a dualizability statement in the $2$-category of kernels. We close with Heyer–Mann&#39;s suave/prim weakening, Aoki&#39;s one-step étale/proper picture, and transmutation to Gestalten.https://ou-liu-red-sugar.github.io/notes/notes/sheaf-cohomology-as-sheafification/Sat, 27 Dec 2025 00:00:00 +0000https://ou-liu-red-sugar.github.io/notes/notes/sheaf-cohomology-as-sheafification/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.https://ou-liu-red-sugar.github.io/notes/notes/synthetic-category-theory-and-type-theory/Wed, 24 Dec 2025 00:00:00 +0000https://ou-liu-red-sugar.github.io/notes/notes/synthetic-category-theory-and-type-theory/<p>This page aims to explain how <strong>type theory</strong> can be understood within the framework of <strong>synthetic category theory</strong>.</p> <p>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.</p> <p>We focus on the following two questions:</p> <ul> <li><strong>Question I.</strong> How should we understand the notion of <strong>isofibration</strong> (hereafter referred to as a <em>fibration</em>) in synthetic category theory?</li> <li><strong>Question II.</strong> Do we still have a <strong>weak factorization system</strong> in this context?</li> </ul> <p>Next, we will answer these questions through the lens of type theory and the categorical perspective of synthetic category theory.</p>https://ou-liu-red-sugar.github.io/notes/notes/stable-doldkan-and-descent/Thu, 27 Nov 2025 00:00:00 +0000https://ou-liu-red-sugar.github.io/notes/notes/stable-doldkan-and-descent/Unified exposition of the stable Dold–Kan correspondence and its application to descent theory in stable categories.