https://ou-liu-red-sugar.github.io/en/Recent content on Ou Liu · NotesHugo -- 0.152.2en-USMon, 17 Aug 2026 00:00:00 +0000https://ou-liu-red-sugar.github.io/en/notes/basic-concepts-on-higher-algebra/Mon, 22 Sep 2025 00:00:00 +0000https://ou-liu-red-sugar.github.io/en/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/en/invest/framework/Mon, 17 Aug 2026 00:00:00 +0000https://ou-liu-red-sugar.github.io/en/invest/framework/A public summary of the fundamental-analysis framework currently being tested in live practice.https://ou-liu-red-sugar.github.io/en/invest/playbook/Mon, 17 Aug 2026 00:00:00 +0000https://ou-liu-red-sugar.github.io/en/invest/playbook/The current operating protocol for an investing research framework under live testing.https://ou-liu-red-sugar.github.io/en/invest/circle-of-competence/Mon, 18 May 2026 00:00:00 +0000https://ou-liu-red-sugar.github.io/en/invest/circle-of-competence/Where I have edge, where I&#39;m learning, where I should fast-skip.https://ou-liu-red-sugar.github.io/en/invest/watchlist/Mon, 18 May 2026 00:00:00 +0000https://ou-liu-red-sugar.github.io/en/invest/watchlist/Candidate pool — names not held but in active monitoring, with explicit trigger conditions.https://ou-liu-red-sugar.github.io/en/notes/sheaf-cohomology-as-sheafification/Sat, 27 Dec 2025 00:00:00 +0000https://ou-liu-red-sugar.github.io/en/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/en/notes/synthetic-category-theory-and-type-theory/Wed, 24 Dec 2025 00:00:00 +0000https://ou-liu-red-sugar.github.io/en/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/en/notes/stable-doldkan-and-descent/Thu, 27 Nov 2025 00:00:00 +0000https://ou-liu-red-sugar.github.io/en/notes/stable-doldkan-and-descent/Unified exposition of the stable Dold–Kan correspondence and its application to descent theory in stable categories.https://ou-liu-red-sugar.github.io/en/notebook/call-payoff/Mon, 01 Jan 0001 00:00:00 +0000https://ou-liu-red-sugar.github.io/en/notebook/call-payoff/Separate the value of the contract at expiry from the result after paying the premium.https://ou-liu-red-sugar.github.io/en/about/Mon, 01 Jan 0001 00:00:00 +0000https://ou-liu-red-sugar.github.io/en/about/Personal notes on investing research, learning, and archived mathematics.