https://ou-liu-red-sugar.github.io/Recent content on Ou LiuHugo -- 0.146.0en-usSat, 25 Apr 2026 00:00:00 +0000https://ou-liu-red-sugar.github.io/notes/notes/continuous-k-theory/algebraic-k-theory/Fri, 24 Apr 2026 00:00:00 +0000https://ou-liu-red-sugar.github.io/notes/notes/continuous-k-theory/algebraic-k-theory/Connective and non-connective algebraic K-theory via cospans and the Waldhausen S-construction; Verdier sequences in stable and non-stable settings; the non-connective extension via iterated Calkin categories.https://ou-liu-red-sugar.github.io/notes/notes/gestalten/presentable-categories/Sun, 28 Dec 2025 00:00:00 +0000https://ou-liu-red-sugar.github.io/notes/notes/gestalten/presentable-categories/This note reviews the basic theory of presentable categories. We focus on κ-compact objects, Ind-completions, accessible categories, and their role in organizing large categories via filtered colimits.https://ou-liu-red-sugar.github.io/notes/notes/continuous-k-theory/compactly-assembled-categories/Fri, 24 Apr 2026 00:00:00 +0000https://ou-liu-red-sugar.github.io/notes/notes/continuous-k-theory/compactly-assembled-categories/Dualizable stable categories from the compactly assembled viewpoint: R-linear categories, dualizability in $\mathsf{Pr}_{\mathrm{st}}^L$, compactly exhaustible objects, the Lurie–Clausen characterisation, and the symmetric monoidal structure of $\mathsf{Pr}^L_{\mathrm{ca}}$.https://ou-liu-red-sugar.github.io/notes/notes/gestalten/presentable-n-categories/Mon, 29 Dec 2025 00:00:00 +0000https://ou-liu-red-sugar.github.io/notes/notes/gestalten/presentable-n-categories/This note introduces the notion of presentable n-categories.https://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/continuous-k-theory/continuous-efimov-k-theory/Fri, 24 Apr 2026 00:00:00 +0000https://ou-liu-red-sugar.github.io/notes/notes/continuous-k-theory/continuous-efimov-k-theory/Extending algebraic K-theory from $\mathsf{Cat}^{\mathrm{rex}}$ to $\mathsf{Pr}^L_{\mathrm{ca}}$: the continuous Calkin category, Verdier cofibre sequences in the large setting, Efimov&#39;s definition of continuous K-theory, its basic properties, and a sketch of the universal property of $K^{\mathrm{cont}}$.https://ou-liu-red-sugar.github.io/notes/notes/gestalten/stefanich-rings/Mon, 29 Dec 2025 00:00:00 +0000https://ou-liu-red-sugar.github.io/notes/notes/gestalten/stefanich-rings/This note discusses the colimit of the sequence \[ \mathsf{CAlg}(1\mathsf{Pr}) \hookrightarrow \mathsf{CAlg}(2\mathsf{Pr}) \hookrightarrow \cdots \hookrightarrow \mathsf{CAlg}(n\mathsf{Pr}) \hookrightarrow \cdots . \] in $1\mathsf{Pr}$, and discuss the $n$-affine, $n$-proper, $n$-suave and $n$-prim maps.https://ou-liu-red-sugar.github.io/notes/notes/gestalten/everything-becomes-affine-under-sufficient-categorification/Mon, 29 Dec 2025 00:00:00 +0000https://ou-liu-red-sugar.github.io/notes/notes/gestalten/everything-becomes-affine-under-sufficient-categorification/This note constructs the six-functor formalism on \(n\mathsf{Pr}_{(-)}\), shows that it is sheafy with respect to fpqc topology, and then extends the theory from affine schemes of derived rings to stacks.https://ou-liu-red-sugar.github.io/wiki/compactly-assembled-category/Sat, 25 Apr 2026 00:00:00 +0000https://ou-liu-red-sugar.github.io/wiki/compactly-assembled-category/A presentable category that is a retract in Pr^L of a compactly generated one. In the stable setting, equivalent to being dualizable in Pr^L_st.https://ou-liu-red-sugar.github.io/wiki/compactly-generated-category/Sat, 25 Apr 2026 00:00:00 +0000https://ou-liu-red-sugar.github.io/wiki/compactly-generated-category/A presentable category that is generated under filtered colimits by its compact (i.e. ℵ₀-compact) objects. Equivalently, the Ind-completion of a small idempotent-complete category with finite colimits.https://ou-liu-red-sugar.github.io/wiki/conventions/Sat, 25 Apr 2026 00:00:00 +0000https://ou-liu-red-sugar.github.io/wiki/conventions/House conventions for this wiki — what &#39;category&#39;, &#39;n-category&#39;, and &#39;classical category&#39; mean here, and the suggested entry structure.https://ou-liu-red-sugar.github.io/wiki/presentable-category/Sat, 25 Apr 2026 00:00:00 +0000https://ou-liu-red-sugar.github.io/wiki/presentable-category/An (∞,1)-category that admits all small colimits and is generated under colimits by an essentially small set of κ-compact objects for some regular cardinal κ.https://ou-liu-red-sugar.github.io/wiki/stable-category/Sat, 25 Apr 2026 00:00:00 +0000https://ou-liu-red-sugar.github.io/wiki/stable-category/A category in which the suspension functor is an equivalence and finite limits coincide with finite colimits — the natural replacement for triangulated categories.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/invest/playbook/Fri, 30 Jan 2026 00:00:00 +0000https://ou-liu-red-sugar.github.io/invest/playbook/Principles, allocation bands, and execution rules for my long-term investing.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.https://ou-liu-red-sugar.github.io/about/Mon, 01 Jan 0001 00:00:00 +0000https://ou-liu-red-sugar.github.io/about/<h1 id="ou-liu-刘欧">Ou Liu (刘欧)</h1> <p>I am a master&rsquo;s student in Mathematics at the <em>University of Regensburg</em>, with interests in <em>motivic homotopy theory</em>, <em>derived algebraic geometry</em>, and <em>higher category theory</em>. Before Regensburg, I studied at <em>Shanxi University</em>, where my undergraduate thesis applied the six-functor formalism to smooth representation theory.</p> <hr> <h2 id="personal">Personal</h2> <ul> <li><strong>Name.</strong> Ou Liu (刘欧)</li> <li><strong>Date of birth.</strong> 2 February 2004 <em>(as commonly observed; the national ID card records 12 January 2004)</em></li> <li><strong>Age.</strong> 22</li> <li><strong>Nationality.</strong> Chinese</li> <li><strong>Languages.</strong> Chinese (native), English (working), German (elementary, improving)</li> </ul> <hr> <h2 id="education">Education</h2> <p><strong>2025 &ndash; Present</strong> · <strong>M.Sc. in Mathematics</strong><br> <em>University of Regensburg</em>, Germany.<br> Research focus on motivic homotopy theory, derived algebraic geometry and higher category theory.</p>https://ou-liu-red-sugar.github.io/wiki/graph/Mon, 01 Jan 0001 00:00:00 +0000https://ou-liu-red-sugar.github.io/wiki/graph/A force-directed graph of all wiki entries and their cross-references.