Mathematics on Ou Liu

Algebraic K-theory

Fri, 24 Apr 2026 00:00:00 +0000

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.

Presentable Categories

Sun, 28 Dec 2025 00:00:00 +0000

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.

Compactly assembled categories

Fri, 24 Apr 2026 00:00:00 +0000

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 $\mathsf{Pr}^L_{\mathrm{ca}}$.

Presentable n-Categories

Mon, 29 Dec 2025 00:00:00 +0000

This note introduces the notion of presentable n-categories.

Basic Concepts on Higher Algebra

Mon, 22 Sep 2025 00:00:00 +0000

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.

Continuous (Efimov) K-theory

Fri, 24 Apr 2026 00:00:00 +0000

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's definition of continuous K-theory, its basic properties, and a sketch of the universal property of $K^{\mathrm{cont}}$.

Stefanich Rings

Mon, 29 Dec 2025 00:00:00 +0000

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.

Everything Becomes Affine Under Sufficient Categorification

Mon, 29 Dec 2025 00:00:00 +0000

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.

A brief introduction to 6-functor formalisms

Fri, 24 Apr 2026 00:00:00 +0000

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

Sat, 27 Dec 2025 00:00:00 +0000

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

Wed, 24 Dec 2025 00:00:00 +0000

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

Thu, 27 Nov 2025 00:00:00 +0000

Unified exposition of the stable Dold–Kan correspondence and its application to descent theory in stable categories.