https://ou-liu-red-sugar.github.io/tags/higher-algebra/Recent content in Higher Algebra on Ou LiuHugo -- 0.146.0en-usFri, 24 Apr 2026 00:00:00 +0000https://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/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/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/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.