范畴论：2025–2026 年度综述

An annual survey of the math.CT arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 120 papers and organised into 44 chapters, it maps the year by the field's research sub-directions.

Book framework (share of the year's output):

• Categorical Quantum Mechanics and Compositional Systems (19.7%) — This subfield applies categorical frameworks, including coalgebras and higher categories, to model quantum coherence, Bayesian inference, and hybrid dynamical systems.
• Topological Spaces and Categorical Duality Structures (16.0%) — This area explores the interplay between topological spaces, locales, and categorical structures like Priestley duality and equivariant networks.
• Abelian Categories and Exact Structure Theory (14.5%) — This direction investigates structural properties of abelian and triangulated categories, focusing on exact completions, cluster structures, and module categories.
• Hopf Algebras and Categorical Algebraic Structures (10.4%) — This subfield studies Hopf algebras, Lie algebras, and their categorical generalizations, including bialgebroids and coalgebraic models in duoidal categories.
• Extriangulated Categories and Homotopical Algebra (8.3%) — This area focuses on the structure of extriangulated categories, Gorenstein envelopes, and homotopy languages within higher algebraic frameworks.
• Monoidal Categories and String Diagram Rewriting (8.0%) — This direction examines symmetric monoidal categories, distributive laws, and algorithmic approaches to string diagram rewriting and uncertainty modeling.
• Higher Categories and Homotopy Coherent Structures (6.9%) — This subfield develops the theory of (∞,1)-categories, operads, and stable homotopy types, with applications to equivariant and motivic contexts.
• Homotopy Type Theory and Synthetic Mathematics (6.1%) — This area bridges category theory and type theory, focusing on simplicial types, synthetic homotopy, and internal languages of locally cartesian closed categories.
• Tensor Triangular Geometry and Spectral Theory (5.3%) — This direction applies tensor triangular geometry to classify tensor ideals and spectra in triangulated categories, extending Balmer's framework.
• Double Categories and Higher Categorical Frameworks (4.8%) — This subfield explores double categories, lax functors, and their applications to open systems and axiomatic quantum field theory.

Each sub-direction is covered across three chapters — how it framed its questions, the strength of its key findings, and its open debates and frontiers — so the reader gets a traceable picture of what this part of the field actually produced over the year.