arXiv subjects

Bangxin Wang

Publications and source records attributed to Bangxin Wang.

3 recordsLinked to original sources

arXiv2026

Crossed-module crossed braided categories

For a crossed module $χ: G \to H$, we introduce the notion of $χ$-crossed braided (resp. ribbon) categories, where the categories are graded by group $G$ and carry an $H$-action. Our definition unifies and generalises several familiar notions: taking $χ= id: G \to G$ with the conjugation action recovers $G$-crossed braided categories; taking $χ: G \to \{*\}$ for abelian $G$ yields $G$-graded braided categories; taking $χ: \{*\} \to G$ leads to braided categories equipped with a $G$-action. The equivalence relation between $χ$-crossed braided categories is typically finer than that between $G$-crossed braided ones. We classify $χ$-crossed braided structures on the category of $G$-graded vector spaces in terms of cohomological data, and give explicit examples for cyclic groups. Given a doubly central algebra with $G$- and $H$-actions in a braided monoidal category, we define a notion of twisted-local modules and show how they give rise to $χ$-crossed braided categories. We furthermore give sufficient conditions so that these categories are additionally $χ$-crossed ribbon or admit an orthogonal $G$-decomposition.

math.CT↗

arXiv2026

Dynamical Systems as Functorial Realisations of Abstract Evolution Shapes

We develop a categorical framework for closed dynamical systems in which the abstract pattern of admissible evolutions is separated from its concrete realisation. A closed dynamical system is formulated as a functor $X\colon S\to C$ from a small category $S$, viewed as an abstract evolution shape, to a coefficient category $C$. By varying $S$ and $C$, this single definition encompasses many important examples including autonomous, non-autonomous, switched, hybrid, and stochastic systems. Within this framework, we introduce invariant subsystems, equilibria, and orbits in functorial terms. We then formulate convergence by combining a cosieve-based intrinsic notion of eventuality on the evolution shape with neighbourhood filters of invariant subsystems. Finally, we establish a categorical Lyapunov principle based on categorical sublevel neighbourhoods. This yields abstract stability and convergence criteria that recover the classical Lyapunov method in standard examples.

math.CT↗

arXiv2024

Hennings TQFTs for Cobordisms Decorated With Cohomology Classes

Starting from an abelian group $G$ and a factorizable ribbon Hopf $G$-bialgebra $H$, we construct a TQFT $J_H$ for connected framed cobordisms between connected surfaces with connected boundary decorated with cohomology classes with coefficients in $G$. When restricted to the subcategory of cobordisms with trivial decorations, our functor recovers a special case of Kerler-Lyubashenko TQFTs, namely those associated with factorizable ribbon Hopf algebras. Our result is inspired by the work of Blanchet-Costantino-Geer-Patureau, who constructed non-semisimple TQFTs for admissible decorated cobordisms using the unrolled quantum group of $\mathfrak{sl}_2$, and by that of Geer-Ha-Patureau, who reformulated the underlying invariants of admissible decorated $3$-manifolds using ribbon Hopf $G$-coalgebras. Our work represents the first step towards a homological model for non-semisimple TQFTs decorated with cohomology classes that appears in a conjecture by the first two authors.

math.GT↗