arXiv subjects

Holiverse Yang

Publications and source records attributed to Holiverse Yang.

4 recordsLinked to original sources

arXiv2026

2-Morita Theory of $E_2$-Algebras and Module Categories

Building on our previous work on 2-Morita equivalence for $E_2$-algebras using topological pictures, we develop a systematic framework for Morita equivalence of topological orders in different dimensions in terms of $n$-Morita categories $\mathrm{Mrt}_{E_n}(\mathcal{C})$. In this framework, various notions of $n$-Morita equivalence are unified as equivalences of objects in $\mathrm{Mrt}_{E_n}(\mathcal{C})$. We also compare the constructions of higher Morita categories due to Haugseng and to Gwilliam--Scheimbauer. For $n=1,2$, we prove that the functor $\mathrm{Mod}_n:\mathrm{Mrt}_{E_n}(\mathcal{C})\to \mathrm{Mrt}_{E_{n-1}}(\mathrm{LMod}^{\mathrm{rep}}(\mathcal{C}))$ is an equivalence, relating the algebraic description of higher Morita theory to its realization in terms of module categories. In this formulation, the notion of a bi-bimodule arises naturally and provides a common framework for local and confined modules. Its explicit orientation data, together with the corresponding fusion rules, clarifies the relations among defects arising from condensation in topological orders.

math-ph↗

arXiv2025

LeanCat: A Benchmark Suite for Formal Category Theory in Lean (Part I: 1-Categories)

While large language models (LLMs) have demonstrated impressive capabilities in formal theorem proving, current benchmarks fail to adequately measure library-grounded abstraction -- the ability to reason with high-level interfaces and reusable structures central to modern mathematics and software engineering. We introduce LeanCat, a challenging benchmark comprising 100 fully formalized category-theory tasks in Lean. Unlike algebra or arithmetic, category theory serves as a rigorous stress test for structural, interface-level reasoning. Our evaluation reveals a severe abstraction gap: the best state-of-the-art model solves only 12.0% of tasks at pass@4, with performance collapsing from 55.0% on Easy tasks to 0.0% on High-difficulty tasks, highlighting a failure in compositional generalization. To overcome this, we evaluate LeanBridge, a retrieval-augmented agent that employs a retrieve-generate-verify loop. LeanBridge achieves a peak success rate of 24.0% -- doubling the performance of the best static baseline. These results empirically demonstrate that iterative refinement and dynamic library retrieval are not merely optimizations but strict necessities for neuro-symbolic reasoning in abstract domains. LeanCat offers a compact, reusable testbed for tracking progress toward reliable, research-level formalization.

cs.LO↗

arXiv2025

2-Morita Equivalent Condensable Algebras and Domain Walls in 2+1D Topological Orders

We classify $E_2$ condensable algebras in a modular tensor category $\mathcal{C}$ up to 2-Morita equivalence. From a physical perspective, this is equivalent to providing a criterion for when different $E_2$ condensable algebras result in the same condensed topological phase in a 2d anyon condensation process. By considering the left and right centers of $E_1$ condensable algebras in $\mathcal{C}$, we exhaust all 2-Morita equivalent $E_2$ condensable algebras in $\mathcal{C}$ and provide a method to recover $E_1$ condensable algebras from 2-Morita equivalent $E_2$ condensable algebras. We also prove that intersecting Lagrangian algebras in $\mathcal{C} \boxtimes \overline{\mathcal{C}}$ with its left and right components generates all 2-Morita equivalent $E_2$ condensable algebras in $\mathcal{C}$. This paper establishes a complete interplay between $E_1$ condensable algebras in $\mathcal{C}$, 2-Morita equivalent $E_2$ condensable algebras in $\mathcal{C}$, and Lagrangian algebras in $\mathcal{C} \boxtimes \overline{\mathcal{C}}$. The relations between different condensable algebras can be translated into their module categories, which correspond to domain walls in topological orders. We introduce a two-step condensation process and study the fusion of domain walls. We also show that an automorphism of an $E_2$ condensable algebra may lead to a nontrivial braided autoequivalence in the condensed phase. As concrete examples, we interpret the categories of quantum doubles of finite groups. We also discuss examples beyond group symmetries. Moreover, our results can be generalized to Witt-equivalent modular tensor categories.

cond-mat.str-el↗

arXiv2023

The boundary phase transitions of the 2+1D $\mathbb{Z}_N$ topological order via topological Wick rotation

In this work, we show that a critical point of a 1d self-dual boundary phase transition between two gapped boundaries of the $\mathbb{Z}_N$ topological order can be described by a mathematical structure called an enriched fusion category. The critical point of a boundary phase transition can be viewed as a gappable non-chiral gapless boundary of the $\mathbb{Z}_N$ topological order. A mathematical theory of the gapless boundaries of 2d topological orders developed by Kong and Zheng (arXiv:1905.04924 and arXiv:1912.01760) tells us that all macroscopic observables on the gapless boundary form an enriched unitary fusion category, which can be obtained by a holographic principle called the ``topological Wick rotation." Using this method, we obtain the enriched fusion category that describes a critical point of the phase transition between the $\mathbf{e}$-condensed boundary and the $\mathbf{m}$-condensed boundary of the $\mathbb{Z}_N$ topological order. To verify this idea, we also construct a lattice model to realize the critical point and recover the mathematical data of this enriched fusion category. The construction further shows that the categorical symmetry of the boundary is determined by the topological defects in the bulk, which indicates the holographic principle indirectly. This work shows, as a concrete example, that the mathematical theory of the gapless boundaries of 2+1D topological orders is a powerful tool to study general phase transitions.

cond-mat.str-el↗