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

TL;DR

This paper introduces LeanCat, a challenging benchmark suite for formal category theory in Lean, revealing significant gaps in current models' abstraction capabilities and demonstrating the effectiveness of retrieval-augmented reasoning methods.

Contribution

The paper presents LeanCat, a new benchmark for formal category theory tasks, and evaluates retrieval-augmented models, showing improved performance and highlighting the importance of iterative reasoning.

Findings

Current models solve only 12% of tasks at pass@4.

Performance drops from 55% on Easy to 0% on High-difficulty tasks.

Retrieval-augmented LeanBridge doubles success rate to 24%.

Abstract

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.