๐ค AI Summary
This work proposes a unified categorical framework for modeling symmetries and equivariant representations in deep learning through a coalgebraic formalism. By embedding invariant structures from the category of sets into the category of vector spaces via functors, and constructing compatible endofunctors to capture equivariance, the paper systematically introduces coalgebraic tools to bridge abstract symmetry specifications with concrete neural network implementations. Within this framework, the authors establish a universal approximation theorem for continuous equivariant functions, applicable to a broad class of symmetry groups. This result provides a rigorous theoretical foundation and realizability guarantee for equivariant deep learning architectures, thereby advancing the principled design of symmetry-aware neural models.
๐ Abstract
Categorical deep learning (CDL) has recently emerged as a framework that leverages category theory to unify diverse neural architectures. While geometric deep learning (GDL) is grounded in the specific context of invariants of group actions, CDL aims to provide domain-independent abstractions for reasoning about models and their properties. In this paper, we contribute to this program by developing a coalgebraic foundation for equivariant representation in deep learning, as classical notions of group actions and equivariant maps are naturally generalized by the coalgebraic formalism. Our first main result demonstrates that, given an embedding of data sets formalized as a functor from SET to VECT, and given a notion of invariant behavior on data sets modeled by an endofunctor on SET, there is a corresponding endofunctor on VECT that is compatible with the embedding in the sense that this lifted functor recovers the analogous notion of invariant behavior on the embedded data. Building on this foundation, we then establish a universal approximation theorem for equivariant maps in this generalized setting. We show that continuous equivariant functions can be approximated within our coalgebraic framework for a broad class of symmetries. This work thus provides a categorical bridge between the abstract specification of invariant behavior and its concrete realization in neural architectures.