[2603.03227] Coalgebras for categorical deep learning: Representability and universal approximation

Computer Science > Machine Learning arXiv:2603.03227 (cs) [Submitted on 3 Mar 2026] Title:Coalgebras for categorical deep learning: Representability and universal approximation Authors:Dragan Mašulović View a PDF of the paper titled Coalgebras for categorical deep learning: Representability and universal approximation, by Dragan Ma\v{s}ulovi\'c View PDF 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...