Category-Equivariant Neural Networks

• Category-Equivariant Neural Networks (CENNs) are defined as neural architectures where layers are natural transformations between feature assignments that encode data and symmetries.
• They generalize traditional equivariant models by unifying group, groupoid, poset, graph, and sheaf neural networks through a categorical formalism that enforces compositional and hierarchical symmetry.
• CENNs provide universal approximation guarantees for equivariant maps, offering rigorous design flexibility and practical implementations across diverse structured data domains.

Category-Equivariant Neural Networks (CENNs) generalize the principle of symmetry-preserving learning to arbitrary categories, unifying group- and groupoid-equivariant networks with architectures respecting broader notions of compositional, relational, or hierarchical symmetry. CENNs are defined as neural networks whose layers, both linear and nonlinear, are natural transformations between functorial "feature assignments" from a category (encoding data and symmetries) into a suitable target such as vector spaces. This categorical formalism subsumes group/groupoid equivariance, poset and lattice equivariance, graph and sheaf neural networks, and encodes equivariance as a generalized naturality property. The CENN paradigm yields both rigorous compositional frameworks for neural architecture design and general universal approximation theorems for equivariant maps (Maruyama, 23 Nov 2025, Gibson et al., 1 Aug 2024).

1. Categorical Formalism and Equivariance as Naturality

CENNs are constructed by encoding symmetries as categories $\mathcal{C}$ whose objects correspond to "types of features" (eg. pixels, cells, or semantic entities), and morphisms capture abstract symmetry transformations (eg. group actions, relational structure, graph adjacency, or cellular face relations). Feature spaces are assigned to objects via a contravariant functor $$F : \mathcal{C} \rightarrow \mathrm{Vect}_\mathbb{R}$$, and network layers are natural transformations $\eta : F \Rightarrow G$. Naturality enforces the equivariance condition: for every morphism $\varphi : X \to Y$,

$G(\varphi) \circ \eta_X = \eta_Y \circ F(\varphi)$

This principle applies to both linear and nonlinear (typically pointwise or local) layers, and is equivalent to classical equivariance when $\mathcal{C}$ is a group or a groupoid (Maruyama, 23 Nov 2025, Gibson et al., 1 Aug 2024).

In topological or measure-enriched settings, $\mathcal{C}$ can be given the structure of a topological category equipped with Radon measures on hom-sets, enabling kernel convolution and integral operators tailored to continuous or structured domains (Maruyama, 23 Nov 2025).

2. Layer Construction: Linear and Nonlinear Category-Equivariant Operators

A general CENN layer is a natural transformation composed from a finite sequence of:

• Category convolutions: Given a family of kernels $$\mathsf{K}_{b \to a}: \mathrm{Hom}_\mathcal{C}(b,a) \times \Omega(a) \to \mathcal{L}\big(E_F(b), E_G(a)\big)$$, linear layers are encoded as integrated (or summed, in discrete cases) transformations over the relevant symmetry space. In groups, this recovers classical steerable or group-convolution layers; for posets, sums over lower ideals; for sheaves, over cellular face inclusions (Maruyama, 23 Nov 2025).
• Scalar-gated nonlinearities: Pointwise or local nonlinearities, such as ReLU or other Lipschitz activations, inserted via natural componentwise application. Equivariance is ensured through the scalar functor mechanism (Maruyama, 23 Nov 2025, Gibson et al., 1 Aug 2024).
• Arrow-bundle lifts/convolutions: Propagate information across morphisms or "edges" in the category, essential for multi-channel and message-passing settings (eg. in graphs, sheaves).

This compositional design yields finite-depth networks closed under equivariant composition, with well-defined functoriality and block structure under category-theoretic decompositions (Maruyama, 23 Nov 2025, Gibson et al., 1 Aug 2024, Pearce-Crump, 2023).

3. Irreducible Decomposition and Piecewise-Linear Structure

If $\mathcal{C}$ admits a semisimple functor category structure (e.g., for finite groups or certain diagram categories), functors $F$ can be canonically decomposed into direct sums of simple ("irreducible") summands. Linear equivariant maps (natural transformations) become block-diagonal along this decomposition; in the case of groups, this is the classical decomposition into isotypic components, and Schur's lemma applies (Gibson et al., 1 Aug 2024, Pearce-Crump, 2023).

Nonlinear activations, such as ReLU, act piecewise-linearly in an appropriate basis. The equivariance of such piecewise-linear maps is guaranteed only under specific representation-theoretic conditions—e.g., for permutation representations in the case of ReLU. For general categories, block compatibility is required in each local chart. This structure enables explicit analysis of network expressiveness and induction of filtrations analogous to Fourier degree in group settings (Gibson et al., 1 Aug 2024).

4. Universal Approximation Theorems in the Categorical Setting

A comprehensive universal approximation theorem holds: for any compact topological category with measures, any pair of contravariant feature functors $X, Y$, the class of finite-depth CENNs (using suitable nonlinearities) is dense in the space of continuous, category-equivariant maps in the compact-open topology. This holds for groups, groupoids, posets, lattices, face categories of CW complexes (sheaves over graphs or cell complexes), and combinations thereof (Maruyama, 23 Nov 2025).

The proof leverages:

• Stone–Weierstrass approximation of scalar-valued functions via category convolutions,
• Realization of these approximants via arrow-bundle convolutions,
• Construction of equivariant maps from scalar gates and composition of natural transformations.

This generalizes the classical feedforward neural network UAT to all categorical symmetry models.

5. Specializations: Groups, Groupoids, Posets, Graphs, and Sheaves

CENNs specialize to standard architectures when categories are chosen appropriately.

• Groups: The single-object category of a group recovers all classical and steerable group-equivariant architectures. Kernels correspond to $G$-steerable filters and group convolutions (Maruyama, 23 Nov 2025, Gibson et al., 1 Aug 2024).
• Groupoids: Extend group action equivariance to settings with varying local symmetry (e.g., homogeneous spaces, objects with differing local group symmetry).
• Posets/lattices: Thin categories with partial order structure yield architectures enforcing hierarchy or ancestry equivariance. Layer operations sum over subordinate features, suitable for domains such as knowledge graphs or inference on hierarchies (Maruyama, 23 Nov 2025).
• Graphs and Sheaves: The face category of a CW complex models the combinatorics of cell complexes. Feature functors assign vector spaces ("stalks") to cells, with restriction maps. Kernels operate over inclusions, and equivariant layers respect sheaf morphisms. This encompasses graph neural networks, cellular sheaf nets, and hierarchical mesh-based networks (Maruyama, 23 Nov 2025).
• Diagrammatic frameworks: Category presentations with string diagrams (eg., partition or Brauer categories) allow for efficient computation and expressive architectural design in the case of classical groups and tensor powers, supporting fast algorithms and conceptual clarity (Pearce-Crump, 2023).

6. Algorithmic Implementations and Practical Considerations

Practical realization of CENNs follows from explicit constructions:

• Equivarification procedures lift standard networks to strictly equivariant ones via right adjoints/coinduced functors, as shown for finite group settings and for generic feedforward nets (Bao et al., 2019).
• Explicit pseudocode is available for set-based groups, convolutional layers, and dense layers. For instance, group-based equivarification creates block-circulant parameter-sharing matrices for dense layers and applies group-wise parallel convolutions for spatial data (Bao et al., 2019).
• Diagrammatic algorithms enable the fast evaluation of equivariant layers by factorizing morphisms into permutations and planar diagrams, drastically improving computational efficiency relative to naive dense representations (Pearce-Crump, 2023).

These procedures guarantee that equivariance is strictly enforced throughout the network and preserve universal approximation within the equivariant function class.

7. Significance, Expressiveness, and Applications

The categorical formalism of CENNs establishes a unifying mathematical framework for equivariant deep learning, encompassing and generalizing all known symmetry-preserving architectures. The key implications are:

• Rigorous symmetry enforcement: All equivariant architectures are viewed as functors or natural transformations, guaranteeing consistent symmetry constraints by construction.
• Expressivity: Universal approximation results state that finite-depth CENNs are sufficient to approximate any continuous equivariant map within their class, generalizing classical function approximation theory (Maruyama, 23 Nov 2025).
• Design flexibility: CENNs permit the direct integration of complex contexts—geometric, compositional, relational—by appropriate category selection (including products of categories for mixed symmetries).
• Interdisciplinary relevance: Applications include geometric deep learning, sheaf-based signal processing, data with hierarchical or relational constraints, combinatorial scientific computing, and domains requiring precise symmetry-aware generalization.

The CENN paradigm demonstrates that symmetry-preserving neural computation can always be phrased, analyzed, and engineered in categorical language, enabling systematic handling of broad classes of symmetry and contextual constraints, and opening new directions for compositional and structured deep learning (Maruyama, 23 Nov 2025, Gibson et al., 1 Aug 2024, Pearce-Crump, 2023, Sangalli et al., 2022, Bao et al., 2019).