CCNN built on cup and cap products matches other simplicial complex neural networks on TopoBench

Proposes using cup and cap products, rather than relying solely on boundary operators and Hodge Laplacians, as the mechanism for moving features and signals across dimensions in topological deep learning. The text states that most TDL approaches are limited in that "features and signals cannot be lifted or lowered across more than one dimension per layer"; introducing cup and cap products targets exactly this constraint. The claim is presented as an architectural design choice, phrased as "we propose the adoption of the cup and cap products", making it a constructive methodological contribution.

Formulates the CCNN architecture, whose aim is to learn node-based variables (0-cochains) while incorporating the many-body interactions present in the data, such as triangles. Information carried by higher-dimensional simplices (edges, triangles, and so on) is brought into the learning of node-level variables rather than remaining confined to node features alone. The text explicitly states "we formulate the Cup and Cap Topological Neural Network (CCNN)" and describes its orientation toward 0-cochains and many-body interactions.

Evaluates CCNN on the TopoBench datasets, with results indicating competitiveness relative to other simplicial complex neural networks. Provides benchmark-level empirical comparison for the proposed architecture rather than leaving it at a formal description. The text reports "revealing its competitiveness with respect to other simplicial complex neural networks"; the abstract gives no specific numbers, dataset sizes, or statistical tests.

The work targets tasks that require modeling higher-order many-body interactions such as triangles and that aim to learn node-based variables (0-cochains), applying to data with simplicial complex structure; its validation is confined to the TopoBench datasets and to comparison with simplicial complex neural network methods. For researchers who want to move features and signals across more than one dimension within a single layer, CCNN offers an architecture option grounded in cup and cap products.

The abstract gives no specific metric values on TopoBench, no dataset composition, no baseline list, and no experimental setup, so the magnitude and conditions of "competitiveness" still require the main text; the concrete computation of cup and cap products in the implementation, along with complexity and numerical stability, is also not described in the abstract; whether the architecture maintains comparable behavior on data distributions beyond TopoBench remains an open question.