We present a novel application of category theory for deep learning. We show how category theory can be used to understand and work with the linear layer functions of group equivariant neural networks whose layers are some tensor power space of $\mathbb{R}^{n}$ for the groups $S_n$, $O(n)$, $Sp(n)$, and $SO(n)$. By using category theoretic constructions, we build a richer structure that is not seen in the original formulation of these neural networks, leading to new insights. In particular, we outline the development of an algorithm for quickly computing the result of a vector that is passed through an equivariant, linear layer for each group in question. The success of our approach suggests that category theory could be beneficial for other areas of deep learning.

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