Combinatorial neural code $\mathscr C\subseteq 2^{[n]}$ is convex if it occurs as the intersection pattern of the convex subsets of $\mathbb R^d$. We relate the emerging theory of convex neural codes to the well-established theory of directional matroids, both categorically and in terms of geometry and computational complexity. On the categorical side, we show that a map that takes an acyclic oriented matroid to code the positive part of its tope is a faithful functor. We adapt the directed matroid ideal introduced by Novik, Postnikov, and Sturmfels to a functor from the category of directed matroids to the category of rings. We then show that the resulting rings naturally map to the neural rings of the matroid neural code.
For the geometry and computational complexity, we show that the code is realized on convex polytopes only if below the code for the partially-order representable directional matroid of the code introduced by Jeffs. Previously published examples of non-convex codes show not under directed matroids, and we construct examples of non-convex codes under non-representable directed matroids. This construction allows us to apply the Mn\”{e}v-Sturmfels universality to show that it is NP-hard to determine whether a combinatorial code is convex.
