A Rigid Category of DNA Secondary Structures

Abstract: We construct a strict pivotal monoidal category $\mathcal{D}{\mathrm{DNA}}$ whose objects are DNA sequences (words over ${A,C,G,T}$) and whose morphisms are isotopy classes of typed noncrossing planar matchings, composed of through-strands and Watson-Crick-typed arcs, in a rectangle with source and target boundaries. The dual of a sequence is its reverse complement, evaluation and coevaluation are canonical duplex pairings, and the snake identities hold by planar isotopy. A bending correspondence identifies each morphism $x \to y$ with a secondary structure on the combined word $x{}^{\vee} y$; in particular, the generalized elements $\varepsilon \to w$ are exactly the non-pseudoknotted secondary structures on $w$. Composition, viewed in this straightened picture, is computed by a zip-and-transfer operation on complementary interfaces, a combinatorial rearrangement of base-pair connectivity of which toehold-mediated strand displacement is a kinetically specific instance. Because $\mathcal{D}{\mathrm{DNA}}$ is rigid monoidal, it shares the categorical backbone of pregroup grammars and the DisCoCat framework for compositional semantics: a strong monoidal functor from a grammatical category to $\mathcal{D}_{\mathrm{DNA}}$ maps grammatical reductions to Watson-Crick base pairing and sentence meanings to secondary structures. We describe this functor and discuss connections to algorithmic self-assembly, composable strand-displacement circuits, and constructive dynamical systems.