Dialogue Categories and Chiralities

Abstract

In this paper, we consider a two-sided notion of dialogue category which we call dialogue chirality and which we formulate as an adjunction between a monoidal category $\mathcal{A}$ of proofs and a monoidal category $\mathcal{B}$ of counter-proofs equivalent to its opposite category ${\mathcal{A}}^{\mathrm{op}(0,1)}$. The two-sided formulation of dialogue categories is compared to the original one-sided formulation by exhibiting a 2-dimensional equivalence between a 2-category of dialogue categories and a 2-category of dialogue chiralities. The resulting coherence theorem clarifies in what sense every dialogue chirality may be strictified to an equivalent dialogue category.