Two-dimensional transducers

We define a bicategory $\mathbf{2TDX}$ whose 1-cells provide a categorification of transducers, computational devices extending finite-state automata with output capabilities. This bicategory is a mathematically interesting object: its objects are categories $\mathcal{A},\mathcal{B},\dots$ and its 1-cells $(\mathcal{Q}, t) : \mathcal{A} \to \mathcal{B}$ consist of a category $\mathcal{Q}$ of `states', and a profunctor $$t : \mathcal{A} \times \mathcal{Q}^\text{op}\times\mathcal{Q} \times (\mathcal{B}^*)^\text{op} \to \mathbf{Set}$$ where $\mathcal{B}^*$ denotes the free monoidal category over $\mathcal{B}$. Extending $t$ to $\mathcal{A}^*$ in a canonical way, to each `word' $\underline a$ in $\mathcal{A}^*$ one attaches an endoprofunctor over the category $\mathcal{Q}$ of states, enriched over presheaves on $\mathcal{B}^*$. We discuss a number of other characterizations of the hom-category $\mathbf{2TDX}(\mathcal{A},\mathcal{B})$; we establish a Kleisli-like universal property for $\mathbf{2TDX}(\mathcal{A},\mathcal{B})$ and explore the connection of $\mathbf{2TDX}$ to other bicategories of computational models, such as Bob Walters' bicategory of `circuits'; it is convenient to regard $\mathbf{2TDX}$ as the loose bicategory of a double category $\mathbb{D}\mathbf{TDX}$: the bicategory (resp., double category) of profunctors is naturally contained in the bicategory (resp., double category) $\mathbf{2TDX}$ (resp., $\mathbb{D}\mathbf{TDX}$); we study the completeness and cocompleteness properties of $\mathbb{D}\mathbf{TDX}$, the existence of companions and conjoints, and we sketch how monads, adjunctions, and other structures/properties that naturally arise from the definition work in $\mathbb{D}\mathbf{TDX}$.