Two-dimensional transducers

We define a bicategory $\mathbf{2}\mathbf{T}\mathbf{D}\mathbf{X}$ 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’ $\underset{‾}{a}$ in ${\mathcal{A}}^{\ast }$ one attaches an endoprofunctor over the category $\mathcal{Q}$ of states, enriched over presheaves on ${\mathcal{B}}^{\ast }$. We discuss a number of other characterizations of the hom-category $\mathbf{2}\mathbf{T}\mathbf{D}\mathbf{X}(\mathcal{A},\mathcal{B})$; we establish a Kleisli-like universal property for $\mathbf{2}\mathbf{T}\mathbf{D}\mathbf{X}(\mathcal{A},\mathcal{B})$ and explore the connection of $\mathbf{2}\mathbf{T}\mathbf{D}\mathbf{X}$ to other bicategories of computational models, such as Bob Walters’ bicategory of `circuits’; it is convenient to regard $\mathbf{2}\mathbf{T}\mathbf{D}\mathbf{X}$ as the loose bicategory of a double category $\mathbb{D}\mathbf{T}\mathbf{D}\mathbf{X}$: the bicategory (resp., double category) of profunctors is naturally contained in the bicategory (resp., double category) $\mathbf{2}\mathbf{T}\mathbf{D}\mathbf{X}$ (resp., $\mathbb{D}\mathbf{T}\mathbf{D}\mathbf{X}$); we study the completeness and cocompleteness properties of $\mathbb{D}\mathbf{T}\mathbf{D}\mathbf{X}$, the existence of companions and conjoints, and we sketch how monads, adjunctions, and other structures/properties that naturally arise from the definition work in $\mathbb{D}\mathbf{T}\mathbf{D}\mathbf{X}$.