Two-dimensional transducers

We define a bicategory 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 A,B, and its 1-cells (Q, t) : A → B consist of a category Q of `states', and a profunctor t : A × Qᵒp × (B^*)ᵒp → Set where B^* denotes the free monoidal category over B. Extending t to A^* in a canonical way, to each `word' a in A^* one attaches an endoprofunctor over the category Q of states, enriched over presheaves on B^*. We discuss a number of other characterizations of the hom-category 2TDX(A,B); we establish a Kleisli-like universal property for 2TDX(A,B) and explore the connection of 2TDX to other bicategories of computational models, such as Bob Walters' bicategory of `circuits'; it is convenient to regard 2TDX as the loose bicategory of a double category DTDX: the bicategory (resp., double category) of profunctors is naturally contained in the bicategory (resp., double category) 2TDX (resp., DTDX); we study the completeness and cocompleteness properties of DTDX, the existence of companions and conjoints, and we sketch how monads, adjunctions, and other structures/properties that naturally arise from the definition work in DTDX.