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Explore the mathematical foundations of computational devices through this colloquium lecture that defines a bicategory 𝟚TDX whose 1-cells categorify transducers - computational devices that extend finite-state automata with output capabilities. Discover how 2-transducers can be characterized as functors of type t : 𝒜 × 𝒬ᵒᵖ × 𝒬 × (ℬ*)ᵒᵖ ⟶ Set, where they represent families of profunctors over a category 𝒬 of states, indexed over input category 𝒜, and enriched over the free monoidal category ℬ* obtained from output category ℬ. Learn about multiple equivalent characterizations of 𝟚TDX(𝒜, ℬ) and examine the Kleisli-like universal property that governs this mathematical structure. Investigate the connections between 𝟚TDX and other bicategories of computational models, including Walters' bicategory of circuits, while understanding how the bicategory of profunctors naturally embeds within this framework. Delve into the double category 𝔻TDX structure, examining its completeness and cocompleteness properties, and explore the monads, adjunctions, and other categorical structures that emerge from these definitions, providing a comprehensive categorical perspective on computational transduction processes.
