In homotopy type theory, the Kleisli category of the exponential comonad on spans is equivalent to the category of V-ary polynomials, yielding a new model of linear logic.
Compact Closed Bicategories
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abstract
A compact closed bicategory is a symmetric monoidal bicategory where every object is equipped with a weak dual. The unit and counit satisfy the usual "zig-zag" identities of a compact closed category only up to natural isomorphism, and the isomorphism is subject to a coherence law. We give several examples of compact closed bicategories, then review previous work. In particular, Day and Street defined compact closed bicategories indirectly via Gray monoids and then appealed to a coherence theorem to extend the concept to bicategories; we restate the definition directly. We prove that given a 2-category T with finite products and weak pullbacks, the bicategory of objects of C, spans, and isomorphism classes of maps of spans is compact closed. As corollaries, the bicategory of spans of sets and certain bicategories of "resistor networks" are compact closed.
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Polynomials in homotopy type theory as a Kleisli category
In homotopy type theory, the Kleisli category of the exponential comonad on spans is equivalent to the category of V-ary polynomials, yielding a new model of linear logic.