M. Stay, "Compact closed bicategories". http://arxiv.org/abs/1301.1053 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 and review previous work. We give the complete definition of a compact closed bicategory, emphasizing the combinatorics. Finally, we prove that each of the examples are compact closed. In particular, we prove that given a 2-category C with finite products and weak pullbacks, the bicategory of objects of C, spans, and isomorphism classes of maps of spans is compact closed. We also prove that given a cocomplete symmetric monoidal category R whose tensor product distributes over its colimits, the bicategory Mat(R) of natural numbers, matrices of objects of R and matrices of morphisms of R is compact closed. As corollaries, the bicategory of spans of sets, the bicategory of relations, and certain bicategories of "resistor networks" are all compact closed. We also give a new proof that the bicategory of small categories, cocontinuous functors between the presheaves on those categories, and natural transformations is compact closed. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ]