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Compact closed bicategories

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Michael Stay

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.

Keywords:
compact, closed, bicategory, span
18D05, 18D15

2010 MSC:
18D05, 18D15

*Theory and Applications of Categories,*
Vol. 31, 2016,
No. 26, pp 755-798.

Published 2016-08-18.

http://www.tac.mta.ca/tac/volumes/31/26/31-26.pdf

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