A new preprint: "Compact closed bicategories"

A new preprint: "Compact closed bicategories"

[Mike Stay]

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


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