The cyclic Chu-construction for closed bicategories with pullbacks, which generalizes the original Chu-construction for symmetric monoidal closed categories, turns out to have a non-cyclic counterpart. Both use so-called Chu-spans as new 1-cells between 1-cells of the underlying bicategory, which form the new objects. Chu-spans may be seen as a natural generalization of 2-cell-spans in the base bicategory that no longer are confined to a single hom-category. This view helps to clarify the composition of Chu-spans.

We consider various approaches of linking the underlying bicategory with the newly constructed ones, e.g. by means of two-dimensional generalizations of bifibrations. In the quest for a better connection, we investigate, whether Chu-spans form a double category. While this turns out not to be the case, we are led to considering a generalization of the construction to paths of 1-cells in the base, leading to two hierarchies of closed bicategories, one for linear paths and one for loops. The possibility of moving beyond paths, respectively, loops of the same length is indicated.

Finally, Chu-spans in *rel* are identified as bipartite state
transition
systems. Even though their composition may fail here due to the
lack of pullbacks in *rel*, basic game-theoretic constructions can
be performed on cyclic Chu-spans. These are available in all
symmetric monoidal closed categories with finite products. If
pullbacks exist as well, the bicategory of cyclic Chu-spans inherits
a monoidal structure that on objects coincides with the categorical
product.

Keywords: closed bicategory, Chu-spans, games, bipartite state transition systems

2000 MSC: 18D99, 91A99

*Theory and Applications of Categories,*
Vol. 17, 2006,
No. 6, pp 103-126.

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