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.
