We define a notion of weak cubical category, abstracted from the structure of n-cubical cospans $x : \wedge^n \to X$ in a category $X$ where $\wedge$ is the `formal cospan' category. These diagrams form a cubical set with compositions $x +_i y$ in all directions, which are computed using pushouts and behave `categorically' in a weak sense, up to suitable comparisons. Actually, we work with a `symmetric cubical structure', which includes the transposition symmetries, because this allows for a strong simplification of the coherence conditions. These notions will be used in subsequent papers to study topological cospans and their use in Algebraic Topology, from tangles to cobordisms of manifolds.
We also introduce the more general notion of a multiple category, where - to start with - arrows belong to different sorts, varying in a countable family, and symmetries must be dropped. The present examples seem to show that the symmetric cubical case is better suited for topological applications.
Keywords: weak cubical category, multiple category, double category, cubical sets, spans, cospans
2000 MSC: 18D05, 55U10
Revised 2007-07-27. Original version at http://www.tac.mta.ca/tac/volumes/18/12/18-12a.dvi
Theory and Applications of Categories,
Vol. 18, 2007,
No. 12, pp 321-347.