Weak cubical categories

[Marco Grandis <grandis@dima.unige.it>]

The following preprint, exposed in my talk at CT2008, Calais, can be
downloaded from my web page

"The role of symmetries in cubical sets and cubical categories
(On cubical categories, I)"  (34 pages)

http://www.dima.unige.it/~grandis/CCat.ps
http://www.dima.unige.it/~grandis/CCat.pdf

With best wishes

Marco Grandis

______________

Abstract. Symmetric weak cubical categories were introduced in
previous papers, as a basis for the study of cubical cospans in
Algebraic Topology and higher cobordism. Such cubical structures are
equipped with an action of the n-dimensional symmetric group on the n-
dimensional component, which has been used to simplify the coherence
conditions of the weak case.

	We give now a deeper study of the role of symmetries. The category
of ordinary cubical sets has a Kan tensor product, which is non
symmetric and biclosed, with left and right internal homs based on
the right and left path functors. On the other hand, symmetric
cubical sets have one path functor leading to one internal hom and a
symmetric monoidal closed structure. Similar facts hold for cubical
and symmetric cubical categories, and should play a relevant role in
the sequel, the study of limits and adjunctions in these higher
dimensional categories. Weak double categories, studied in four
papers with R. Pare, are a cubical truncation of the present structures.

	While constructing examples of cubical categories, we also
investigate a 'rewriting' procedure of reduction to canonical forms,
which allows one to quotient a weak symmetric cubical category of
cubical spans (resp. cospans), and obtain a strict symmetric cubical
category of 'cubical relations' (resp. 'cubical profunctors').

______________