A preprint on Weak cubical categories

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

The following preprint is available on my server:


M. Grandis, Higher cospans and weak cubical categories (Cospans in
Algebraic Topology, I)
Dip. Mat. Univ. Genova, Preprint 552 (2007).

in pdf and ps:

http://www.dima.unige.it/~grandis/wCub.pdf

http://www.dima.unige.it/~grandis/wCub.ps



Comments and suggestions are appreciated.

Marco Grandis

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Abstract. We define a notion of weak cubical category, abstracted
from the structure of n-cubical cospans  x: \Lambda^n ---> X  in a
category  X,  where  \Lambda  is the 'formal cospan' category. These
diagrams form a cubical set with compositions in all directions,
which are computed with 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.

Mathematics Subject Classifications: 18D05, 55U10

Key words: weak cubical category, multiple category, double category,
cubical sets, spans, cospans.

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