The cubical singular complex , or , of a topological space has an additional structure of compositions.
Let be an -tuple of positive integers and
be the map Then a subdivision of type of a map is a factorisation ; its parts are the cubes where is an -tuple of integers with , and where is given by
We then say that is the composite of the cubes and write . The domain of is then the set .
The composite is in direction if is the only and we then write the composite is in the directions , if , are the only and we then write
for and The aim is to follow matrix conventions in writing double compositions.
These definitions and notations are useful for showing how the singular cubical complex allows expression for algebraic inverses to subdivision, something seemingly very difficult either simplicially or globularly. The implications of this advantage for weak category theory seem not to have been investigated.
A cubical set with connections and compositions and inverses is a cubical set with connections in which each has partial compositions and unary operations , satisfying the following axioms.
If , then is defined if and only if , and then for :
If , then is defined and
We have for and whenever both sides are defined:
These relations are called the interchange laws, and both sides of this equation may be written:
If further is a cubical set with connections then we also require
These last two equations are called the transport laws and the right hand sides are also written respectively
They can be interpreted as saying that turning left, or right, with your arm outstretched, is the same as turning left, or right.
It is easily verified that the singular cubical set of a space satisfies these axioms if are defined by
whenever ; and
The above list of relations may seem formidable, but they all express simple geometric ideas most of which have been well used in some form or another in algebraic topology.
Notice also that in the singular cubical complex of a space, the interchange and transport laws hold exactly.
We also get a (strict) notion of cubical omega-category with connections by assuming that all compositions give a category structure with source and target maps and identity maps , and also for all
These are important cancellation laws for the connections. They can be interpreted as saying that turning left and then right, or vice versa, leaves you facing the same way. They were introduced by C.B. Spencer for double categories (see below).
In the omega-groupoid case, the can be recovered from the , and vice versa, by using the inverses, assumed to arise from the groupoid structure.
The main result of the second paper below is that (strict) cubical omega-groupoids with connections are equivalent to crossed complexes. It is easy to construct a functor from the former to the latter; the hard work is to show that such an omega-groupoid may be functorially reconstructed from the crossed complex it contains.
This work is used in the third paper to construct and apply a strict cubical homotopy groupoid of a filtered space.
The main result of the fourth paper below is that (strict) cubical omega-categories with connections are equivalent to strict globular omega-categories.
For more information, see Nonabelian Algebraic Topology..
C.B. Spencer, “An abstract setting for homotopy pushouts and pullbacks”, Cahiers Topologie G'eom. Diff'erentielle, 18 (1977), 409-429.
R. Brown and P.J. Higgins, The algebra of cubes, J. Pure Appl. Alg.+, 21 (1981), 233–260.
Brown, R. and Higgins, P.~J. Colimit theorems for relative homotopy groups. J. Pure Appl. Algebra 22~(1) (1981) 11–41.
F. Al-Agl, R. Brown and R. Steiner, Multiple categories: the equivalence between a globular and cubical approach, Advances in Mathematics, 170, (2002), 71–118.