Infinitesimal cubical structure, and higher connections

The purpose of the present note is to experiment with a possible framework for the theory of “higher connections”, as have recently become expedient in string theory; however, our work was not so much to find a framework which fits any existing such theory, but rather to find notions which come most naturally of their own accord. The approach is based on the combinatorics of the “first neighbourhood of the diagonal” of a manifold, using the technique and language of Synthetic Differential Geometry (SDG), as in [8], and notably in [11]. The present note may be seen as a sequel to the latter, and also to [14]. The basic viewpoint in [11] is that connections (1-connections) take value in groupoids (a viewpoint which goes back to Ehresmann), and that they in effect may be seen as morphisms in the category of reflexive symmetric graphs, noting that any groupoid has an underlying such. To go beyond this into higher dimensions, one would like to consider some kind of higher groupoids to receive the values of the connections (see e.g. [18]), and we also need a higher dimensional version of the notion of reflexive symmetric graph. This led us to pass into the cubical, rather than into the simplicial world. The passage into this world depends on having a cubical complex associated to any manifold, in analogy with the simplicial complex of “infinitesimal simplices” that one derives out of the first neighbourhood of the diagonal. This latter simplicial complex is known to be the carrier of a theory of “combinatorial differential forms”, as in [8], [11], [13], and in [1]. The observation that opens up for a similar cubical complex is that infinitesimal simplices in a manifold canonically give rise to infinitesimal parallelepipeda. This hinges on the possibility of forming affine combinations of the points in an infinitesimal simplex, a possibility first noted in [12], see also [13]. On the algebraic side, the kind of “higher” groupoid which fits the bill are essentially the “ω-groupoids” of Brown and Higgins [2], or their truncation