Combinatorics of the first neighbourhood of the diagonal
by
Anders Kock
University of Aarhus, Denmark

The consideration of the k'th neighbourhood of the diagonal of a manifold M was initiated by Grothendieck to import notions from differential geometry into the realm of algebraic geometry. These notions were re-imported into differential geometry by Malgrange. Grothendieck and Malgrange utilzed the notion of ringed space (a space equipped with a structure sheaf of functions). The only points of the (underlying space of) M_(k) are the diagonal points (x, x) with x in M. But it is worthwhile to describe mappings to and from M_(k) as if it consisted of ``pairs of k-neighbour points (x, y)'' (write x ~ <sub>k</sub> y for such a pair; such x and y are ``point proches'' in the terminology of A. Weil). The introduction of topos theoretic methods has put this ``synthetic'' way of speaking onto a rigourous basis, and we shall freely use it.

A differential 1-form \omega on M is thus defined to be a map \omega from M₍₁₎ to R, vanishing on the diagonal. So \omega(x, y) makes sense whenever x ~ ₁ y; and \omega(x, x) = 0 for all x in M. Unravelling the definition of M₍₁₎ in terms of its structure sheaf almost immediately reveals that such an \omega is an element of the Kähler differentials \Omega¹ (M) = I/I². There is no linearity requirement on \omega; and \omega(x, y) = -\omega(y, x) is automatic.

One can go on and define a k-form on M as an element of the k'th exterior power of \Omega¹. This is the classical approach in algebraic geometry. But there is an alternative, more geometric/simplicial approach to the theory of differential forms, which we shall expound.

It is based on the consideration of the space M_[k] of ``infinitesimal k-simplices''. This space is the ``set'' of k+1-tuples (x₀ , ... , x_k ) of points from M, with x_i ~ ₁ x_j for all i, j = 0, ... , k. We shall call the simplex degenerate if two of its vertices x_i and x_j are equal. Then the geometric/synthetic/combinatorial approach to differential forms is based on

Theorem There is a bijective correspondence between functions \omega from M_[k] to R vanishing on degenarate simplices, and classical differential k-forms on M.

Note that there is no multilinearity or alternating requirement on \omega. - Since the M_[k]'s jointly form a simplicial ``set'', a differential k-form may be seen as a k-cochain, and there is therefore a formula for its coboundary; for instance, if \omega is a 1-form, d\omega is the 2-form given by

d\omega(x, y, z) = \omega(x, y) + \omega(y, z) + \omega(z, x)      *

It can be proved to correspond to the classical exterior derivative, under the correspondence of the theorem. But it generalizes in a more seamless way to differential forms with values in Lie groups G more general than R. For instance, * is replaced by

d\omega(x, y, z) = \omega(x, y) ·\omega(y, z) ·\omega(z, x).

This aspect of the theory is well suited for a treatment of the theory of connections in principal G-bundles P or groupoids PP^-1. A connection in a bundle E on M is, in the synthetic theory, a law Ñ which to a pair of 1-neighbour points x ~ ₁ y in M associates a ``parallel transport'' map Ñ_yx from the x-fibre of E to its y-fibre.

We shall describe the relationship between gauge-P valued forms, connections, and curvature, in synthetic/combinatorial terms. The curvature R_Ñ of Ñ is a 2-form (in the above sense) on M with values in a suitable group bundle on M, and is given by

RÑ (x, y, z) = Ñ(x, y)   o  Ñ(y, z)  o   Ñ(z, x),

- We shall utilize these descriptions to provide an explicit construction of a connection in a principal G-bundle P on M, out of the data of a G-valued Cech-cocycle for the bundle P, and an R-valued partition of unity on M; this is based on the possibility of forming arbitrary affine combinations of the vertices of an infinitesimal simplex.

Date received: March 26, 2002