Combinatorics of the first neighbourhood of the diagonal

Abstract, Krynica April-May 2002

Anders Kock

The consideration of the k’th neighbourhood of the diagonal of a manifold M

M_{(k)} ⊆ M × 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 ∈ 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 ∼_{k} 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 ω on M is thus defined to be a map ω : M_{(1)} → R, vanishing on the diagonal. So ω(x, y) makes sense whenever x ∼_{1} y; and ω(x, x) = 0 for all x ∈ M. Unravelling the definition of M_{(1)} in terms of its structure sheaf almost immediately reveals that such an ω is an element of the Kähler differentials Ω^{1}(M) = I/I^{2}. There is no linearity requirement on ω; and ω(x, y) = −ω(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 Ω^{1}. 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]} ⊆ M^{k+1}

of “infinitesimal k-simplices”. It is the “set” of k + 1-tuples (x_{0}, …, x_{k}) 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 ω : M_{[k]} → R vanishing on degenarate simplices, and classical differential k-forms on M.

Note that there is no multilinearity or alternating requirement on ω. – 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 ω is a 1-form, dω is the 2-form given by

dω(x, y, z) = ω(x, y) + ω(y, z) + ω(z, x).  (1)

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, (1) is replaced by

dω(x, y, z) = ω(x, y) · ω(y, z) · ω(z, x).  (2)

This aspect of the theory is well suited for a treatment of the theory of connections in principal G-bundles P or groupoids P P^{−1}. A connection in a bundle E → M is, in the synthetic theory, a law ∇ which to a pair of 1-neighbour points x ∼_{1} y in M associates a “parallel transport” map ∇_{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_{∇} of ∇ is a 2-form (in the above sense) on M with values in a suitable group bundle on M, and is given by

R_{∇}(x, y, z) = ∇(x, y) ∘ ∇(y, z) ∘ ∇(z, x),

– We shall utilize these descriptions to provide an explicit construction of a connection in a principal G-bundle P → M, out of the data of a G-valued Čech-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.