Synthetic Differential Geometry

Abstract

Proposition 8.1. Assume M is infinitesimally linear. For any vector field X : M × D → M on M , we have (for any m ∈ M) ∀(d 1 , d 2) ∈ D(2) : X(X(m, d 1), d 2) = X(m, d 1 + d 2) (8.4) Proof. Note that the right hand side makes sense, since d 1 + d 2 ∈ D for (d 1 , d 2) ∈ D(2). Both sides in the equation may be viewed as functions

Principal bundles, groupoids, and connections

“…This is not completely obvious, since Ω differs from the exterior derivative dω of the classical connection form ω by a "correction term" 1/2 [ω, ω] involving the Lie Bracket of g; or, alternatively, the curvature form comes about by modifying dω by a "horizontalization operator" (this "modification" also occurs in the treatment in [17]). The fact that this "correction term" (or "modification") does not come up in our context can be explained by Theorem 5.4 in [8] (or see [7] Theorem 18.5); here it is proved that the formula dω(x, y, z) = ω(x, y) · ω(y, z) · ω(z, x) already contains this correction term, when translated into "classical" Lie algebra valued forms. The Theorem has the following Corollary, which is essentially what [17] call the infinitesimal version of the Gauss-Bonnet Theorem (for the case where G = SO(2)): Corollary 7.2.…”

“…This is motivated by Synthetic Differential Geometry (SDG), cf. [7], and more recently [12] and [3], where the notion of connection (infinitesimal parallel transport) and differential form is elaborated in these terms. In particular, let us remind the reader how the notion of "differential form on a manifold M with values in the Lie algebra of a Lie group G" is paraphrased synthetically.…”

“…To cope with the mathematical complications arising in this extension process, Breen and Messing [2], [3] found it helpful to utilize some of the formal or "synthetic" method elaborated in the last decades (as in e.g. [7]). The present note hopefully also provides a contribution to such "synthetic gauge theory", by combining it more firmly with the theory of groupoids.…”

“…e.g. [7]. The point, however, is: the set theoretic descriptions are good enough to convey the idea about the essence of the constructions and the geometry.…”

“…By [7] I.18, or in more detail, [8], there is a bijective correspondence between G-valued k-forms θ on a manifold P (where G is a Lie group, say P −1 P ), and differential k-forms θ, in the classical sense, with values in the Lie algebra g of G (i.e. multilinear alternating maps T P × P ... × P T P → g).…”