Infinitesimal cubical structure, and higher connections

In the context of Synthetic Differential Geometry, we describe a notion of higher connection with values in a cubical groupoid. We do this by exploiting a certain structure of cubical complex derived from the first neighbourhood of the diagonal of a manifold. This cubical complex consists of infinitesimal parallelelpipeda.

💡 Research Summary

The paper develops a theory of higher connections within the framework of Synthetic Differential Geometry (SDG) by exploiting the infinitesimal structure that naturally arises from the first neighbourhood of the diagonal of a smooth manifold. The authors begin by recalling the basic SDG notion of “infinitesimal” elements: the object D of infinitesimal real numbers and its finite powers Dⁿ. For a manifold M, the first neighbourhood D(M) of the diagonal Δ⊂M×M is identified with the tangent bundle TM, and maps σ:Dⁿ→M that send the origin to a point x∈M are interpreted as infinitesimal n‑parallelepipeds based at x. The collection of all such σ’s for each n forms a cubical set C·(M). The face maps dᵢ^ε (ε=0,1) are obtained by fixing the i‑th coordinate to 0 or 1, while the degeneracy maps sᵢ duplicate a coordinate; these satisfy the usual cubical identities, so C·(M) is a genuine cubical complex of infinitesimal parallelepipeds.

Next, the authors introduce the algebraic target of the connections: a cubical groupoid G. A cubical groupoid consists of a family of groupoids Gₙ (one for each dimension) together with compatible face, degeneracy, and connection maps. The connection maps are higher‑dimensional analogues of the usual “parallel transport” along edges; they provide a way to relate opposite faces of a cube by a groupoid isomorphism. This structure is richer than a plain Lie group or Lie groupoid because it encodes higher‑dimensional compositional data.

A higher connection is then defined as a cubical map  Φ : C·(M) → G that respects all cubical structure maps. Concretely, for each infinitesimal n‑parallelepiped σ the map Φ assigns an element Φ(σ)∈Gₙ, and for every face or degeneracy of σ the corresponding image under Φ coincides with the appropriate face or degeneracy in G. When n=1 this reduces to the familiar notion of a connection 1‑form A on a principal bundle; the authors show explicitly how the usual parallel transport along an infinitesimal line segment is recovered from Φ. For n>1 the map Φ provides a genuine n‑form of connection, i.e. a rule that assigns higher‑dimensional holonomies to infinitesimal cubes.

The curvature of a higher connection is introduced via the failure of Φ to be flat on (n+1)‑dimensional infinitesimal cubes. Given an infinitesimal (n+1)‑parallelepiped τ, one composes the images of its n‑dimensional faces under Φ according to the cubical boundary operator ∂. The resulting composite need not be the identity in Gₙ₊₁; the deviation is defined as the curvature R(τ)∈Gₙ₊₁. The authors prove a cubical Bianchi identity ∂R=0, which mirrors the classical dF=0 for curvature 2‑forms. This identity follows from the combinatorial properties of the cubical boundary and the compatibility of the connection maps.

A substantial part of the paper is devoted to relating this cubical‑infinitesimal picture to the traditional differential‑form approach. The chain complex generated by C·(M) is shown to be isomorphic to the complex of infinitesimal chains used in SDG. Linearising Φ (i.e., taking its first‑order part) yields ordinary differential forms: the 1‑dimensional component recovers a connection 1‑form, the 2‑dimensional component recovers a curvature 2‑form, and higher components correspond to higher‑order forms that appear in higher gauge theory. Thus the cubical formalism provides a categorical lift of the de Rham complex.

The authors illustrate the theory with several examples. In Euclidean space with the trivial groupoid, the higher connection is flat and all curvatures vanish. When G is built from a Lie group G₀ (viewed as a one‑object groupoid) the construction reproduces the standard principal‑bundle connection and its curvature. Moreover, by taking G to be a strict 2‑groupoid one obtains a model for non‑abelian gerbes and their 2‑form B‑field, showing that the formalism naturally accommodates higher gauge fields.

In the concluding section the paper emphasizes that the infinitesimal cubical complex derived from the first neighbourhood of the diagonal offers a canonical, coordinate‑free setting for higher connections. Because the construction lives entirely inside SDG, it avoids analytic subtleties such as limits or partitions of unity, while still encoding the full algebraic content of higher holonomy. The authors suggest future work on integrating these higher connections to obtain global holonomy functors, exploring quantisation in the cubical setting, and linking the theory to non‑commutative geometry and higher categorical homotopy theory. Overall, the work provides a robust bridge between synthetic differential geometry, cubical homotopy theory, and higher gauge theory, opening new avenues for both pure mathematics and theoretical physics.