On two-Dimensional Holonomy

Categorification is an influential idea in many areas of mathematics, and in geometry it is natural to try and think about categorifying the notions of holonomy and parallel transport in terms of higher categorical generalisations of the notions of loop, Lie group and connection on a principal bundle, in the spirit of Baez and Schreiber [B, BS]. In this article we construct a framework for 2-dimensional, or surface, holonomy along such lines. The based loops on a manifold M are replaced by what we call the (strict) thin fundamental categorical group of M , P 2 (M, * ), a monoidal category whose objects are rank-1 homotopy classes of based loops on M and whose morphisms are rank-2 homotopy classes of homotopies between based loops (or 1-parameter families of loops). Here, rank-1 homotopy, at the level of loops, means that the loops are homotopic in such a way that the rank of the differential of the homotopy between them is less than or equal to 1, i.e. the homotopies between loops do not sweep out area. Similarly, at the level of morphisms, two homotopies between based loops are rank-2 homotopic, if they themselves are homotopic in such a way that the rank of the differential of the homotopy between them is less than or equal to 2, i.e. the homotopy between homotopies does not sweep out volume. For precise definitions, we refer to subsections 1.3.1 and 1.3.2. The Lie group is replaced by a categorical Lie group, C(G), naturally obtained from a crossed module of Lie groups G = (∂ : E → G, ⊲). The connection on a principal G-bundle P over M is replaced by what we call a categorical connection, consisting of a connection 1-form on P with values in g, the Lie algebra of G, together with a 2-form on P with values in e, the Lie algebra of E, satisfying some conditions including the well-known "vanishing of the fake curvature". Locally these conditions correspond to Baez and Schreiber's [BS] local formulation for a 2-connection. The Maurer-Cartan structure equation and Bianchi identity for the curvature of an ordinary connection are shown to have natural analogues for the curvature 3-form of the categorical connection. Also a variant of the Ambrose-Singer theorem (which plays a crucial role in our construction of categorical holonomy) generalises to a higher-order version.

A categorical holonomy is defined to be a (strict monoidal) smooth functor from P 2 (M, * ) to C(G). The main result that we prove (Thm. 39) is how a categorical connection gives rise to a categorical holonomy. The underlying geometrical idea is to lift the 1-parameter family of loops into P , horizontally in one direction, namely the direction along the loops, and to use this lift to pull back the forms of the categorical connection, which are then integrated suitably.

Note that the appearance of a principal G-bundle P with a connection is natural in the context of categorical holonomies. This is because, at the level of the set of objects of P 2 (M, * ), i.e. π 1 1 (M, * ), the thin homotopy classes of based loops on M , any smooth functor P 2 (M, * ) → G gives rise to a smooth group morphism π 1 1 (M, * ) → G, and therefore it defines a principal G-bundle over M with a connection [CP, MP]. The whole construction is carried out using the language and methods of differential geometry and principal bundles, thereby avoiding working with infinitedimensional path spaces, which was an approach taken in [BS]. The construction is coordinate-free from the outset, since we use forms defined on P . We remark that the construction of P 2 (M, * ) is of interest in its own right in defining a strict thin fundamental categorical group of a manifold (previously only a weak version was known). This part of our construction is very similar to that of [BHKP, HKK].

We study the relation between our construction and non-abelian gerbes, as in [BrMe], and 2-bundles, as in [BS], in 2.4.6. Note that each 2-bundle with structure 2-group coming from a crossed module of the form (Ad : G → Aut(G), ⊲), where ⊲ denotes the obvious left action of Aut(G) in G, is naturally a non-abelian gerbe. Let G = (∂ : E → G, ⊲) be a Lie crossed module. Our construction corresponds to a particular case of G-2-bundles, for which the E-valued transition functions are trivial, and therefore the G-valued transition functions satisfy the usual cocycle condition for a principal G-bundle. Although G-2-bundles are a natural way to approach two-dimensional holonomy, we emphasise that our main goal is the definition of categorical group maps P 2 (M, * ) → C(G). This is a very strong condition, and we argue that it should force the G-2-bundle with connection to be of the special form considered here -see 2.4.6 where this point is elaborated.

At the end of this article we define the notion of Wilson sphere, which means that a categorical holonomy, taking values in the kernel of ∂ : E → G, can be associated to embedded spheres in a manifold, up to acting by an element of the group G.

1.1 Resumé

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