A classic result in the foundations of Yang-Mills theory, due to Barrett [Int. J. Theor. Phys. **30**, 1171–1215 (1991)], establishes that given a “generalized” holonomy map from the space of piece-wise smooth, closed curves based at some point of a manifold to a Lie group, there exists a principal bundle with that group as structure group and a principal connection on that bundle such that the holonomy map corresponds to the holonomies of that connection. Barrett also provided one sense in which this “recovery theorem” yields a unique bundle, up to isomorphism. Here we show that something stronger is true: with an appropriate definition of isomorphism between generalized holonomy maps, there is an equivalence of categories between the category whose objects are generalized holonomy maps on a smooth, connected manifold and whose arrows are holonomy isomorphisms, and the category whose objects are principal connections on principal bundles over a smooth, connected manifold. This result clarifies, and somewhat improves upon, the sense of “unique recovery” in Barrett’s theorems; it also makes precise a sense in which there is no loss of structure involved in moving from a principal bundle formulation of Yang-Mills theory to a holonomy, or “loop,” formulation.

## REFERENCES

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In what follows, we limit attention to manifolds that are smooth, Hausdorff, and paracompact, and will no longer state these assumptions explicitly.

Everything discussed here is independent of the particular choice of reparameterization of the composition. The standard method is to define $\gamma 2\u2022\gamma 1(t)={\gamma (2t)for\u2009t\u226412\gamma (2(t\u221212))for\u2009t\u226512$.

Note that the (generalized) holonomy maps *H*_{Γ,u} determined in this way really do depend on the choice of *u* ∈ *P*, even for connected manifolds; changing base point, even within a fiber, yields a holonomy map that is conjugate in *G* to the one we began with, so if *u*_{2} = *u*_{1}*g*, then *H*_{Γ,u2}(*γ*) = *g*^{−1}*H*_{Γ,u1}(*γ*) *g* for any *γ* ∈ *L*_{π(u1)}. In the sequel, we make precise a sense in which these are nonetheless isomorphic holonomy maps.

For another version of this worry, used to question the significance of Barrett’s result, see Healey.^{19}

See the discussion at http://math.ucr.edu/home/baez/qg-spring2004/discussion.html.

**81**, 1073–1091 (

**83**(5) (2016).