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Functoriality of Principal Bundles and Connections

Perhaps the most important contribution of gauge theory to general mathematics is to point out the importance of association functors. Emphasizing category theory we characterize association functors by two of their natural properties and use this characterization to establish an equivalence between the category of principal bundles and a suitably defined category of functors. From the point of view of differential geometry we detail the specialization of non-linear or Ehresmann to principal and linear connections and discuss the widely known and very useful universality of principal curvature in order to characterize the vector bundles in the image of a given association functor. This is the second version.


Introduction
Principal bundles and their association functors play a fundamental role in differential geometry and mathematical physics. Spin structures in pseudo-Riemannian geometry are defined right away as special principal bundles and the basic tenet of harmonic analysis is that the canonical association functor of a pointed homogeneous space is an equivalence of categories to the category of homogeneous fiber bundles. Last but not least the choice of principal bundle corresponds to the choice of vacuum sector in quantum field gauge theories. Nevertheless principal bundles tend to obfuscate calculations due to some inevitable arbitrariness, as one can see for example in Cartan geometries and in the botched proof of Blunder 5.24 in the otherwise excellent reference [LM]. Explicit calculations are more easily done using only the existence of association functors and the universality of curvature, arguably one of the most useful theorems in all of differential geometry.
En nuce this article brings these reservations against the use of principal bundles to a point: We show that a principal bundle GM over a manifold M is completely determined by its association functor Ass GM . Conversely every functor F from a suitable category of model fibers to the category of fiber bundles over M satisfying two more or less self-evident axioms agrees with the association functor for some principal bundle over M. Under natural transformations the class of all such functors F becomes the category GTS M of gauge theory sectors over M, which turns out to be equivalent to the category PB M of principal bundles over the manifold M.
Category theory is usually not considered to be of particular importance to differential geometry, the text books [KMS] and [L] as well as the article [SM] are notable exceptions to this rule. Besides the characterization of associated vector bundles as geometric vector bundles in the sense of [SW] the common differential geometer may find little of interest in this article. Our main motivation for studying categorical properties of principal bundles nevertheless is the need to formulate the proper analogue of the concepts of principal bundles and connections in non-commutative geometry along the lines of [D1], [D2] and [D3]. Every definition of quantum bundles with quantum connections like the one presented in [Sa] will necessarily reflect functorial properties of principal bundles in classical differential geometry.
In order to provide a more detailed outline of this article we consider a Lie group G and the category MF G of manifolds F endowed with a smooth left action ⋆ : G × F −→ F under smooth G-equivariant maps. Every principal G-bundle GM over a manifold M defines a functor from the category MF G of model fibers to the category FB M of fiber bundles over M which we may promote to a functor Ass ω GM : MF G −→ FB ∇ M to the category of fiber bundles with connections in the presence of a principal connection ω on GM. This association functor maps Cartesian products in MF G to Cartesian products in FB ∇ M and maps a manifold F endowed with the trivial G-action to the trivial fiber bundle M × F . Our first main theorem stipulates that these two properties already characterize association functors as the reader can appreciate in Theorem 5.1 Let us now consider the category PB ∇ M of principal bundles with connections over M: Objects are triples ( G, GM, ω ) formed by a Lie group G and a principal G-bundle GM over M endowed with a principal connection ω, while morphisms are tuples ( ϕ grp , ϕ ) consisting of a parallel map ϕ : GM −→ĜM of the underlying principal bundles, which is equivariant over the homomorphism ϕ grp : G −→Ĝ of Lie groups. The canonical factorization of the model homomorphism ϕ grp entails a factorization of ϕ ϕ : GM pr −→ GM/ ker • ϕ grp pr −→ GM/ ker ϕ grp ϕ −→ĜM into a parallel projection, a covering and a parallel injective immersion. In this sense every morphism in the category PB ∇ M of principal bundles with connections over M is a product of just three basic types: The removal of a connected isospin subgroup, a covering pr of principal bundles, a generalized spin structure, and a holonomy reduction ϕ.
In order to translate this description of generalized spin structures and holonomy reductions as basic type morphism between principal bundles into a truly functorial description we consider the category GTS ∇ M of gauge theory sectors with connections over M. Its objects are tuples ( G, F ) of a Lie group G together with a functor F : MF G −→ FB ∇ M satisfying the assumptions of Theorem 5.1. A morphism ( ϕ grp , Φ ) between two such gauge theory sectors is a natural transformation Φ : F • ϕ * grp −→F between the functors twisted by the pull back ϕ * grp : MFĜ −→ MF G of the action along the homomorphism ϕ grp : G −→Ĝ of Lie groups: Corollary 5.2 (Association Functor as Equivalence of Categories) For every smooth manifold M the association functor Ass provides us with an equivalence of categories from the category PB ∇ M of principal bundles to the category GTS ∇ M of gauge theory sectors with connections: In particular two principal G-bundles endowed with principal connections on M are isomorphic via a parallel, G-equivariant homomorphism of fiber bundles, if and only if their association functors are naturally isomorphic.
A direct consequence of Corollary 5.2 is that association functors are not in general full functors, this is they are not surjective on morphisms, simply because the action pull back functor ϕ * grp : MFĜ −→ MF G is not a full functor unless the image of G inĜ is dense. In other words there will be more parallel smooth homomorphisms of associated fiber bundles than there are G-equivariant smooth maps between their model fibers unless the principal connection ω has dense holonomy group.
According to Corollary 5.2 a spin structure on an oriented pseudo-Riemannian manifold ( M, g ) can be defined as a functor extending the association functor MF SO(T ) −→ FB ∇ M determined by the oriented orthonormal frame bundle of M to a functor MF Spin(T ) −→ FB ∇ M still satisfying the assumptions of Theorem 5.1, the corresponding spinor bundle $M is simply the image of the irreducible Clifford module under the extended functor. A fundamental problem in differential geometry related to spin structures is to characterize the vector and fiber bundles in the image of a given association functor. A partial answer to this problem is given in Proposition 4.6, which opens the way to an axiomatic characterization of spinor bundles and highlights the universality of principal curvatures. This paper breaks down into five sections. Section 2 is a leisurely introduction to non-linear or Ehresmann connections on fiber bundles; we relate their curvature to the commutator of iterated covariant derivatives and discuss how non-linear connections specialize to principal and linear connections. In Section 3 we generalize objects of group type in categories with Cartesian products to principal objects. Association functors are studied in Section 4, the universality of curvature is formulated in Proposition 4.4. Having proved Theorem 5.1 in Section 5 we define the category of gauge theory sectors and establish the equivalence of categories formulated in Corollary 5.2.
The research project described in this article was inspired by the first part of the article [N] and can be seen as a direct analogue of this work in the framework of differential instead of algebraic geometry, moreover we address the additional complications brought about by the presence of connections.

Fiber Bundles and Non-Linear Connections
Perhaps the single most important concept in differential geometry is the notion of connections or the closely related notion of covariant derivatives on a vector or more general on a fiber bundle over a fixed manifold M. In this section we will modify the standard category FB M of fiber bundles over M to a category more useful for our study, the category FB ∇ M of fiber bundles with non-linear connections over M. Moreover we will discuss principal and linear connections in the framework of this category.
In general a fiber bundle over a manifold M with model fiber manifold F is a manifold F M endowed with a smooth projection map π : F M −→ M, which is locally trivializable. The preimage of a point p ∈ M under π is called the fiber of the bundle over p, it is a submanifold F p M := π −1 ( p ) ⊂ F M of the total space F M diffeomorphic to the model fiber F . Fiber bundles over M are the objects in the category FB M , morphisms in this category are smooth maps ϕ : F M −→F M between the total spaces which commute with the respective projectionsπ • ϕ = π and thus map the fibers of F M to the fibers ofF M. Terminal objects in the category FB M correspond to diffeomorphisms π :M −→ M thought of as fiber bundles over M with single point fiber.
The Cartesian product of two fiber bundles F M andF M in FB M is called the fibered product in differential geometry F M × MF M and it is defined as the equalizer of π • pr L andπ • pr R in the manifold product F M ×F M.
In order to study connections in the context of category theory we prefer the following definition: Definition 2.1 (Non-linear Connections on Fiber Bundles) A non-linear connection on a fiber bundle F M over a manifold M is a field P ∇ ∈ Γ( F M, End T F M ) of projections (P ∇ ) 2 = P ∇ on the tangent bundle T F M such that its image distribution equals the vertical foliation: Every non-linear connection P ∇ on a fiber bundle F M allows us to define the first order differential operator which is the non-linear analogue of the classical definition of covariant derivatives on vector bundles. Somewhat annoyingly this covariant derivative D ∇ X f contains the redundant information f = π F M • D ∇ X f , where π F M denotes the vertical tangent bundle projection Vert F M −→ F M, the simplicity of linear and principal connections stems from the fact that we can get rid of this redundancy altogether, the reduced covariant derivative ∇ X f captures only the partial derivatives of the section f .
The Nijenhuis or curvature tensor of a non-linear connection P ∇ on a fiber bundle F M over a manifold M is the horizontal 2-form R ∇ on the total space F M of the fiber bundle with values in the vertical tangent bundle defined for two arbitrary vector fields X, Y on F M by: In particular the curvature R ∇ measures exactly the failure of the horizontal distribution ker P ∇ ⊆ T F M associated to P ∇ to be integrable. An interpretation of the curvature tensor along classical lines as a commutator of covariant derivatives is shown in [SaW] Definition 2.2 (Parallel Homomorphisms between Fiber Bundles) A parallel homomorphism between fiber bundles F M andF M over the same manifold M endowed with connections P ∇ and P∇ respectively is a homomorphism ϕ : F M −→F M of fiber bundles such that the following diagram commutes: The constraintπ • ϕ = π characterizing homomorphisms of fiber bundles in the category FB M readily implies ϕ * ( Vert F M ) ⊂ VertF M, hence the homomorphism ϕ of fiber bundles is parallel, if and only if ϕ * maps the horizontal distribution of F M to the horizontal distribution ofF M: Modifying the category FB M we define the category FB ∇ M of fiber bundles with connection over M, in this category morphisms are parallel homomorphisms of fiber bundles.
In the resulting category terminal objects are still diffeomorphisms considered as fiber bundles with single point fibers endowed with the zero connection P ∇ = 0. Besides terminal objects the category FB ∇ M has Cartesian products: The fibered product F M × MF M of two fiber bundles F M and F M over M with connections P ∇ and P∇ carries the product connection where t −→ f t and t −→f t are smooth curves in F M andF M subject to the fibered product constraint π(f t ) =π(f t ) for all t. In light of all these definitions the Cartesian product with the base manifold M becomes a functor from the category MF of smooth manifolds to the category FB ∇ because every trivial fiber bundle M × F over M comes along with the trivial connection P triv , namely the projection to the tangent bundle of F : Evidently the horizontal distribution T M × F ⊂ T (M × F ) is an integrable foliation with leaves M × {f } for every trivial connection P triv , in consequence R triv = 0 vanishes necessarily. The product functor M× defined in equation (3) will feature prominently in Sections 3 and 5.
Having discussed general non-linear connections on fiber bundles in some detail we now want to specialize to principal and linear connections in the second part of this section. Recall first of all that a principal bundle modeled on a Lie group G is a smooth fiber bundle GM with model fiber G endowed with a smooth right ρ, fiber preserving action of G on its total space GM. Also it is possible to define the affine product \ : GM × M GM −→ G.
The automorphism group bundle of a principal bundle GM over a manifold M is the Lie group bundle Aut GM over M defined by with the bundle projection π Aut GM : Aut GM −→ M, ( p, ψ ) −→ p. In mathematical physics the Fréchet-Lie group Γ( M, Aut GM ) of all global sections of the automorphism bundle is called the gauge group of GM. The fiber of the Lie group bundle Aut GM over a point p ∈ M is a Lie group Aut p GM isomorphic, although not canonically so, to the original group G, in particular its Lie algebra aut p GM ∼ = g is isomorphic to the Lie algebra of G. All these Lie algebras assemble into a smooth Lie algebra bundle aut GM, whose global sections Γ( M, aut GM ) form the Fréchet-Lie algebra of the gauge group Γ( M, Aut GM ) of the principal bundle GM.

Definition 2.3 (Principal Connections)
A principal connection on a principal G-bundle GM over a manifold M is a non-linear connection P ∇ on the fiber bundle GM, which is invariant under the right action of G on GM in the sense that the right translations R γ : GM −→ GM, g −→ gγ, are parallel automorphisms for all γ ∈ G.
In difference to general fiber bundles the vertical tangent bundle of a principal bundle GM is trivializable by This allows to establish the following well-known result

Lemma 2.4 (Principal Connection Axiom)
On every principal G-bundle GM the association P ∇ ←→ ω characterized by ω := vtriv • P ∇ induces a bijection between principal connections in the sense of Definition 2.3 and g-valued 1-forms ω on GM satisfying the axiom Cartan's Second Structure Equation [B] is a convenient description of the image of the composition of the curvature tensor R ∇ with the vertical trivialization vtriv in terms of the exterior derivative of the connection form . The strategy persued for linear connections on vector bundles V M follows the model of principal connections closely. The tangent bundle of a vector space is canonically trivializable T V ∼ = V × V by taking actual derivatives and this becomes via This map can be used to project out ∇ X v := vtriv(D ∇ X v) the redundant information from the covariant derivative D ∇ X v of a section v ∈ Γ( M, V M ): Definition 2.5 (Linear Connections on Vector Bundles) A linear connection on a vector bundle V M on M is a non-linear connection P ∇ on V M such that the reduced covariant derivative is R-bilinear: In [SaW] it is showed a proof of the following lemma.

Lemma 2.6 (Characterization of Linear Connections) A non-linear connection P ∇ on a vector bundle gives rise to an
and only if the multiplication by every λ ∈ R is a parallel endomorphism:

Principal Objects in Categories
In every category C with terminal objects and Cartesian products the notion of a group so fundamental to all of mathematics can be generalized to the notion of a group like object in C . In this section we take this beautiful idea to characterize homogeneous spaces with trivial stabilizers, generally known as principal homogeneous or affine group spaces, in terms of their structure morphisms. Moreover we apply this characterization of affine group spaces to the category FB ∇ M of fiber bundles with connections over a manifold M in order to characterize principal bundles with principal connections.
A group like object in a category C with terminal objects and Cartesian products is an object G ∈ OBJ C together a choice of structure morphisms in C called the multiplication, the inverse and the neutral element respectively with an arbitrary fixed terminal object * such that the three diagrams all commute, where e = ǫ • term equals the composition of ǫ with the terminal morphism term : G −→ * . In the category Set of sets for example the terminal objects are sets with exactly one element, hence ǫ : * −→ G essentially corresponds to an element of G. In turn the commutative diagrams above convert respectively into the associativity, the existence of a neutral element and the existence of inverses axiom in the definition of a group. In other words group like objects in Set are just plain groups.
In categories more complicated than Set the classification of group like objects can be simplified by the use of functors: Every covariant functor F : C −→Ĉ , which maps terminal objects to terminal objects and preserves Cartesian products, maps group like objects in the category C to group like objects inĈ . The standard forgetful functor MF −→ Set from manifolds to sets for examples maps a group like object in MF to a group, albeit a Lie group whose multiplication and inverse are smooths maps.
In the same vein group like objects G in the category Grp of groups carry two different group structures, one for being an object in Grp and the other due to the forgetful functor Grp −→ Set. It is a rather insightful exercise to verify that these two group structures actually agree so that G is necessarily abelian, because its multiplication m : G × G −→ G is a morphism in Grp. In consequence the fundamental group π 1 ( G, e ) of a topological group G is always abelian, because the functor π 1 maps terminal objects to terminal objects and preserves Cartesian products.
With these examples of the usefulness of functors in combination with a categorical definition of groups in mind we want to describe the concept of an affine group or principal homogeneous space in terms of category theory. Given a group like object G in a category C we define a (right) principal Gobject to be an object X ∈ OBJ C endowed with two structure morphisms in C called action and left division respectively such that the action diagrams and the following diagrams encoding simple transitivity all commute: In these diagrams pr L and pr R denote the projections to the leftmost and rightmost factor respectively, moreover m : G × G −→ G denotes multiplication in G and e : X −→ G the composition of ǫ with the terminal morphism term : X −→ * . Left principal objects can be defined in complete analogy simply by switching left and right factors. Intuitively, a principal object is essentially the group object itself, where we have forgotten the neutral element, in fact every group like object G in a category C becomes a principal object over itself under the two structure morphisms ρ := m and \ := m • (ι × id). In the category Set of sets for example a principal object over a group G is a set X endowed with a right action ρ : X × G −→ X, (x, g) −→ xg, due to the commutative diagrams in (7) and an additional application \ : X × X −→ G, (x, y) −→ x −1 y, such that the following two axioms are met for all x, y ∈ X and g ∈ G which reflect the commutative diagrams in (8). In consequence the right action of G on X is transitive with trivial stabilizers, once we have declared an arbitrary point x ∈ X to be the neutral element a principal object X = ∅ becomes indiscernible from the group G. In linear algebra for example it would be appropriate to define an affine space to be a principal object V = ∅ under the additive group underlying a vector space V over a field K.
The obvious diffeomorphism GM × M (M ×G) ∼ = GM × G of fiber bundles provides GM with a fiber preserving right action ρ : GM × G −→ GM such that each fiber G p M becomes a principal G-object in the category Set, this is to say that the action ρ is simply transitive on fibers. For every γ ∈ G the element morphism γ : { * } −→ G in the category MF induces moreover a parallel homomorphism in the category FB ∇ M of fiber bundles which is just the right multiplication R γ : GM −→ GM, g −→ gγ. In turn the non-linear connection P ∇ present on the object GM in FB ∇ M arises from a principal connection ω in the sense of Definition 2.3.
A general group like object in the category FB M of fiber bundles over a manifold M is just a bundle of Lie groups over M, a fiber bundle GM endowed with the structure of a Lie group on each fiber such that the multiplication m : GM × M GM −→ GM, the inverse ι : GM −→ GM and the neutral element section ǫ : M −→ GM are smooth. Somewhat stronger is the concept of a Lie group bundle: A bundle GM of Lie groups, which can be trivialized locally by group isomorphisms. Evidently this stronger condition is necessary and sufficient for the existence of a non-linear connection P ∇ , under which GM becomes a group like object in the category FB ∇ M .

Association Functors and Principal Bundles
Principal bundles are in a sense universal fiber bundles, every given principal bundle induces myriad fiber bundles with a large variety of model fibers over the same base manifold. The construction of all these fiber bundles is functorial in nature and best thought of as a functor, the association functor Ass ω GM , from a suitably defined category MF G of model fibers to the category FB ∇ M of fiber bundles with connections over a manifold M. In this section we study the more important properties of association functors, the universality of principal connections and their curvature and characterize all vector bundles in the image of a fixed association functor.
Besides the categories FB M and FB ∇ M of fiber bundles we are interested in the category MF G of manifolds F acted upon by a fixed Lie group G under smooth G-equivariant maps ϕ : F −→F as morphisms. Terminal objects are one point manifolds { * } and Cartesian products in the category MF G see G acting diagonally on the Cartesian product F ×F of the manifolds underlying two objects F andF . Interestingly the category MF G comes along with a canonical endofunctor, the tangent bundle endofunctor which sends an object F ∈ OBJ MF G to the tangent bundle of its underlying manifold considered as a manifold T F in its own right, on which the Lie group G acts by the differential of its characteristic action ⋆ on F : In order to define the tangent bundle functor on morphisms we observe that the differential ϕ * : T F −→ TF of a G-equivariant map ϕ : F −→F is again G-equivariant and this observation suggests T ϕ := ϕ * . It should be noted that the Lie group G provides a distinguished object in the category MF G , namely its Lie algebra g := T e G considered just as a manifold endowed with the adjoint representation Ad : G × g −→ g. The infinitesimal action links this distinguished object to the tangent bundle endofunctor: Lie group G on a smooth manifold F . The infinitesimal action of the Lie algebra g of the group G associated to this smooth action ⋆ is defined by where t −→ γ t with γ 0 = e represents the tangent vector d dt 0 γ t ∈ g. En nuce the infinitesimal action is a natural transformation from the endofunctor g× to the tangent bundle endofunctor. In fact ⋆ inf : g × F −→ T F is G-equivariant and thus a morphism in MF G for all objects F due to for all f ∈ F and all tangent vectors X = d dt 0 γ t at γ 0 = e, moreover ⋆ inf intertwines with the differential ϕ * of every G-equivariant smooth map ϕ : F −→F in the identity ϕ * (X ⋆ inf f ) = X ⋆ inf ϕ(f ).
which is well-defined in terms of representatives (g, f ) of the equivalence class [g, f ]. In other words Ass GM : Of course this isomorphism is motivated by , whenever the representative curve t −→ [ g t , f t ] for a vertical tangent vector to GM × G F has been chosen such that g t stays in the fiber of g 0 for all t. In concrete examples the automorphism group bundle Aut GM is usually more readily identified than the principal bundle GM itself due to its omnipresent action on associated fiber bundles. Consider the orthonormal frame bundle of a pseudo-Riemannian manifold ( M, g ) for example where T is a pseudo-euclidean model vector space of the correct signature and O( T ) acts from the right by precomposition ( p, F ) γ = ( p, F • γ ). For a general principal bundle GM we can use the same idea to identify the automorphism group bundle Aut GM as a Lie group bundle over M with the image of a group object in the category MF G . Letting G act on itself by conjugation ⋆ : G × G −→ G, (γ, g) −→ γgγ −1 , we obtain in fact a group object G ad ∈ OBJ MF G , whose image under the association functor is a Lie group bundle Ass GM ( G ad ) over M acting G-equivariantly on GM by for all γ ∈ G and all g,ĝ ∈ GM in the same fiber. In particular Aut p GM is isomorphic, but not naturally so, to the Lie group G in every p ∈ M.
Under this identification Ass GM ( G ad ) = Aut GM of Lie group bundles the natural action of Aut GM on associated fiber bundles GM × G F pointed out in Remark 4.3 becomes the functorial extension of the original action ⋆ considered as a G-equivariant smooth map ⋆ : G ad × F −→ F . In the same vein the functor Ass GM converts the infinitesimal action of Definition 4.1 considered as a G-equivariant map which in turn becomes the infinitesimal action associated to Remark 4.3: Before we proceed to prove the universality of principal connections and their curvature we want to digress a little to discuss the gauge principle, a fundamental principle in the study of principal bundles allowing us to translate calculations on GM to statements about M. In its most basic formulation the gauge principle is the assertion that we have a canonical bijection between the horizontal differential forms η ∈ Ω • hor ( GM, V ) on GM with values in some representation V of G satisfying R * γ η = γ ⋆ η for all γ ∈ G and general differential forms on the base manifold M with values in the associated vector bundle GM × G V . Explicitly this gauge principle reads GP[ η ] p ( X 1 , . . . , X r ) := [ g, η g (X 1 , . . . ,X r ) ] for arbitrary liftsX 1 , . . . ,X r ∈ T g GM of the argument tangent vectors X 1 , . . . , X r ∈ T p M to an arbitrary point g ∈ G p M in the fiber over p ∈ M. Due to horizontality the resulting differential form GP[ η ] does not depend on the choice of lifts and the assumption R * γ η = γ ⋆ η ensures that GP[ η ] does not depend on the choice of g ∈ G p M either. The gauge principle converts the curvature 2-form Ω ∈ Ω 2 hor ( GM, g ) of Cartan's Second Structure Equation (5) into a 2-form on M with values in aut GM: Proposition 4.4 (Universality of Principal Curvature) Every choice of a principal connection ω on a principal G-bundle GM allows us to promote the association functor Ass GM : MF G −→ FB M to a functor to the category of fiber bundles over M with non-linear connections: In other words ω induces a natural connection ∇ on GM × G F for every G-manifold F ∈ OBJ MF G . The curvature R ∇ of this induced connection is determined by the infinitesimal action of the Lie algebra bundle aut GM and the 2-form R ω ∈ Ω 2 ( M, aut GM ). More precisely for all local sections f ∈ Γ loc ( M, GM × G F ) and all X, Y ∈ Γ( M, T M ) we find: Proof: By definition GM × G F is the quotient of the Cartesian product GM × F by a free right action of the Lie group G. In turn the canonical projection pr : GM × F −→ GM × G F defines a tower of fiber bundles over M, which becomes U × (G × F ) pr −→ U × F π −→ U in a local equivariant trivialization of GM. The central idea of the proof is to choose the connection P ∇ on GM × G F such that pr is parallel with respect to the product P ω × P triv of the principal connection ω on GM and the trivial connection P triv on M × F . For this purpose we consider a curve t −→ ( g t , f t ) in GM × F and choose a curve t −→ γ t in G with γ 0 = e representing the tangent vector d dt 0 γ t = ω g 0 ( d dt 0 g t ) ∈ g. The Principal Connection Axiom 2.4 ensures and so t −→ g t γ −1 t represents a horizontal tangent vector. In turn because the first summand is horizontal and the second vertical in GM × F . Projecting this identity to equivalence classes in GM × G F we find due to the Definition 4.1 of the infinitesimal action and the choice of the curve t −→ γ t . In light of the isomorphism (10) the right hand side denotes a vertical tangent vector to GM × G F and so the latter formula defines a non-linear connection P ∇ on the fiber bundle GM × G F . With respect to this non-linear connection P ∇ the canonical projection pr : GM × F −→ GM × G F is parallel, because it maps horizontal tangent vectors d dt 0 [ g t , f 0 ] with ω g 0 ( d dt 0 g t ) = 0 to horizontal vectors. The construction of P ∇ is natural in the category MF G as well: The functorial extension Ass GM ( ϕ ) : due to the infinitesimal equivariance X ⋆ inf ϕ(f ) = ϕ * (X ⋆ inf f ). In order to calculate the curvature of the connection P ∇ we use the fact that in a tower of fiber bundles like (14) with a parallel submersion pr the curvature of the image connection P ∇ is just the image of the preimage connection P ω × P triv under the differential pr * . Using arbitrary liftsX,Ỹ ∈ T g GM of tangent vectors X, Y ∈ T p M to a point g ∈ G p M we calculate in this way ∈ Ω 2 ( M, aut GM ) is the 2-form with values in aut GM the gauge principle (12) associates to Ω : One of the most important properties of association functors is that they intertwine the actions of smooth functors on the categories Rep G and VB M . A smooth functor is an endofunctor S : Vect × R −→ Vect × R of the category of finite dimensional vector spaces under linear isomorphisms such that This extension to representations makes the classification of smooth functors an exercise in the representation theory of general linear groups: Every smooth functor is naturally isomorphic S ∼ = S 1 ⊕ . . . ⊕ S r to a finite direct sum of Schur functors S 1 , . . . , S r twisted by density lines [FH].
In the same vein every smooth functor S extends naturally to an endofunctor of the category VB ∇ M of vector bundles with connections over a manifold M. The smoothness of S allows us to define a differentiable structure on the disjoint union of vector spaces obtained by applying S fiberwise to obtain a new vector bundle S V M over M; every connection ∇ on the original vector bundle V M extends naturally to a connection ∇ S on S V M by the requirement that parallel transport with respect to this connection along an arbitrary curve t −→ p t in the manifold M is simply the image of parallel transport with respect to ∇ under the functor S. Because parallel transport in associated vector bundles is essentially the image of parallel transport in the principal bundle GM itself, every association functor Ass ω GM intertwines the two extensions of a smooth functor S to the categories Rep G of representations and VB ∇ M of vector bundles with connections: Classically the vector bundles of the form S T M on a manifold M with a smooth functor S are called pseudotensor bundles, their sections pseudotensors, and they comprise exactly the natural vector bundles of order one. Some modern authors however seem to confuse the classical concept of tensors with the property of having a value defined at every point.

Lemma 4.5 (Properties of Association Functors)
Consider a principal G-bundle GM over a manifold M endowed with a principal connection ω and the corresponding association functor from the category MF G of manifolds endowed with smooth G-actions to the category FB ∇ M of fiber bundles over M endowed with non-linear connections: 1. The association functor Ass ω GM preserves Cartesian products: 2. On the full subcategory MF ⊂ MF G of manifolds with trivial Gaction the association functor Ass ω GM agrees with the product functor: Proof: Of course all three statements of this lemma are easily proved directly by unwrapping all the definitions made above; the second statement for example is an elaborate description of the trivial fiber bundle isomorphism whenever G acts trivially on F and thus effectively only on the first factor of GM × F in the construction of the quotient GM × G F . This fiber bundle isomorphism is evidently natural, it is compatible with all the fiber bundle homomorphisms induced by smooth maps ϕ : F −→F between manifolds F andF with trivial G-action. Nevertheless we think the lemma is quite interesting, because the third is actually a consequence of the first two statements. Combining the existence of additive inverses and the unity axiom ∀v : 1 · v = v into the axiom ∀v : v + (−1) · v = 0 we see that only three structure maps are needed to formulate all axioms for a vector space object V in a category C in terms of commutative diagrams provided we have specified a field object K: In the category MF G for example we may take the manifold R with the trivial G-action as the field object K = R triv , the corresponding vector space objects are smooth representations of the Lie group G over R.
On the other hand the first and second statement of the lemma assert that the association functor Ass ω GM preserves Cartesian products and agrees with the product functor M × on the full subcategory MF ⊂ MF G . In consequence Ass ω GM sends terminal objects in MF G to terminal objects in FB ∇ M and a representation V to a fiber bundle V M := GM × G V with three parallel structure maps, the zero section 0 : M −→ V M and: According to Lemma 2.6 the parallelity of the scalar multiplication map alone suffices to force the non-linear connection P ∇ on V M ∈ OBJ FB ∇ M to be a linear connection in the sense of Definition 2.5.
Historically the concept of principal bundles and principal connections arose from Cartan's beautiful idea of moving frames, which asserts that every vector bundle V M with connection ∇ lies in the image of the association functor Ass ω GM for some principal bundle with connection. A suitable choice for the principal bundle GM is the frame bundle with model vector space V which is a principal GL V -bundle over M with right multiplication given by precomposition (p, F ) γ = (p, F • γ). The tautological diffeomorphism is a parallel isomorphism for the principal connection on GL(M, V M) constructed from the parallel transport PT ∇ t : V p 0 M −→ V pt M with respect to ∇ along the curve t −→ p t ; the principal connection axiom of Lemma 2.4 is particularly easy to verify using this definition for ω.
In consequence of this moving frames argument it does not make too much sense to ask, whether or not a vector bundle with connection is in the image of some association functor. The appropriate answer to this question for an association functor fixed in advance is definitely more interesting and was given in the master thesis of one of the authors. A closely related concept is the concept of geometric vector bundles defined in [SW]: which is a representation of the Lie algebra aut p GM at every p ∈ M with the additional property that the curvature of the given connection ∇ agrees with the pointwise action of the curvature R ω ∈ Ω 2 ( M, aut GM ): Proof: Consider to begin with the vector bundle V M := GM × G V associated to a representation V of the Lie group G. According to our discussion of the infinitesimal action following Definition 4.1 the composition is G-equivariant and thus gives rise to a parallel R-bilinear map, which is a representation ( ⋆ inf ) p of the Lie algebra aut p GM on V p M in every point: Conversely assume that ⋆ inf : aut GM × M V M −→ V M is a parallel representation of the Lie algebra bundle aut GM on a vector bundle V M with a linear connection P ∇ . According to equation (11) the fiber Lie group Aut p GM is isomorphic to G in every point p ∈ M and so simply connected, in consequence the infinitesimal action ( ⋆ inf ) p of its Lie algebra aut p GM integrates to a representation of the Lie group Aut p GM on the vector space V p M. Though slightly technical it is straightforward to prove that the integrated representation depends smoothly on the point p ∈ M the details of this argument are left to the reader. In addition to the vector bundle V M with its connection P ∇ we consider the vector bundle GM × G V associated to some representation V of G endowed with the linear connection P ω induced by the principal connection ω in Proposition 4.4. The two connections determine a linear connection P (ω,∇) on the vector bundle Hom( GM × G V, V M ) characterized by the fact that its parallel transport PT (ω,∇) t along an arbitrary curve t −→ p t makes the following diagram commute for all linear maps F : where PT ω t and PT ∇ t are the parallel transports along the same curve with respect to P ω and P ∇ .
The principal idea of the proof is now to construct a parallel and actually flat vector subbundle of the vector bundle Hom( GM × G V, V M ) over M. For this purpose we consider the family of vector subspaces of the fibers In order to show that this family of subspaces is the family of fibers of a vector subbundle of Hom( GM × G V, V M ) we observe that the parallel transport in both vector bundles GM × G V and V M along a curve t −→ p t is equivariant over the parallel transport with respect to the Lie group connection P ω on the automorphism bundle Aut GM induced by ω. More precisely we find is parallel by assumption. In consequence the parallel transport PT (ω,∇) with respect to the linear connection P (ω,∇) specified in diagram (17) By assumption the underlying manifold M is (simply) connected, and hence all vector subspaces [ Hom Aut GM ( GM × G V, V M ) ] p have the same dimension. With parallel transport depending smoothly on the curve we conclude that Hom Aut GM ( GM × G V, V M ) is a genuine vector subbundle of , moreover it is a parallel subbundle as it is invariant under parallel transport along arbitrary curves.
On the other hand the curvature of the linear connection P (ω,∇) on the vector bundle Hom( GM × G V, V M ) is determined by the universality of principal curvature discussed in Proposition 4.4, namely it holds true that for all tangent vectors X, Y ∈ T p M and F ∈ Hom p ( GM × G V, V M ). Due to equivariance the curvature of the connection P (ω,∇) restricted to the parallel vector subbbundle Hom Aut GM ( GM × G V, V M ) vanishes identically, put differently Hom Aut GM ( GM × G V, V M ) is a flat vector bundle over M under the restriction of the connection P (ω,∇) . In the argument presented so far the actual choice of the representation V did not play any role. In order to make a diligent choice we fix a frame g ∈ G p M over a point p ∈ M and consider the Lie group isomorphism which is essentially the Lie group bundle isomorphism (11) restricted to the fiber of p. This Lie group isomorphism allows us to pull back the integrated representation (16) of Aut p GM on the vector space V := V p M to a smooth representation ⋆ : G × V −→ V enjoying the critical property that is an equivariant vector space isomorphism under Aut p GM in the sense: In consequence the fiber of the vector bundle Hom Aut GM ( GM × G V, V M ) over the chosen point p ∈ M contains the vector space isomorphism Φ, which translates under parallel transport along arbitrary curves with respect to the flat connection P (ω,∇) into a parallel, globally defined section Φ on the simply connected manifold M. Evaluation of this parallel section in the points of M converts it into a parallel isomorphism of vector bundles: Every association functor is in a sense a reproducing functor, there exists in its source category an object, whose image in its target category is isomorphic to the principal bundle defining the association functor in the first place. Based on this simple observation we characterize the association functors among all functors from MF G to FB ∇ M in this section, moreover we establish an equivalence of categories between the category of principal bundles and a suitably defined category of functors called gauge theory sectors.
Consider the smooth action of a given Lie group G on its underlying manifold by left multiplication ⋆ : G × G −→ G, (γ, g) −→ γg, which defines an object G left ∈ OBJ MF G in the category of G-manifolds. The image of G left under the functor Ass ω GM is isomorphic as a fiber bundle to GM and the inverse isomorphism g −→ [g, e] is easily verified to be parallel with whenever d dt 0 g t is horizontal in the sense ω( d dt 0 g t ) = 0. This reproducing property of Ass ω GM lies at the heart of the proof of the following theorem: Theorem 5.1 (Characterization of Association Functors) Consider a covariant functor F : MF G −→ FB ∇ M from the category of G-manifolds to the category of fiber bundles with connection over M. If the functor F preserves Cartesian products and agrees with the product functor on the full subcategory MF ⊂ MF G of manifolds with trivial G-action, then F is naturally isomorphic to the association functor corresponding to some principal G-bundle GM endowed with a principal connection ω.
Proof: Consider a functor F : MF G −→ FB ∇ M from the category of Gmanifolds to the category of fiber bundles over M endowed with non-linear connections, which preserves Cartesian products and agrees with the product functor M× : MF −→ FB ∇ M on the full subcategory of trivial Gmanifolds. At least three different objects in the domain category MF G of the functor F have underlying manifold equal to the Lie group G: The difference between these three objects in MF G resides in their actions, which is by left multiplication γ ⋆ g := γg and conjugation γ ⋆ g := γgγ −1 respectively for G left and G ad , whereas G acts trivially on G triv . Every terminal object in the category MF G is a zero-dimensional manifold point { * } with necessarily trivial G-action, hence F maps it to the terminal object M × { * } in the category FB ∇ M . In other words the functor F maps terminal objects to terminal objects and preserves Cartesian products and in consequence turns group like and principal objects in the category MF G into group like and principal objects in the category FB ∇ M . With G acting by automorphisms on both G ad and G triv both objects are group like objects in the category MF G under the multiplication and inverse inherited from G. The significance of the group like object F( G ad ) in the category FB ∇ M may be somewhat obscure at this point, the group like object F( G triv ) = M × G however is just the trivial G-bundle over M endowed with the trivial connection. Moreover the original Lie group multiplication defines G-equivariant structure maps in analogy to definition (6) by means of ρ(g,ĝ) := gĝ and \(g,ĝ) := g −1ĝ , which naturally enough turn G left into a G triv -principal object in the category MF G . According to Lemma 3.1 the image of G left is a principal G-bundle GM := F( G left ) over the manifold M endowed with a principal connection ω. In passing we observe that the group like object G ad acts G-equivariantly on G left via and this action identifies the group like object F( G ad ) in the category FB ∇ M with the gauge group bundle Aut GM of GM by means of the action: It remains to show that the original functor F is naturally isomorphic to the association functor Ass ω GM . For this purpose we consider a general object F ∈ OBJ MF G ; replacing its G-action by the trivial G-action on the same underlying manifold we project it to an object F triv ∈ OBJ MF in the subcategory of manifolds with trivial G-action. The G-equivariant map is actually an isomorphism in MF G with inverse ( g, f ) −→ ( g, g ⋆ f ), which fits for an arbitrary element γ ∈ G into the commutative diagram in the category MF G , where ρ γ : G left −→ G left , g −→ gγ, denotes the right multiplication by γ and ⋆ the original G-action characterizing the object F thought of as a G-equivariant (sic!) map ⋆ : G left × F triv −→ F . Writing the right multiplication ρ γ in the category MF G as a composition factorizing over the element morphism γ : { * } −→ G triv in the subcategory MF ⊂ MF G we conclude that F( ρ γ ) : GM −→ GM agrees with the right multiplication R γ : GM −→ GM, g −→ gγ, in the principal bundle GM induced by F( ρ ) : GM × G −→ GM, because F preserves Cartesian products and agrees with the product functor M× on the trivial G-manifolds { * } and G triv . In consequence the commutative diagram (19) translates under the functor F into the following commutative diagram in the category FB ∇ M with F M := F( F ), because F preserves Cartesian products, hence preserves projections and agrees on manifolds with trivial G-action like F triv with the product functor M×. The parallel homomorphism F( ⋆ ) : GM × F −→ F M thus descends to the quotient F( ⋆ ) : GM × G F −→ F M of GM × F by the right G-action defining the associated fiber bundle GM × G F , which lets γ ∈ G act by R γ × (γ −1 ⋆). It goes without saying that the projection pr R : GM × M F M −→ F M factors through the quotient of GM × M F M by the right G-action on the principal bundle GM, the commutative diagram (20) ensures moreover that the quotient diagram is actually an isomorphism in the category FB ∇ M , moreover the construction of this parallel fiber bundle isomorphism F( ⋆ ) : Ass ω GM F −→ F( F ) for a given object F ∈ OBJ MF G is natural under morphisms in MF G and comprises a natural isomorphism F( · ) : Ass ω GM −→ F of functors. In order to press the point of Theorem 5.1 home let us define two rather special categories associated to a smooth manifold M. Objects in the category PB ∇ M of principal bundles with connections over M are triples ( G, GM, ω ) formed by a Lie group G and a principal G-bundle GM over M endowed with a principal connection ω. Every morphism between two such objects ( ϕ grp , ϕ ) : ( G, GM, ω ) −→ (Ĝ,ĜM,ω ) consists of a parallel homomorphism ϕ : GM −→ĜM of fiber bundles which is G-equivariant over the Lie group homomorphism ϕ grp : G −→Ĝ.
Of course the association functors Ass ω GM • ϕ * grp and AssωĜ M are explicitly specified on morphisms in Definition 4.2 and both vertical arrows turn out to be the right multiplication [ g,Γ ] −→ [ g,Γγ ] byγ ∈Ĝ. In turn we find ϕ(gγ) = Φ(Ĝ left ) [ gγ,ê ] = Φ(Ĝ left ) [ g, ϕ grp (γ) ] = ϕ(g) ϕ grp (γ) for all g ∈ GM and γ ∈ G and conclude that ϕ is equivariant over ϕ grp . Eventually we consider for an arbitrary objectF ∈ OBJ MFĜ the orbit map orbf :Ĝ left −→F ,γ −→γ ⋆f , associated to an elementf ∈F as a morphism in the category MFĜ with associated commutative diagram: Definition 4.2 provides again an explicit description of the two vertical arrows and the top arrow reads [ g,γ ] −→ [ ϕ(g),γ ], the commutativity of the diagram thus implies that Φ(F ) is given by [ g,f ] −→ [ ϕ(g),f ]. In other words the two natural transforms Φ and Φ ϕ agree on arbitrary objects and so the functor Ass is full, this is surjective on morphisms. In order to show that Ass is injective on morphisms or faithful the reader may simply verify that the equivariant map GM −→ĜM defined in equation (21) equals ϕ in case we start with the natural transformation Φ = Φ ϕ .
Mutatis mutandis the arguments presented in this section work without taking connections into account: A functor F : MF G −→ FB M is naturally isomorphic to the association functor Ass GM for some principal bundle GM, if and only if F preserves Cartesian products and agrees with the product functor M× : MF −→ FB M on the full subcategory of trivial Gmanifolds. Suitably defined categories of principal bundles and gauge theory sectors then turn the association functor into an equivalence of categories: