A Framework for Geometric Field Theories and their Classification in Dimension One

In this paper, we develop a general framework of geometric functorial field theories, meaning that all bordisms in question are endowed with geometric structures. We take particular care to establish a notion of smooth variation of such geometric structures, so that it makes sense to require the output of our field theory to depend smoothly on the input. We then test our framework on the case of $1$-dimensional field theories (with or without orientation) over a manifold $M$. Here the expectation is that such a field theory is equivalent to the data of a vector bundle over $M$ with connection and, in the nonoriented case, the additional data of a nondegenerate bilinear pairing; we prove that this is indeed the case in our framework.


Introduction
Inspired by work of Witten [16], Segal and Atiyah pioneered the mathematical description of quantum field theories as functors [1,12]. More precisely, they described a d-dimensional quantum field theory Z as a functor that assigns to a closed (d − 1)-manifold Y a vector space Z(Y ) and to a d-dimensional bordism X from Y to another closed d-manifold Y a linear map Z(X) : Z(Y ) → Z(Y ). Moreover, Z is required to be a symmetric monoidal functor, which means that Z applied to a disjoint union of manifolds of dimension d − 1 or d corresponds to the tensor product of the associated vector spaces or linear maps. Segal's paper focused on conformal field theories, which means that the manifolds involved come equipped with conformal structures, while Atiyah discusses topological field theories, where the manifolds are smooth, but not equipped with any additional geometric structure.
Our first goal in this paper is to develop a general framework for geometric field theories. This involves a general definition of a "geometric structure" G on d-dimensional manifolds, which then leads to the definition of a symmetric monoidal bordism category GBord whose morphisms are d-dimensional bordisms equipped with a G-structure. This is much more general than the conformal structures considered by Segal or the rigid structures based on the action of a Lie group G on a d-dimensional model space M of Stolz and Teichner [13]. Then we essentially follow [13] to define G-field theories. As discussed at length in that paper, it is crucial to ensure the smoothness of the field theories; intuitively, this means in particular that the operator Z(X) : Z(Y ) → Z(Y ) associated to a bordism X from Y to Y depends smoothly on the bordism X. At a technical level, this means that we need "family versions" of the bordism category GBord and the target category Vect of suitable vector spaces whose objects and morphisms are now families of the originally considered objects/morphisms, parametrized by arXiv:2001.05721v2 [math.DG] 25 Jul 2021 smooth manifolds. In [13] this is implemented by considering GBord and Vect as categories internal to the 2-category of smooth stacks, but it has become clear that, for technical reasons, it is easier to construct and to work with the complete Segal object in smooth stacks that should be thought of as the "nerve" of the internal category we considered before, as is done in the preprint [2]. We carry out our constructions for non-extended field theories. It is possible to define extended field theories using an extension of our approach, via d-fold (or d-uple) Segal objects. For one-dimensional field theories, which are the main object of study in this article, these distinctions are irrelevant.
The second goal of this paper is to check whether this abstract and involved definition yields something sensible in the simplest cases, namely 1-dimensional (oriented) topological field theories over a manifold M . In other words, the geometric structure on 1-manifolds X is simply a smooth map γ : X → M , or such a map γ plus an orientation on X. There are actually two versions of each of these results, depending on whether all vector spaces involved are real or complex. For the field theories, the two flavors come from the choice of the target category as the category (of families of) real or complex vector spaces. Similarly, the vector bundles over M considered can be real or complex. Theorem 1.1 is certainly the expected result. The basic idea is that a vector bundle E → M with connection ∇ determines a 1-dimensional field theory Z over M which associates to a point x (interpreted as an object of GBord) the vector space Z(x) = E x given by the fiber over x, and to a path γ : [a, b] → M (interpreted as a morphism in GBord from γ(a) to γ(b)) the linear map Z(γ) : Z(x) → Z(y) given by parallel translation along the path γ.
In fact, there are closely related results in the literature, in work of Freed [4,Appendix A], and, in particular, in the papers by Schreiber and Waldorf [10] and by Berwick-Evans and Pavlov [2], whose title indeed seems a statement of our first theorem. Indeed, our framework is closely related to that of the latter paper; however, our goal to give a general definition of geometric bordism categories leads to a different bordism category even in dimension one, as explained below (see Section 2.1). In [3,10], invariance under "thin" homotopies plays a prominent role. These concepts turned out to be not relevant to the present paper, as we were able to prove the main results (in particular Proposition 4.3) without such assumptions. This paper is organized as follows. In Section 2, we give a more detailed exposition of our construction and discuss the differences to the papers cited above. Afterwards, in Section 3, we define our notion of geometry, and use it to define a smooth category of geometric bordisms, for any geometry, in any dimension. Starting in Section 4, we restrict to the case of field theories in dimension one. In particular, we prove a version of the classification Theorems 1.1 and 1.2 under the technical assumption that "families of vector spaces" are finite-dimensional vector bundles. As discussed in Section 2.4 for a geometric field theory Z, unlike for topological field theories, the vector space Z(Y ) associated to an object Y of the bordism category GBord is typically not finite-dimensional. This in turn leads to the requirement that the vector spaces Z(Y ) need to be equipped with a topology or a "bornological structure" (see Appendix A) in order to formulate the requirement that the operator Z(X) : Z(Y ) → Z(Y ) associated to a bordism X from Y to Y depends "smoothly" on X. As also explained in Section 2.4, the appropriate notion of an "S-family of (topological or bornological) vector spaces" needs to be more general Figure 1. An object of C n comprises the d-manifold X and the marked hypersurfaces Y 0 , . . . , Y n . than locally trivial bundles over the parameter spaces S, namely sheaves of O S -modules. These are the objects of the target category appropriate for general field theories, and so we consider the version of Theorems 1.1 and 1.2 for that target category as the main result of this paper. This is proved in Section 5 (see Theorem 5.2 and Remark 5.3 for the precise statement).

Discussion of the results
In this section we provide an informal overview of our framework of geometric field theories, a discussion of our motivations, and comparisons to the existing literature.

Bordisms in the Segal approach
In the presence of geometric structures, it is difficult to perform the gluing of bordisms along their boundaries in a systematic way, as needed to define composition in geometric bordism categories; see for instance the discussion in the introduction of [2]. Here the idea of Segal objects comes to the rescue, as it allows one to instead consider only decomposition of bordisms along hypersurfaces, which is unproblematic. In this approach, a category C is encoded by its nerve, that is, the simplicial set C, where C 0 is the set of objects and, for n ≥ 1, C n is the set of chains of n composable morphisms; composition and identity morphisms in C determine the simplicial structure maps between these sets. To describe, at least roughly, the d-dimensional bordism category in this way, we let C n consist of d-dimensional manifolds X together with a collection of compact hypersurfaces Y k ⊂ X, for k = 0, . . . , n, as in Figure 1. This encodes a chain of n composable bordisms, the kth of them being the portion of X lying between Y k−1 and Y k . Composition is encoded by forgetting the marked hypersurfaces. To build in a geometry G (for instance, orientations, Riemannian metrics, or maps to a background manifold M ), we just ask that X is endowed with that additional structure.
In particular, objects of C (i.e., elements of C 0 ) consist of a d-dimensional manifold X with a marked compact hypersurface Y , instead of just the (d − 1)-dimensional manifold Y . Now, this set of objects is much larger than what we would like to have, since the portion of X far away from Y should be irrelevant. This issue can be dealt with by promoting C 0 from a set to a groupoid ; we add isomorphisms that establish suitable identifications between the pairs Y ⊂ X. (The same approach applies later on, as we work fibered over Man, by promoting a certain sheaf to a stack.) This shifts the problem to making a choice of such isomorphisms.
The choice made in [2] is to say that morphisms in C 0 are maps ϕ : Y → Y that have an extension to a diffeomorphism between open neighborhoods of Y and Y in X and X . This makes the concrete embedding of the hypersurface immaterial and ensures that the set of isomorphism classes of objects is precisely the set of (d − 1)-dimensional manifolds, without any extra data. The issue with this approach is that, while it works in the special case at hand, it does not generalize to arbitrary geometries G, since we are not allowed to restrict a G-structure on X to one on the hypersurface Y . Moreover, even for those G which make sense in any dimension and allow restricting to hypersurfaces, it may not be true that a G-isomorphism is determined by its data on a hypersurface.
Our choice for morphisms in C 0 is designed to accommodate for any geometry G, in the sense of Section 3.3, and is as follows. First we remark that one of our axioms for a geometry is that one can always restrict it to an open subset of a G-manifold X. We then decree that a morphism between two pairs X ⊂ Y and X ⊂ Y in C 0 is determined by a G-isomorphism defined on an open neighborhood U ⊆ X of Y , the underlying smooth map of which sends Y to Y ; we identify two such G-isomorphisms defined on, say, U and U if they coincide on some smaller neighborhood V ⊂ U ∩ U of Y . Concisely, morphisms in C 0 are germs of G-isometries at the marked hypersurfaces.
Further stages of the simplicial object C are constructed in a similar fashion.

Points versus germs of paths
Our definition of C 0 raises another difficulty, which is generally unavoidable from our point of view: The set of isomorphism classes of objects is huge, as each different germ of the geometric structure determines its own isomorphism class. This is already true for the case of one-dimensional bordisms over a target manifold M , which is considered in this paper. Here, an object in the bordism category can no longer be pictured as a point of M (or, more generally, a finite collection of such); instead, objects are germs of paths in M , a much larger space.
The main results of our paper (Theorems 1.1 and 1.2) say that, at least in the one-dimensional case, this does not make a difference: A field theory Z ∈ 1-TFT(M ) is completely blind to the germ information, and its value on objects of C contains no more data than that of a vector bundle over M , as expected. This can be seen as a "reality check" for our definition of geometric field theories.
A typical heuristic argument as to why the germs do not matter is that the space of germs of paths in M deformation retracts to M . A field theory Z indeed defines a vector bundle on this space of germs, viewed as a diffeological space. However, at this level of generality, the familiar homotopy invariance of vector bundles breaks down. So, instead, we will use the data assigned by Z to higher simplicial levels to show that Z| C 0 is determined by a vector bundle on M .

Building in smoothness
A second technical layer in our framework comes from the need to formalize the idea that our field theories should be smooth. This is already explained in detail in [13] and adapted to the Segal approach in [2]. The idea here is that a smooth category C is a complete Segal object in the 2-category of (symmetric monoidal) stacks over the site of manifolds; compare this with the preliminary description of above, where C was a Segal object in the 2-category of (symmetric monoidal) groupoids. Thus, for each integer n ≥ 0 and each smooth manifold S (a "parameter space"), we have a groupoid C n,S of S-families of chains of n composable morphisms; this data is functorial in the variables n and S.
To promote the bordism category to a smooth category, we need to fix the meaning of "Sfamily of bordisms" X/S. In a nutshell, this will be defined to be a submersion π : X → S such that each fiber π −1 (s) is a bordism in the previous sense. It remains to explain what a geometry G is in this new context. Before, G could be defined, technically, to be a sheaf or a stack on the site Man of smooth manifolds; thus, to each X, corresponds a set (or groupoid) G(X) of Gstructures on X. To extend this to families, we introduce, in Section 3, a new site of families of d-dimensional manifolds, denoted Fam d . Its objects are submersions π : X → S with ddimensional fibers. A d-dimensional geometry is now simply a sheaf or stack on the site Fam d . To illustrate this, consider the geometry G of (fiberwise) Riemannian metrics; if X/S ∈ Fam d , then G(X/S) is the set of inner products on the vertical tangent bundle Ker(T π : T X → T S).

Appropriate families of vector spaces
To promote the codomain of our field theories to smooth categories, we must, likewise, specify what we mean by an S-family of vector spaces. It is well known that for a topological field theory Z the vector space Z(Y ) associated to any object Y of a topological bordism category is finite-dimensional. So it is natural to declare that an S-family of vector spaces is simply a finite-dimensional, locally trival smooth vector bundle over S, and we will indeed consider exclusively this case in Section 4. Notice that with this choice, a field theory Z, as a particular example of a smooth functor, will assign to an S-family X/S of bordisms a linear map Z(X/S) of vector bundles over S -an S-family of linear maps.
For geometric field theories, the vector spaces Z(Y ) are typically not finite-dimensional. For example, the quantum mechanical description of a particle moving in a compact Riemannian manifold N is given by a 1-dimensional Riemannian field theory Z which associates to the object given by Y = {0} ⊂ (− , ) = X (with the standard Riemannian metric on the interval (− , )) the "vector space of functions on N ". Let X/S be an S-family of bordisms such that for every s ∈ S, the fiber X s is a bordism from Y to Y (where Y and Y do not depend on s). Then the smoothness requirement for Z in particular says that the maps S → Z(Y ) given by s → Z(X s )v are smooth, for all v ∈ Z(Y ). If Z(Y ) is infinite dimensional, a topology (or a bornological structure) on Z(Y ) is needed to define a smooth map with target Z(Y ). In quantum mechanics, the vector spaces are traditionally equipped with a Hilbert space structure; for instance, in the case of a particle moving in a Riemannian manifold, the vector space of functions on N is interpreted as L 2 (N ), the Hilbert space of square-integrable functions. However, as discussed in [13,Remark 3.15] there are difficulties formulating the smoothness of the functor Z if the target category is built from families of Hilbert spaces; instead, topological (or bornological) vector spaces are used. In the quantum mechanics example, it is the space C ∞ (N ) of smooth functions on N , equipped with its standard Fréchet topology.
It might seem appealing to define an S-family of topological or bornological vector spaces to be a locally trivial bundle of such spaces over S with smooth transition functions. Such a definition, though, has very undesirable consequences for the topology of the space of field theories for a fixed geometry G; namely, for any object Y of GBord, the isomorphism type of the topological vector space Z(Y ) is invariant on the path component of Z in the space of field theories. The heuristic reason is that if there is a path of field theories Z t , t ∈ [0, 1], then we have a family Z t (Y ) of topological vector spaces parametrized by [0, 1]. If we interpret that to mean a locally trivial vector bundle, then in particular Z 0 (Y ) and Z 1 (Y ) are isomorphic. This is, in general, an unexpected feature of field theories. For instance, it is conjectured by Stolz and Teichner [13] that supersymmetric Euclidean field theories provide cocycles for certain cohomology theories; in particular, to 1|1-dimensional correspond K-theory classes. But the dimension of a vector bundle representing a K-theory class is not an invariant of the class (only ist virtual dimension), so this should also not be the case at the field theory level.
We choose to deal with this by dropping the local triviality condition and defining an Sfamily of topological (or bornological) vector spaces to be a sheaf of such spaces over S which is a module over the sheaf of smooth functions on S. This includes vector bundles over S by associating to a vector bundle its sheaf of sections. It then becomes a fact requiring proof that, under the additional assumption that a field theory is topological, all the families of vector spaces involved turn out to be locally trivial.

Homotopy invariance considerations
One focus in Berwick-Evans and Pavlov [2] is to endow the category of smooth categories (dubbed C ∞ -categories there) with a model structure. This lets one conclude that the space of field theories is insensitive to fine details in the definitions, as long as everything remains weakly equivalent. It also lets one compute with simplified models of the bordism category, since all that matters is that the cofibrancy condition is met, and this is easy in their model structure. We make no attempt to address questions of homotopy meaningfulness in this paper; rather, our focus is on the techniques dealing with the geometric situation.
Lastly, we remark that our bordism category 1-Bord or (M ) possesses an obvious forgetful map to the bordism category of [2]. Since in the model structure on the category of C ∞ -categories considered there, weak equivalences are just fiberwise equivalences of groupoids, the discussion above shows that this forgetful map is not a weak equivalence in this model structure. Therefore, our result does not follow from that in [2].

Smooth functors and geometric field theories
Functorial field theories are functors from an appropriate bordism category to a suitable target category. The bordisms in the domain category might come equipped with geometric structures, in a sense to be clarified in this section. After providing examples of geometric structures in Section 3.1 and recalling the language of fibered categories and stacks in Section 3.2, we provide a general definition of "geometries" in Section 3.3. In Section 3.6 we construct the geometric bordism category GBord, which is an example of a smooth category, a concept defined in Section 3.4. In the final Section 3.7 we define geometric field theories as "smooth functors" from GBord to a suitable smooth target category.

Examples of geometries
The goal of this subsection is to define what we mean by a geometry on smooth manifolds of a fixed dimension d, see Definition 3.7. To motivate that abstract definition, we begin by listing well-known structures on manifolds that will be examples of "geometries", and distill their common features into our Definition 3.7.
Examples 3.1. The following are examples of "geometries" on a d-manifold X which we would like to capture in an abstract definition: 1. A Riemannian metric or a conformal structure on X.

2.
A reduction of the structure group of the tangent bundle of X to a Lie group G equipped with a homomorphism α : G → GL(d). More explicitly, such a structure consists of a principal G-bundle P → X and a bundle map α X : P → Fr(X) to the frame bundle of X (whose total space consists of pairs (x, f ) of points x ∈ X and linear isomorphisms f : The bundle map α X is required to be G-equivariant, where the right action of g ∈ G on Fr(X) is given by ( Interesting special cases of reductions of the structure group include the following: (a) A GL + (d)-structure on X is an orientation on X (here and in the following three examples, the group G is a subgroup of GL(d), and α : G → GL(d) is the inclusion map). (b) An SL(d)-structure on X is a volume form on X. (c) An O(d)-structure on X is a Riemannian metric on X. (d) An SO(d)-structure is Riemannian metric plus an orientation. (e) A Spin(d)-structure on X is a Riemannian metric plus a spin structure (here α is the composition of the double covering map Spin(d) → SO(d) and the inclusion SO(d) → GL(d)).

3.
A rigid geometry is specified by a d-manifold M (thought of as a "model manifold") and a Lie group G acting on M (thought of as "symmetries" of M). Given this input, a (G, M)-structure on X is determined by the following data, which we refer to as a (G, M)-atlas for X: • an open cover {X i } i∈I of X, • embeddings φ i : X i → M for i ∈ I (the charts of the atlas), • group elements g ij ∈ G for X i ∩ X j = ∅ which make the diagram commutative and satisfy a cocycle condition (these are the transition functions for the atlas). Two (G, M)-atlases related by refinement of the covers involved define the same (G, M)-structure on X (as in the case of smooth atlases for X defining the same smooth structure). Alternatively, analogous to smooth structures, (G, M)-structures on X can be defined as maximal (G, M)-atlases for X. If X, X are two manifolds with (G, M)structure, a morphism between them consists of a smooth map f : Some concrete examples of rigid geometries are as follows: (a) For G = SO(d) R d , the Euclidean group of isometries of M = R d , a (G, M)-structure on X can be identified with a flat Riemannian metric on X.
on X consists of a flat Riemannian metric together with a spin structure. (c) For M = S d and G = Conf(S d ), the group of conformal transformations of the sphere, a (G, M)-structure on X is a conformal structure on X. (d) For M = R d and G = Aff(d), the affine group, a (G, M)-structure on X is an affine structure on X. (e) If M is a simply connected manifold of constant sectional curvature κ and isometry group G, then a (G, M)-structure on X is a Riemannian metric on X of constant curvature κ.
Rigid geometries as described above are closely related to the notion of pseudogroups, as developed by Cartan. The main difference is that the action of G on M is not required to be faithful (as, e.g., in the case of the spin group Spin(d) acting on R d through SO(d)).
The above notion was introduced by Stolz and Teichner, with an eye on supersymmetric field theories (see [13,Section 2.5]).

4.
A smooth map X → M to some fixed manifold M .

5.
A principal G-bundle over X (for a fixed Lie group G), or a principal G-bundle over X with connection.
Remark 3.2. In a physics context, the manifold X is typically the relevant spacetime manifold and the geometry on X is needed for the construction of some field theory. For example, a Riemannian metric on X allows the construction of the scalar field theory whose space of fields is the space C ∞ (X) of smooth functions on X and whose action functional is the energy functional given by S(f ) := X df 2 vol (here vol is the volume form determined by the Riemannian metric. A fermionic analog of this field theory consists of fields which are spinors on X; its action functional is based on the Dirac operator. The construction of this field theory requires a Riemannian metric and a spin structure on X, i.e., a reduction of the structure group to Spin(d).
In many of the Examples 3.1, the geometries on a fixed d-manifold X form just a set (in particular, in the cases (1), (2a), (2b), (2c), (2d), (3a), (3c), (3d), (3e) and (4)). In other cases, e.g., (2e), (3b), (5), there is more going on: these geometric structures can be interpreted as the objects of a groupoid which contains non-identity morphisms. For example, for a fixed group G, the principal G-bundles P → X over X form a groupoid, with the morphisms from P to P being the G-equivariant maps that commute with the projection maps to X (this is Example 3.1 (5)).
This suggests to think of a G-structure on X as an object of a groupoid G(X) associated to X (which might be discrete in the sense that the only morphisms in that groupoid are identity morphisms, as in our examples (1), (2a), (2b), (2c), (2d), (3a), (3c), (3d), (3e) and (4)). A crucial feature of all our Examples 3.1 is that the data of a geometry is local in X in a sense to be made precise. For example, a Riemannian metric on X is determined by prescribing a Riemannian metric g i on each open subset X i belonging to an open cover {X i } i∈I of X in such a way that these metrics g i , g j coincide on the intersection X i ∩ X j . In other words, the Riemannian metrics on X form a sheaf. The same statement is true in our other examples of geometric structures G(X), where the groupoid G(X) is discrete.
In the case of non-discrete groupoids, for example if G(X) is the groupoid of principal Gbundles over X (as in Example 3.1(5)), it is still true that G(X) is local in X, but it is harder to formulate what that means. The idea is that for any open cover {X i } i∈I of X the groupoids associated to intersections of the X i determine the groupoid G(X) of principal bundles over X, up to equivalence. This is expressed by saying the groupoids G(X) form a stack on the site of manifolds. For the precise definition of stack we refer the reader to Vistoli's survey paper [15, cf. Definition 4.6], but it should be possible to follow our discussion below without prior knowledge of stacks. In fact, we hope that the following might motivate a reader not already familiar with stacks to learn about them.

Digression on stacks
Our first example of a stack will be the stack Vect of vector bundles. We note here the relevant structures.
• For a fixed manifold X let Vect(X) be the category of whose objects are smooth vector bundles E → X over X and whose morphisms from E → X to E → X are smooth maps F : E → E which commute with the projection to X and whose restriction F x : E x → E x to the fibers over x ∈ X is a linear map for each x ∈ X.
• Let Vect be the category whose objects are vector bundles E → X over some manifold X and whose morphisms from E → X to E → X are pairs of smooth maps f , f for which the diagram is commutative, and the restrictionf x : E x → E f (x) off to the fiber over x is linear for all x ∈ X. Abusing notation, we often simply write φ : E → E for such a morphism φ = f , f .
There is an obvious functor p : Vect → Man to the category of smooth manifolds that sends a vector bundle E to its base space. The category Vect(X) of vector bundles over X is the fiber of p, i.e., the subcategory of Vect consisting of all objects whose image under p is X and all morphisms of Vect whose image under p is the identity of X. The functor p : Vect → Man has two interesting properties: Existence of cartesian lifts. Given a smooth map f : X → X and a vector bundle E over X , we can form the pullback bundle E := f * E over X, which is the domain of a tautological morphism φ = f , f : E → E . The vector bundle map φ has the property that, for each x ∈ X, the linear map of fibersf x : E x → E f (x) is an isomorphism. Morphisms with this property are called cartesian, and the vector bundle morphism φ : E = f * E → E is also referred to as the cartesian lift of the morphism f : X → X .
While the characterization of cartesian vector bundle morphisms φ : E → E as those which restrict to fiberwise isomorphisms is hands-on and concrete, it is more common to characterize them by the universal property we describe now. The advantage is that universal properties make sense in any category.
A vector bundle morphism φ : E → E is cartesian if, for any vector bundle map φ : E → E and any map f : p(E ) → p(E ) such that p(φ ) = p(φ) • f , there exists a unique vector bundle map φ : E → E with p(φ ) = f . This property can be expressed succinctly by saying that, given a commutative diagram consisting of the solid arrows in the diagram below, there exists a unique morphism φ indicated by the dashed arrow that makes the whole diagram commutative.
Here, X = p(E), f = p(φ), etc., and by commutativity of the squares we mean that applying the functor p : Vect → Man to the top morphism in Vect gives the bottom morphism in Man: Descent property. Let f i : X i → X be a collection of morphisms in Man that is a cover of X in the sense that all the f i are open embeddings and the union of the images f i (X i ) is all of X. Then the category Vect(X) can be reconstructed, up to equivalence, from the categories Vect(X i ), Vect(X i ∩ X j ), and Vect(X i ∩ X j ∩ X k ) and the restriction functors between them. More precisely, from the diagram given by these categories and the restriction functors between them one can construct the descent category Vect({X i → X}) associated to the cover {X i → X} (see [15,Section 4.1.2]) and a restriction functor which is an equivalence.
It turns out that the existence of pullbacks and the descent property can be formulated quite generally for functors p : F → S as follows. Definition 3.3 (cartesian morphism, fibration, prestack). Let p : F → S be a functor. A morphism φ in the category F is cartesian if it satisfies the universal property expressed by the diagram (3.1); see also [15,Definition 3.1]. The functor p : F → S is called a Grothendieck fibration and F a category fibered over S if, for any morphism f : X → X in S and object E ∈ F with p(E) = X, there is a cartesian morphism φ : E → E with p(φ) = f ; see also [15,Definition 3.1]. We say that a fibration F → S is a prestack if every morphism of F is cartesian.
Remark 3.4. That F → S is a prestack means that F → S is fibered in groupoids, meaning that the subcategory F(S) of F lying over id S is a groupoid for every object S of S. Conversely, it turns out that a category fibered in groupoids is automatically a prestack [15,Proposition 3.22]. To any prestack corresponds a presheaf of groupoids on S, sending S to F(S). This sheaf depends on a choice of pullback for every object F of F along every morphism f : S → T in S (i.e., a cartesian arrow φ in F with target F and p(φ) = f ), but it is unique up to unique isomorphism.
As discussed above, for the functor p : Vect → Man the cartesian morphisms in Vect are the vector bundle morphisms φ : E → E that restrict to isomorphisms on fibers. Moreover, given a smooth map f : X → X between manifolds and a vector bundle E over X, the tautological bundle map φ from the pullback bundle E := f * E to E is a cartesian lift of f . Hence p : Vect → Man is a Grothendieck fibration; in other words, Vect is fibered over Man.
The discussion of the descent property for p : Vect → Man above was based on the definition of a cover {X i → X} of a manifold X. So, before discussing descent in a general category S, we need to clarify what is meant by a "cover" of an object X of S. A Grothendieck topology on S is the assignment, to each object X of S, of a collection of sets of morphisms {X i → X}, called covers of X, so that the following conditions are satisfied: 1. If Y → X is an isomorphism, the set {Y → X} is a cover.
2. If {X i → X} is a cover and Y → X is any morphism, then the pullbacks X i × X Y exist, and the collection of projections 3. If {X i → X} is a cover and, for each index i, we have a cover {Y ij → X i }, the collection of composites {Y ij → X i → X} is a cover of X (here j varies in a set depending on i).
A category equipped with a Grothendieck topology is called a site.
A prestack F → S over a site S is called a stack, if, for every cover {X i → X} of every object X in S, the functor (3.2) is an equivalence.
For a stack, the groupoids F(X i ) associated to the patches X i , together with transition data on double and triple intersections, determine F(X). In fact, this definition makes sense for general fibered categories, not only those fibered in groupoids (i.e., prestacks). In this paper, we will use the unqualified term stack only for those stacks which are fibered in groupoids, and say stack of categories in the general case, when not all morphisms need to be cartesian. Similar to the case of sheaves, to any prestack, there is a canonically associated stack, its stackification [6, p. 18].
Let Man d be the category of d-dimensional manifolds and smooth maps. We will always consider the Grothendieck topology on Man d of jointly surjective open embeddings. Precisely, a collection {X i → X} is a cover, if each map X i → X is an open embedding and the images of the X i in X cover X. We end our digression on stacks with a few more general remarks. Definition 3.8 (fibered functors, base-preserving natural transformations). Let F, G → S be two Grothendieck fibrations. A functor H : F → G is called a fibered functor if it commutes strictly with the projections to S and sends cartesian morphisms to cartesian morphisms. A natural transformation ξ : H → K between fibered functors is base-preserving if, for any object x ∈ F, the morphism ξ x : H(x) → K(x) maps to an identity morphism in S. Definition 3.9 (categories of (pre-)stacks). For each site S, we get a 2-category PSt S of Grothendieck fibrations, fibered functors and base-preserving natural transformations. The full subcategory of stacks will be denoted by St S . We will omit the subscript when S = Man is the site of smooth manifolds.

Geometries in families
The preliminary Definition 3.7 satisfactorily captures the contravariance and locality aspects of a geometric structure. However, as discussed in Section 2.3, it is crucial to work with families of smooth manifolds. In particular, we need to talk about geometric structures on families of dmanifolds. This is formalized in Definition 3.12 below by replacing the category Man d in the preliminary definition by the category Fam d of families of d-dimensional manifolds, equipped with a suitable Grothendieck topology.  To turn Fam into a site, we declare a cover of the object X/S ∈ Fam to be a collection of morphisms {X i /S i → X/S} i∈I such that the images of the X i form an open cover of X. This satisfies the axioms of a Grothendieck topology: it is clear that covers of covers determine a cover, and we check the existence and stability of base changes (condition (2) of Definition 3.5) in the following lemma.
Lemma 3.11. If {X i /S i → X/S} is a cover and Y /T → X/S is any morphism, then the fiber products (X i /S i ) × (X/S) (Y /T ) exist and determine a cover of Y /T .
Both are manifolds since the maps X i → X and S i → S are submersions: The first by the requirement that the X i form an open cover of X; to see that the latter is, observe that the composition X i → X → S is a submersion, which equals X i → S i → S, hence S i → S must be a submersion, too. We now have a diagram as follows: The dashed map is obtained by the cartesian property of the bottom face (containing S i and T ), which implies that all faces of the cube commute. To see that Y i → T i is a submersion, let (s i , t) ∈ T i = S i × S T be a point that is the image of the point (x i , y) ∈ Y i = X i × X Y . Further, consider a tangent vector to (s i , t), represented by a pair of paths (γ S i , γ T ), where γ S i : I → S i , γ S i (0) = s i , γ T : I → T , γ T (0) = t are paths that coincide after passing to S. Since Y is a submersion, we can find a lift γ Y : I → Y (with I ⊂ I) of γ T with γ Y (0) = y. Letγ X be the image of this lift to X. By construction, γ X (0) lies in the image of the map X i → X, hence (because X i is an open embedding), we can find a path γ X i : is an open embedding. To see that it is injective, let (x i , y) and (x i , y ) ∈ X i × X Y = Y i be two points that are mapped to the same point in T i × T ×Y . First, it follows that x i and x i are mapped to the same point in X; since X i → X is an embedding, we have that x i = x i . Secondly, since (x i , y) and (x i , y ) coincide in T i × T Y , we must have y = y . This shows injectivity.
is a submersion by a similar argument as above. To see that it is also an immersion, take a tangent vector represented by a tuple (γ X i , γ Y ) of curves γ X i : I → X i and γ Y : I → Y that coincide after passing to X. Then the pushforward of this tangent vector is represented by Assume that this pushforward is zero, i.e.,γ S i (0) = 0 andγ Y (0) = 0. Then because γ X i and γ Y are mapped to the same curve γ X : I → X and X i → X is an embedding, we have that alsȯ γ X i (0) =γ X (0) =γ Y (0) = 0. Hence the tangent vector represented by (γ X i , γ Y ) is zero as well and Y i → T i × T Y is an immersion. In total, we obtain that it is an open embedding.
It only remains to see that Y i /T i has the universal property of the fiber product (X i /S i )× (X/S) (Y /T ). In other words, given an arbitrary family Z/U and maps Z/U → X i /S i , Z/U → Y /T which agree on X/S, there exists a unique morphism Z/U → Y i /T making the diagram commute. By the cartesian property of two of the squares of the cube (3.3), there exist unique maps Z → Y i and U → T i . We claim that these determine a morphism in Fam, that is, the diagram commutes and the top map is a fiberwise open embedding. Both maps Z → T i agree when postcomposed with T i → S i respectively T i → T , so they agree by the universal property of T i = S i × S T . The map Z → Y i is an a fiberwise open embedding because so is its composition Z → Y i → Y with another fiberwise open embedding. Hence this morphism fits in as the dashed morphism in (3.4). By uniqueness of the maps Z → Y i and U → T i , this is the unique morphism with this property.
The above lemma shows that our notion of cover defines a Grothendieck topology, which turns Fam into a site. By restricting to those families X/S, where the fibers X s , s ∈ S are all d-dimensional, we get a subcategory, Fam d . Since our covers do not mix fiber dimensions, restricting to d-dimensional covers turns also Fam d ⊂ Fam into a site. This allows to talk about sheaves and stacks on Fam d .
We are now ready to give the main definition of this section.
Definition 3.12 (geometry). A d-dimensional geometry is a stack G on the site Fam d of families of manifolds with d-dimensional fibers. By an S-family of G-manifolds we will mean a family X/S ∈ Fam d together with an object of G(X/S).
To each family X/S are associated a natural relative tangent bundle T (X/S) = Ker(T X → T S) and variations: the relative cotangent bundle T ∨ (X/S) = Coker T ∨ S → T ∨ X , their tensor powers, etc. For emphasis, we sometimes call their sections fiberwise vector fields, differential forms, etc. This allows us to define fiberwise versions (or "families") of many familiar structures, such as Riemannian metrics, symplectic and complex structures, connections on a principal bundle, and so on. For instance, a Riemannian metric on X/S is a positive-definite section of the second symmetric power of T ∨ (X/S). A family of connections on a vector bundle V → X is a differential operator ∇ : C ∞ (V ) → C ∞ T ∨ (X/S) ⊗ V satisfying a version of the Leibniz rule involving the fiberwise exterior derivative d : C ∞ (X) → Ω 1 (X/S). Thus, ∇ allows us to perform parallel transport only along the fibers of the submersion X → S.
It is now mostly straightforward to adapt Examples 3.1 to geometries in families. We spell this out in two cases.

A morphism lying over
is the map induced by f and the action by g.
The composition of morphisms is determined by composition of theḡ. Note that every morphism is cartesian.
The usual stackification procedure [6, p. 18] exactly recovers the more concrete definition of a rigid geometry given in [13, Definition 2.33]: A section of the stack G over X/S, which we call a (G, M)-atlas, is given by the following data: (1) a cover {X i /S i → X/S} i∈I , (2) fiberwise embeddings φ i : X i → M for each i ∈ I (the charts of the atlas), and (3) transition functions g ij : S i × S S j ⊃ p(X i × X X j ) → G relating appropriate restrictions of φ i and φ j , and satisfying a cocycle condition. Morphisms between atlases based on the same cover {X i /S i } i∈I are given by collections of maps h i : S i → G, i ∈ I, which interpolate the charts X i → M. Moreover, atlases related by a refinement of covers must be declared equivalent; this is taken care of by the stackification machinery.

Simplicial prestacks and smooth categories
It is well known (see, e.g., [11,Section 2]) that a simplicial set C : ∆ op → Set is equivalent to the nerve of a category if, and only if, the Segal maps . . , n, is the morphism sending 0 → i − 1 and 1 → i, and fiber products are taken over the maps d * 0 , d * 1 : . (For references, see, e.g., [9,11].) This observation allows us to internalize the notion of a category in other ambient (higher) categories. In this paper, we would like to talk about categories endowed with a notion of "smooth families" of objects and morphisms. Thus, we take as ambient the 2-category PSt of prestacks on Man.

Definition 3.15 (simplicial prestack). A simplicial prestack (on manifolds) is a pseudofunctor
Remark 3.16. Here the simplex category ∆ is regarded as a 2-category with only trivial 2morphisms, and all constructions are performed in the realm of bicategories. That C is a pseudofunctor then means that for two composable morphisms η, κ in ∆, the induced morphisms of stacks κ * η * and (ηκ) * agree only up to a coherent natural isomorphism, which is part of the data of C.
A smooth category will be a simplicial prestack satisfying suitable conditions. Before introducing them, we fix some terminology. Condition (2) below assures that the simplicial set n → h 0 C n (S) is equivalent to the nerve of a category C (where h 0 C n (S) is the set of isomorphism classes of objects in C n (S)); we call an object of C 1 (S) an equivalence if it represents an invertible morphism in C. (2) the degeneracy map C 0 → C 1 gives an equivalence of the domain with the full substack of equivalences in C 1 .
We will refer to morphisms and 2-morphisms between smooth categories as smooth functors and smooth natural transformations, respectively.
The above conditions are modeled on complete Segal spaces, which extends the nerve construction explained above, to give a model for (∞, 1)-categories (see [9] for further details). Condition (1) is the Segal condition, (2) is the completeness condition. In other words, smooth stacks are complete Segal objects in St.
Remark 3.18. The fiber products appearing in the definition (which are taken using d * 0 , d * 1 : C 1 → C 0 ) are in the bicategorical sense, that is, they are what is sometimes called a homotopy fiber product.
Example 3.19 (smooth categories from smooth stacks). Our most interesting examples of smooth categories will be the geometric bordism categories constructed below. However, to get the first examples, we now provide a way to construct a smooth category from a smooth stack. This is a version of Rezk's classification diagram construction [9].
Let C be a stack of categories (so that C(S) does not need to be a groupoid). In our applications, C will be the stack of vector bundles or a stack of sheaves of C ∞ -modules as in Section 5.2. We then construct a smooth category C • from this input as follows.
Objects of C n lying over S ∈ Man are tuples (C n , . . . , C 0 ; f n , . . . , f 1 ), where the C j are objects of C(S) and f j : C j−1 → C j are morphisms in C(S) (i.e., morphisms in C covering the identity on S). Morphisms from an object (C n , . . . , C 0 ; f n , . . . , f 1 ) over S to an object (C n , . . . , C 0 ; f n , . . . , f 1 ) over T covering f : S → T are tuples (α n , . . . , α 0 ), where α j : C j → C j are cartesian arrows covering f such that the diagram commutes. The simplicial structure of C • is so that face maps perform composition of morphisms and degeneracies insert identities. More explicitly, a morphism κ : This gives a (strict) functor C • : ∆ op → St Man , and it is obvious that it satisfies the Segal condition.
Definition 3.20 (strictness). We say that a simplicial prestack C is strict if it is a strict functor, that is, the natural isomorphisms κ * η * ∼ = (ηκ) * are all identities. We say that a smooth functor between two strict smooth categories is strict if it commutes on the nose with the structure maps in ∆, as a natural transformation of strict functors.
The following is an easy structure result for the examples just constructed, whose proof we omit.
that restricts a smooth natural transformation to the corresponding 2-morphisms of stacks at simplicial level zero is injective. Moreover, the map that restricts to simplicial level one is an isomorphism.

Symmetric monoidal structures
To talk about field theories, we need to endow our smooth categories with symmetric monoidal structures. This requires first to define symmetric monoidal stacks. Definition 3.22 (monoidal fibered category). A symmetric monoidal fibered category is a fibered category F → S with a (fibered) tensor product functor ⊗ : F × S F −→ F, a (fibered) unit functor : S → F (where S → S denotes the trivial fibered category), together with a collection of natural transformations (an associator α : . These data are required to be compatible in the sense that they turn each fiber category F(S) into a symmetric monoidal category. A symmetric monoidal stack is a symmetric monoidal fibered category that is also a stack.
Symmetric monoidal smooth stacks together with (strong) symmetric monoidal functors and natural transformations form a bicategory which we shall denote St ⊗ Man . Definition 3.23 (symmetric monoidal smooth category). A symmetric monoidal structure on a simplicial prestack C : Man . Symmetric monoidal smooth functors and natural transformations are likewise defined as 1-and 2-morphisms of simplicial objects in symmetric monoidal stacks. A symmetric monoidal smooth category is a smooth category (Definition 3.17) with a symmetric monoidal structure.
Example 3.24. If, in Example 3.19, we start with a symmetric monoidal smooth stack as input, the result will naturally be a symmetric monoidal smooth category.

Geometric bordism categories
Let G be a geometry for d-dimensional manifolds, i.e., a stack on Fam d . In this section, we will define our symmetric monoidal smooth category (Definition 3.23) of G-bordisms, denoted by GBord. We start by defining a smooth symmetric monoidal stack GBord n for every object n ∈ Z ≥0 . Afterwards, we define maps of symmetric monoidal stacks κ * : GBord n −→ GBord m for every morphism κ : [m] → [n] in ∆. These maps will satisfy (κ • η) * = η * • κ * , so that we obtain a strict functor GBord : ∆ op → St ⊗ . At the end of the subsection, we comment on the smooth category property of GBord.
Definition 3.25 (the stack GBord n ). An object of the stack GBord n lying over a manifold S consists of the following data: (O2) A G-structure on X/S, that is, an object G X/S of G(X/S).
We will abuse notation and abbreviate X/S for objects, keeping in mind that the collection {ρ a } and the object G X/S are also part of the data. The subspace X n 0 defined above will be called the core of X/S.
Morphisms of the stack GBord n are going to be equivalence classes of maps, where a map lying over a morphism f : S → T consists of the following data: (M1) An open neighborhood U of the core X n 0 such that for some ε > 0, and (M2) A smooth map F : U → Y covering f , which is a fiberwise open embedding onto a neighborhood of the core Y n 0 , such that for each 0 ≤ a ≤ n, there exist positive smooth functions ζ a on U such that F * ρ a = ζ a ρ a | U .
We declare two maps to be equivalent if they have a common restriction to a smaller neighborhood of the core. Here a restriction of a map (f, Remark 3.26. The condition relating F * ρ a and ρ a | U in (M1) is equivalent to saying that these two functions have the same sign. We express it in terms of the positive function ζ a so that our definition still makes sense, without change, in the supermanifold case.
Remark 3.27. To simplify the presentation, we implicitly choose a cleavage for the stack G (i.e., a preferred choice of pullback arrows) in order to define "restrictions" of objects in G such as G X/S | U/S . Explicitly, this means a triangle G X/S ← G U/S → G Y /T consisting of an object G U/S together with an arrow G U/S → G X/S covering the inclusion U/S → X/S and an arrow G U/S → G Y /T covering the map (f, F ) : U/S → Y /T . These data are unique up to unique isomorphism.
Morphisms are composed as follows. Suppose that, in addition to the map (f, F, ϕ) described above, we are given a second map (f , F , ϕ ) starting at Y /T . Choose a subset V ⊂ U satisfying (M1) and such that is a representative for the composition. It is straightforward to show that this operation on morphisms is independent of the choice of representatives.
Each of the stacks GBord n is a symmetric monoidal stack with the tensor product given by fiberwise disjoint union, that is, X/S ⊗ Y /S := (X Y )/S. A G-structure on X/S ⊗ Y /S is obtained by the stack property, using the obvious cover {X → X Y, Y → X Y }. Moreover, it is clear that the pullback maps κ * are monoidal, so that GBord is a symmetric monoidal smooth category.
This concludes the construction of a category GBord n with a projection onto Man, where an object X/S; ρ 0 , . . . , ρ n ; G X/S is mapped to S. Remark 3.28. We think of an object in GBord n (pt) as a sequence of n composable bordisms, where the indiviual bordisms are the sets X a a−1 , a = 1, . . . , n, defined in (3.6). These are dmanifolds with boundary by requirement (b), unless we are in the degenerate case where ρ a−1 and ρ a have common zeros. Objects in GBord n (S) are thought of as families of such bordisms, parametrized by S.
Example 3.29. For n = 0, we just have one cut function ρ 0 , and the core X 0 0 = ρ −1 0 (0) is a codimension 1 submanifold of X, by condition (O3)(b) above. This condition also implies that X 0 0 intersects the fibers of X → S transversally and hence determines a (d − 1)-dimensional submanifold of each fiber X s . Similarly, for n = 1, the core X 1 0 determines, for each s ∈ S such that ρ 0 (s) < ρ 1 (s) everywhere, a d-manifold X 1 0 ∩ X s with boundary. It is compact, by condition (O3)(c), thus a bordism between X 0 0 ∩X s and X 1 1 ∩X s . In this way, GBord 0 and GBord 1 comprise families of bordisms and their boundaries. However, in the presence of a nontrivial geometry G, there is the additional data of a neighborhood of the core together with a geometric structure on this neighborhood. This prohibits us to simply work directly with the cores. Proof . We have to show that GBord n is a prestack and that it satisfies descent.
First, we show that GBord n is a fibered category. To this end, let f : S → S be a morphism, and let B = X/S; ρ 0 , . . . , ρ n ; G X/S be an object of GBord n over S. Let X = S × S X be the fiber product over S. This is an object of Fam over S , and we obtain a morphism (f, F ) : X /S → X/S in Fam. Set ρ a = (F ) * ρ a . Now B = X /S ; ρ 0 , . . . , ρ n ; F * G X /S is an object of GBord n over S , and (f, F, ϕ) : B → B is a morphism in GBord n which is a cartesian lift of f .
To show that GBord n is a prestack, by [15,Proposition 3.22], it now suffices to show that for any manifold S, GBord n (S) is a groupoid. Let (id, F, ϕ) : X/S; ρ 0 , . . . , ρ n ; G X/S → Y /S; ρ 0 , . . . , ρ n ; G Y /S be a map covering the identity, where F is defined on some open neighborhood U of the core X n 0 . Because F | U is a fiberwise diffeomorphism onto its image covering the identity, it is in fact a diffeomorphism onto its image. Since G is a stack on Fam and ϕ : . It is clear that both compositions of these maps are restrictions of the identity map, hence equivalent to the identity. This shows that each morphism in GBord n (S) is invertible, hence it is a groupoid.
To verify the descent property, let S be a manifold and {S i } i∈I a covering family of S. The objects of the category GBord n ({S i → S}) of descent data are tuples The light shaded regions depict the neighborhoods (ρ i 0 ) −1 (−ε, ε) of the core (X i ) 0 0 (drawn as a thick line). When attempting to glue, the difficulty is to shrink X i toX i in such a way that the corresponding restrictionF of F mapsX 1 | S12 diffeomorphically toX 2 | S12 .
GBord n (S) → GBord n ({S i → S}) is given by restriction, i.e., by pullback along the inclusions S ij → S i , respectively S i → S. It is clear that this functor is fully faithful, as morphisms are locally determined and glue.
To see that it is essentially surjective, let {B i } i∈I , {F ij } i,j∈I be an object as above. The maps F ij are defined on some open neighborhood U i of the core of X i | S ij and come with morphisms ϕ ij : . We need to show that the B i glue together to a bordism B over S.
To begin with, we assume that the given cover of S consists of two elements S 1 and S 2 , with S 12 = S 1 × S S 2 . Let F : U/S 12 → F (U )/S 12 be the gluing diffeomorphism. Let χ 1 , χ 2 be a smooth partition of unity, i.e., non-negative functions with supp(χ i ) ⊂ S i and χ 1 + χ 2 = 1. We denote the lifts of χ i to X 1 and X 2 via the projections X 1 → S 1 and X 2 → S 2 by the same letter. Now for a = 0, 1, . . . , n, consider the functions defined bỹ Let ε > 0 be as in assumption (M3) on F and set LetF be the restriction of F to U 1 ∩X 1 . Observe that by construction, Hence,X 1 andX 2 glue together, viaF 12 , to a family of manifolds over S. Since by construction, for each a, the functions ρ 1 a and ρ 2 a coincide over S 12 , they combine to give functions ρ 0 , . . . , ρ n on X.
be the restriction of the objects of G contained in the data of B i and letφ be the corresponding restriction of the gluing isomorphism ϕ in G convering F (which then coversF ). Since G satisfies descent, these objects GX 1 /S 1 , GX 2 /S 2 glue together to an object G X/S over X/S. In total, we obtain an object B = X/S; ρ 0 , . . . , ρ n ; G X/S over S, together with maps B → B i compatible with the gluing isomorphisms. In other words, we have found a preimage to the given object in GBord n ({S i → S}).
Next, assume that the cover {S i } i∈I is locally finite. Then I must be countable, and we can identify I = N. By replacing the functions ρ i a by λ i · ρ i a for suitable constants λ i > 0, we may achieve that the value ε > 0 in the condition (M3) on the cut functions F ij can be chosen as ε = 1 for all i and j. This replaces B i by isomorphic objects in GBord n and hence does not change the object in the category of descent data. Observe here that since the cover is locally finite, there are only finitely many non-trivial transition functions F ij for each fixed i, and hence only finitely many constraints for each λ i . Hence the modification is indeed possible. Now, using the previous case, we may one by one glue together the corresponding bordisms B 1 , . . . , B k to a bordism B (k) . Since the cover is locally finite, this may be achieved in such a way that over each compact subset of S, the bordisms B (k) are all canonically equivalent for k large, hence they all glue together to a bordism B over S.
For a general cover {S i } i∈I , we choose a locally finite subcover, which leads to an equivalent category of descent data.
Remark 3.31. The modification ρ i a → λ i ρ i a in the proof above was made to deal with the following technical problem: Without this change, the construction of B (k+1) might require a smaller choice of ε > 0 than the one needed in the construction of B (k) . Therefore, the total space X (k+1) of B (k+1) may be strictly smaller than that of B (k) and might degenerate in the limit k → ∞.
The stacks GBord n form a simplicial object in an obvious way: Given an order-preserving map κ : [n] → [m], we obtain a functor κ * : GBord m → GBord n , given by removing or duplicating the functions ρ a . Forgetting a cut function has the interpretation of gluing bordisms along the boundary determined by it. Regarding the morphisms, we would like to say that they remain the same under κ * . Precisely, what happens is the following. Notice that a map in GBord m is also a map in GBord n . Hence, if a morphism in GBord m is represented by a map F , we let κ * [F ] be again the morphism represented by F , but with respect to the equivalence relation in GBord n . This makes sense because any two maps which are equivalent in GBord m are also equivalent in GBord n , as follows directly from the definition. Proof . To verify Definition 3.17(1), we have to show that the maps are equivalences for each manifold S and each n ∈ N. This is clear, since gluing bordisms over a fixed parameter space is straightforward.
To verify Definition 3.17(2), we consider the categories h 0 GBord(S) determined by the Segal set n → h 0 GBord n (S), for each manifold S. Objects B of the stack GBord 1 (S) then determine morphisms [B] in h 0 GBord(S), and we have to analyze which objects give rise to invertible morphisms this way. We claim that for such a bordism B = (X/S; ρ 0 , ρ 1 ; G X/S ), the morphism [B] is invertible if and only if B is "thin", meaning that ρ 0 and ρ 1 have the same zero set. Given this claim, the completeness condition Definition 3.17(2) follows, since B is isomorphic to , the object c * B 2 = (X/S; ρ 0 , ρ 2 ; G X/S ) must be isomorphic, as an object of GBord 1 (S), to the thin bordism id M = (X/S; ρ 0 , ρ 0 ; G X/S ). Here c : [1] → [2] maps 0 → 0 and 1 → 2. Therefore, c * B 2 must be thin, i.e., ρ 0 and ρ 2 have the same zero sets. This implies that also ρ 1 must have the same zero set, which implies that both B and B −1 are thin as well. Conversely, if B is thin, then, as mentioned above, it is isomorphic to B = (X/S; ρ 0 , ρ 0 ; G X/S ), which implies that [B] = id [M ] . In particular, [B] is invertible.
We conclude this section by giving some examples of geometric bordism categories. Example 3.33 ("no geometry"). If we choose G to be the trivial stack on Fam d (whose fibers are single points), we get the realization of the d-dimensional bordism category in the world of smooth categories, to be denoted d-Bord. Objects of d-Bord n consist simply of a family X/S of manifolds parametrized by S, together with cut functions ρ 0 , . . . , ρ n . In this case, morphisms from X/S to Y /T are just given by smooth fiber-preserving maps F that are defined on a neighborhood U of the core, are fiberwise open embeddings and send X b a to Y b a ; two such maps F , F (defined on U , U ) are identified if they coincide on a smaller neighborhood V ⊂ U ∩ U of X n 0 .
Example 3.34 (bordisms over a manifold). If G is the d-dimensional geometry represented by a manifold M , as in Example 3.13, we get the category d-Bord(M ) of bordisms over M . Clearly, this specializes to the previous example if one takes M to be a point. This is our main example in the second part of this paper (where moreover d = 1).
Example 3.35 (orientations). If G is the d-dimensional geometry of fiberwise orientations, as in Example 3.14, we denote the resulting smooth category by d-Bord or . In this case, the fibers X s of an object X/S ∈ d-Bord or n carry orientations, and the maps F are required to be orientationpreserving when restricted to the fibers. That this is a condition rather than additional data for F reflects the fact that G is a sheaf on Fam d , i.e., discrete as a stack.

Geometric field theories
Conceptually, a field theory should be a symmetric monoidal functor from a suitable bordism category to the category of vector spaces. To put this concept into our setup, we need our source and target categories, as well as the functor, to be smooth. As source we take GBord for some geometry G, as defined above. To specify the target, we need to fix a notion of "smooth family" of vector spaces. Initially, we will study field theories taking values in the smooth category Vect of finite-dimensional vector bundles on Man, obtained by applying the procedure of Example 3.19 to the stack (of categories) of finite-dimensional vector bundles. Later, in Section 5, we will consider the more general case of C ∞ -modules. These categories are symmetric monoidal using the tensor product of vector spaces, respectively modules. Definition 3.36 (geometric field theories). Let G be a d-dimensional geometry. A (d-dimensional) field theory with geometry G is a symmetric monoidal smooth functor A morphism of field theories is a smooth, symmetric monoidal natural transformation.
Denoting by Fun ⊗ the groupoid of functors between smooth categories, together with smooth, invertible natural transformations, we denote by the groupoid of functorial field theories for a given geometry G.
Field theories as functors between smooth categories are complicated objects, due to the fact that stacks form a 2-category. A field theory Z consists of the following data. First, for every object [n] ∈ ∆, there is a map of stacks Z n : GBord n −→ Vect n .
However, since stacks form a 2-category, we cannot expect that these strictly commute with the structure maps in ∆; instead, for each morphism κ : is another map in ∆, then the corresponding 2-morphisms have to satisfy the coherence condition Visually, this is depicted by (Here, we are using the strictness of GBord and Vect, that is, the fact that (κη) * and η * κ * are equal ; otherwise the identification of the two diagrams would involve, additionally, the coherence data η * κ * ∼ = (κη) * .) Fortunately, in our setting, the data of a field theory can be simplified considerably, as the following lemma shows. Proof . Fix an integer n ≥ 0 and X ∈ B n , and denote by κ i : and note that we have natural isomorphisms By our assumption on V, Z n (X) ∈ V n is a chain of morphisms in the stack of which V is the nerve. Towards defining the functor Z n : B n → V n , set Z n (X) to be chain of morphisms V 0 → · · · → V n such the diagram below commutes: The above diagram also fixes the effect of Z n on morphisms of B n , if we insist that the collection ζ 0 , . . . , ζ n defines a natural transformation ξ n : Z n → Z n . It remains to show that the collection {Z n } defines a strict smooth functor, that is, the diagram commutes strictly for every morphism κ : [m] → [n] in ∆. Now, κ * Z n (X) is the chain of morphisms obtained from Z n (X) by appropriate compositions or insertion of identities. On the other hand, Z m (κ * X) is a chain of morphisms of the form By strictness of B as a simplicial object, we have κ . It remains to see that the morphisms in the chains (3.8) and (3.9) are identical. Consider the commutative diagram below: We will be done if we show that the ith composite vertical map is equal to ζ κ(i) , since in this case we can replace the top row by (3.8) and still have a commutative diagram. But this fact is simply the coherence condition (3.7), applied to the case η = κ i . Remark 3.38 (field theories as strict functors). Since our bordism categories, as well as the smooth category of vector bundles, satisfy the assumptions of the above lemma, it follows that we make no mistake by defining field theories as strict symmetric monoidal functors Z : GBord → Vect, and their morphisms as strict natural transformations. We will work in this context in the next section, which simplifies our life considerably.

Classification of one-dimensional field theories
In this section, we discuss the classification of one-dimensional field theories over a manifold M . Let us briefly discuss the classical case (with ordinary categories and M = pt) in order to see what to expect. The one-dimensional (ordinary) bordism category 1-Bord is easy to describe. The objects, compact zero-dimensional manifolds, are just finite collections of points. To understand the morphisms, one needs the classification of compact, connected one-dimensional manifolds with boundary; this classification is very simple (say, using Morse theory). Apart from the circle, which is the only closed example, we have two elbows (the one with two incoming boundary components and zero outcoming boundary components, as well as its dual) and the interval (with one incoming and one outgoing boundary component).
We briefly recall the well-known construction of one-dimensional field theories from vector spaces.
Construction 4.1 (unoriented 1-TFTs). In the unoriented case, a field theory can be obtained from the data of a finite-dimensional vector space V over K = R or C together with a symmetric nondegenerate bilinear form β as follows: 1. To a collection of k points, we assign the k-fold tensor product V ⊗k . In particular, the ground field K corresponds to the empty set. 2. To the interval, we assign the identity homomorphism on V .
3. To the elbow with two incoming boundary components, we assign the bilinear form β.

4.
To the elbow with two outgoing boundary components, we assign τ : where ε = ±1 depending on the signature of β.

5.
To the circle, we assign the number n = dim(V ).
There are several things to check in order to see that this defines a field theory: For example, one has to check that β • τ = n, as well as the snake identity Conversely, any one-dimensional field theory Z determines such a pair (V, β): just set V = Z(pt) and β to be the value of the elbow with two incoming boundary components. Then it follows from the snake identity (4.1) that V must be finite-dimensional and β must be nondegenerate. Moreover, the fact that β must be symmetric follows from the observation that the elbows have an automorphism that switches the two boundary components. The above construction can be upgraded to an equivalence of categories where Vect ∼ β is the groupoid of finite-dimensional vector spaces equipped with a nondegenerate symmetric bilinear form, with maps being isometries.
Things change if we equip our bordisms with non-discrete data. In the following, we will consider the geometry where objects X/S come equipped with a smooth map γ : X → M , where M is some fixed target manifold, as in Example 3.34. As a first approximation, we can think of objects of the bordism category as points in M , while morphisms are essentially paths in M . The corresponding smooth category is denoted by 1-Bord(M ), and the category of field theories will be denoted by It turns out that this groupoid is equivalent to the groupoid Vect ∼ ∇,β (M ), the objects of which are finite-dimensional vector bundles over M with a fiberwise nondegenerate symmetric bilinear form and a compatible connection, and the morphisms of which are connection-preserving isometries. The remainder of this section is dedicated to the proof of this result. First, in Section 4.1, we explain how to construct elements of 1-TFT(M ) from a vector bundle with connection and bilinear form, in a functorial way, cf. Proposition 4.4 below. While a little tricky in the detail, this is more or less the standard construction. The main work of the proof is done in Section 4.2, where we restrict our attention to the path subcategory of the bordism category, where all issues already arise. First, we restrict to the case that paths have sitting instants near the marked points (see Definition 4.8 for the precise definition), which is rather standard. The main new idea is then to reduce the general case to this one using so-called modification functions. The proof is then finished in Section 4.4. Finally, in Section 4.5, we comment on the oriented case.
In order to prove Theorem 4.2, one needs a result that reconstructs a connection from parallel transport data. To set up the one we use, denote by C ∞ ([0, 1], M ) the set of smooth maps from [0, 1] to M . It has a natural (infinite-dimensional) smooth manifold structure modelled on a nuclear Fréchet space, and there are smooth evaluation maps for t ∈ [0, 1]. Thus, given a vector bundle V over M , we can form the pullback bundles ev * t V . The tensor product ev * 0 V ∨ ⊗ev * 1 V is the vector bundle over C ∞ ([0, 1], M ) whose fiber at a path γ is given by Hom V γ(0) , V γ(1) . Finally, given 0 ≤ a ≤ b ≤ 1, we let s a,b be the smooth map from Proof . For v ∈ T p M , let γ ∈ C ∞ ([0, 1], M ) be a path such that γ(t) = p andγ(t) = v for some t ∈ (0, 1]. For any section u of V , set noting that P (s t−ε,1 γ)u(γ(t − ε)) ∈ V p for each ε, hence differentiation makes sense. We proceed to show that this definition is independent of the choice of γ and t and defines a connection on V . In fact, in a local trivialization of the bundle V , we have where we used that (s t,t γ)(t) ≡ p and P i j (s t,t γ) = δ i j , since P maps constant paths to the identity. Now notice the vector field d dε | ε=0 s t−ε,t γ along the constant path s t,t γ is given by where p denotes the path constant equal to p and V denotes the T p M -valued function V (t) = tv.
Observe that ω i j defines a matrix of one-forms on T p M . Then hence ∇ v is a connection with Christoffel symbols ω i j . Finally, fix γ ∈ C ∞ ([0, 1], X) and u 0 ∈ γ(0) and let u(t) := P (s 0,t γ)u 0 . Then we have u(0) = u 0 and using (s t−ε,t γ) · (t) = tγ(t), we obtain Hence u(t) is the parallel transport of u 0 along γ.

Construction of field theories from vector bundles
Let M be some fixed target manifold. In this section, we construct a field theory from the data of an object (V, ∇, β) ∈ Vect ∼ ∇,β (M ). More precisely, we prove the following proposition. Breaking down our general definition to this special case, an object of 1-Bord(M ) n lying over a manifold S is given by a family X/S of one-dimensional manifolds X s , s ∈ S, together with a map γ : X → M and functions ρ 0 , . . . , ρ n : X → R which cut out codimension-one submanifolds The properness assumption (O3c) implies that the restrictions X a a ∩X s to the fibers are compact, i.e., finite collections of points. More generally, we have the following result. Proof . Let s ∈ S and x ∈ X a a ∩ X s , and denote by π : X → S the projection. Since dρ a | Xs (x) = 0, X a a intersects the fiber X s transversally, that is, dπ(x) is an isomorphism when restricted to the tangent space T x X a a . Therefore, π| X a a is a local diffeomorphism. Furthermore, by assumption, X a a is proper over S, meaning that π| X a a is a proper map. However, a proper local diffeomorphism is a covering map the fibers of which have at most finitely many points (possibly zero).
To prove Proposition 4.4, we start by constructing a field theory from the data of a vector bundle with non-degenerate bilinear form and compatible connection.
Let (V, ∇, β) ∈ Vect ∼ ∇,β (M ). Our goal is to construct a field theory Z V,∇,β , which will be the value of (V, ∇, β) under the functor Φ in Proposition 4.4. Recall that the smooth functor Z V,∇,β , as a morphism of simplicial objects, will consist of a sequence of stack maps 1-Bord(X) n → Vect n , n ∈ ∆. At the nth simplicial level, we must have Z V,∇,β (X/S; ρ 0 , . . . , ρ n ; γ) = (W 0 , . . . , W n ; f 1 , . . . , f n ) for some vector bundles W a over S and vector bundle maps f a : W a−1 → W a . To define W a , for each a = 0, . . . , n, set firstW a := (γ| X a a ) * V , which makes sense since, for each a = 0, . . . , n, X a a is a codimension-one submanifold of X. By Lemma 4.5, X a a is a finite covering of S, hence any small enough open U ⊂ S is covered by U 1 , . . . , U k ⊂ X a a such that the projection map π provides diffeomorphisms π| U j : U j → U . Hence we can set where W a | U = C if k = 0. These vector bundles glue together to a vector bundle W a over S.
To define f a for each a = 1, . . . , n, consider the subsets X a a−1 . Let Y 1 , . . . , Y k be the connected components of X a a−1 | U , where U ⊂ S is a small connected open as above for both a − 1 and a. The map f a | U will be the tensor product of maps f j a , where f j a is determined by the connected component Y j . Each f j will be a vector bundle map where Z runs over the connected components of Y j ∩ X a−1 a−1 and Z runs over the connected components of Y j ∩ X a a (by possibly making U smaller, we can assume that the projection map is a diffeomorphism to U when restricted to any one of these sets Z and Z ). The tensor product f a | U := f 1 a ⊗ · · · ⊗ f k a is then indeed a vector bundle map W a−1 | U → W a | U , by definition (4.3). Now, each Y j is, essentially, either a circle bundle or an interval bundle over U ; this only fails to be the case if ρ a−1 (x) = ρ a (x) at some points x ∈ X a a−1 | U . To address this issue, we let which is an open subset of U . The complement U \ U j • is the set where Y j is a "thin bordism", in the sense that Note in particular that for s ∈ U \ U j • , Y j | s consists of finitely many points.
• is a fiber bundle whose fibers are compact one-dimensional manifolds with boundary (thus, either intervals or circles). Moreover, if Y j • is a circle bundle, then U j Proof . By construction, the total space Y j • is a compact manifold and the projection π| Y j • is a proper submersion, hence a fiber bundle (with possibly empty fibers). The last statement follows from the fact that circle bundles cannot degenerate, by the requirements on the functions ρ a .
By possibly shrinking U further, we may assume moreover that all these bundles are trivial.
We now define f j case by case.
Suppose first that Y j • is a circle bundle, so that Y j • = Y j and U j • = U . In this case, Y j ∩ X a−1 a−1 = Y j ∩ X a a = ∅, hence we have to produce a vector bundle map from the trivial line bundle to itself, that is, a function on U . Choose a trivialization ϕ : U × S 1 → Y j . Now set where P (ϕ) is the parallel transport around the loops ϕ s : S 1 → Y j ⊂ X given by ϕ s (t) = ϕ(s, t) with respect to the pullback connection of γ * V → X. We claim that f j is independent of the choice of ϕ. Ifφ is another trivialization of Y j that induces the same orientation on the fibers and agrees with ϕ at the basepoint 1 ∈ S 1 , then it is just a reparametrization of ϕ, and our claim follows from the invariance of parallel transport under reparametrizations. Without the assumption on basepoints, P (ϕ s ) and P (φ s ) are conjugates for each s ∈ U , so the trace f j is still independent of the choice of ϕ. Finally, ifφ induces the opposite orientation, then P (φ s ) = P (ϕ s ) −1 . This yields the same trace, since P (ϕ) preserves the bilinear form β; the calculation is where b 1 , . . . , b n is a generalized orthonormal basis for β and we used the symmetry of β (note that β is not assumed to be Hermitian in the complex case).
Suppose now that Y j • is a bundle of intervals. In this case, we have the following lemma. Proof . If U j • = U , the lemma is clear, because then Y j is an interval bundle over U , which must be necessarily trivial: it admits two nowhere agreeing sections, given by the zero sets of ρ a−1 , respectively ρ a . In general, over U , our bordism is isomorphic to a bordism of the form ((R × U )/U ; ρ 0 , ρ 1 ; γ), where ρ i (t, s) = t − τ i (s), i = 0, 1, for smooth functions τ 0 , τ 1 : U → R with τ 0 ≤ τ 1 .
In particular, this means that the paths ϕ s : [0, 1] → X given by ϕ s (t) := ϕ(s, t) map to the fibers Y j | s , and, for s ∈ U \ U j • , ϕ s is constant (since for such s, Y j • is a collection of finitely many points).
Let P (ϕ) be the vector bundle isomorphism between the bundles (γ • ϕ • (id × i)) * V , i = 0, 1, over U given over s ∈ U by parallel translation along the path ϕ s . Now notice that ϕ • (id × i), i = 0, 1 is a section of π : X → S, with image contained in either X a−1 a−1 or X a a ; hence For all s ∈ U , since X a a−1 | s is a (possibly degenerate) interval, V j ∩ X a−1 a−1 | s has either zero, one or two elements. Correspondingly, V j ∩ X a a | s has two, one or zero elements. In either case, we can use the bilinear form β to turn P (ϕ) into a morphism of the required form (4.4). This defines f j in this case. As before, we use the parametrization independence as well as the fact that parallel transport preserves β in order to show that this definition is independent of the choice of ϕ.
This defines Z V,∇,β on objects. On morphisms in 1-Bord(M ), we declare that Z V,∇,β acts by pullbacks in the obvious way. This concludes the definition of Z V,∇,β . Of course, there are several things to check in order to show that this is a field theory. However, all checks can be made pointwise, hence are very similar to the classical arguments outlined above (cf. Construction 4.1).
Proof of Proposition 4.4. Of course, we set Φ(V, ∇, β) = Z V,∇,β for (V, ∇, β) ∈ Vect ∼ ∇,β (M ), where Z V,∇,β is the field theory constructed above. We now discuss how Φ acts on morphisms in Vect ∼ ∇,β (M ). To this end, let (V, ∇, β) and (V , ∇ , β ) be vector bundles on M with connection and a compatible bilinear form, and let α : V → V be a vector bundle isomorphism preserving these additional structures. We now define the smooth natural transformation First, we look at the simplicial level zero. If X/S = (X/S; ρ 0 ; γ) is a single point, meaning that X 0 0 has connected fibers (in other words, π : X 0 0 → S is a diffeomorphism), we have Hence we can set in this case. Any object in 1-Bord(M ) 0 can, at least locally, be uniquely decomposed into a union of single-point-objects just discussed, and hence the requirement that η α is symmetric monoidal determines it on all of 1-Bord(M ) 0 . By Lemma 3.21, any smooth natural transformation η : Z V,∇,β → Z V ,∇ ,β is determined by its component η 0 at the simplicial level zero; however, it is not clear that η α defined above on simplicial level zero indeed extends to all higher simplicial levels. Here, again by Lemma 3.21 it suffices to consider the simplicial level one. To this end, let X/S = (X/S; ρ 0 , ρ 1 ; γ) be an object of 1-Bord(M ) 1 and write Z V,∇,β (X/S) = (W 0 , W 1 ; f 0 ) and Z V,∇,β (X/S) = (W 0 , W 1 ; f 0 ). We have to check that the diagram  For any natural transformation η : Z V,∇,β → Z V ,∇ ,β , the component η pt M is a vector bundle isomorphism V → V . In particular, if α : V → V is a vector bundle isomorphism preserving connections and bilinear forms, tracing through the above definitions shows that η α pt M = α. Hence if η α = η α , this implies α = α ; in other words, Φ is faithful. Conversely, it is easy to check that Φ η pt M = η for any natural transformation η, so that Φ is also full.

Functors from the path category
Let M be a fixed target manifold. In this section, we restrict our attention to the smooth path category of M , a certain subcategory of 1-Bord(M ) which is somewhat easier to describe. Definition 4.8 (smooth path category). Write Path(M ) for the full smooth subcategory of 1-Bord(X) consisting of those objects (X/S; ρ 0 , . . . , ρ n ; γ) ∈ 1-Bord(M ) n such that each of the sets X b a , 0 ≤ a ≤ b ≤ n, defined in (3.6) has connected fibers. Let moreover Path c (M ) be the full subcategory of Path(M ) consisting of those objects (X/S; ρ 0 , . . . , ρ n ; γ) such that γ is fiberwise constant in a neighborhood of X a a for each 0 ≤ a ≤ n.
Objects in Path(M ) n over S ∈ Man can be thought of as S-families of paths in M with n + 1 marked points, while the full subcategory Path c (M ) n consists of those paths that have sitting instants at the marked points. Notation 4.9 (standard objects). We denote by (γ; τ 0 , . . . , τ n ) := (R × S)/S; ρ 0 , . . . , ρ n ; γ ∈ Path(M ) n the object where R × S → S is the projection onto the second factor, γ : R × S → M is a smooth map, and the cut functions ρ 0 , . . . , ρ n are given by ρ j (x, s) = x − τ j (s) for smooth functions τ j : S → R satisfying τ 0 ≤ · · · ≤ τ n .
Remark 4.10. Going through the definition of the morphisms in 1-Bord(M ) shows that morphisms between standard objects over S = pt are (equivalence classes of) diffeomorphisms F of R, which must be orientation preserving, as they need to preserve the sign of the cut functions ρ i .  where W a := (γ • (τ a × id)) * V is a vector bundle over S and, for each s ∈ S, P j (s) is the parallel transport via ∇ along the path t → γ(t, s), t ∈ [τ j−1 , τ j ]. To a morphism between two standard objects (γ; τ 0 , . . . , τ n ) and (γ ; τ 0 , . . . , τ n ) we assign the identity; this is well-defined to extend to higher simplicial levels is precisely the condition that α preserves connections. Hence if α : V → V is a connection-preserving isomorphism of vector bundles, we get a natural transformation η α : Z V,∇ → Z V ,∇ . This yields a functor The fundamental result is now the following.   this means that we can (and will) assume in the future that Z(γ, a) is equal to Z(η, a) for paths that are equal near a.
In this section, we will prove the following weaker version of Theorem 4.13, which states that Φ is an equivalence when considered as a functor to Fun(Path c (M ), Vect). The proof of Theorem 4.13 will then be completed by Proposition 4.21 from Section 4.3, which reduces the general case to the one just below. We start our preparations for the proof of the above proposition with a couple of lemmas, for which we fix a smooth functor Z : Path(M ) → Vect. The following lemma uses Notation 4.9. Proof . Invertibility can be checked pointwise, hence we may assume that γ is a single path (i.e., a family over the point). We have Z(γ; a, a) = id as Z(γ; a, a) Suppose that b 0 < ∞. Then, since the set of b such that Z(γ; a, b) is not invertible is a closed set, the infimum is actually a minimum. Therefore Z(γ; a, b 0 ) is not invertible, but Z(γ; a, b) is invertible for each b < b 0 . Now On the other hand, since Z(γ; b 0 , b 0 ) = id, the linear map Z(γ; b, b 0 ) is invertible for b close enough to b 0 . This leads to a contradiction to the assumption that Z(γ; a, b 0 ) is not invertible, as for such b close to b 0 , the right hand side is a composition of two invertible maps. Hence we must have b 0 = ∞, which proves the lemma. Proof . Let U a be a small neighborhood in which γ is constant and let G : R → R be a diffeomorphism satisfying Since G has the same germ at a as as γ • T a has a sitting instant at t = 0; here by abuse of notation, γ(0) denotes the constant path equal to γ(0). From an orientation preserving diffeomorphism F , we get a commutative diagram Now both γ • T a and γ • F −1 • T F (a) have a sitting instant at t = 0, so the assumption from Remark 4.15 on Z tells us that the two vector spaces in the right column agree; for any diffeomorphism F and all a ∈ R. Using this equivariance property, we obtain the following lemma.
where the T γ,a are defined as in (4. This clearly defines a natural transformation Z →Z. ThatZ([F ] a ) acts as the identity also follows directly from (4.8).
We are now in a position to prove the main result of this section.
Proof of Proposition 4.16. That res •Φ is full and faithful is shown just as in the proof of Proposition 4.4. It therefore remains to show that res •Φ is essentially surjective. Moreover, it suffices to consider the functor on the subcategory of standard objects. Let Z : Path c (M ) • → Vect be a strict smooth functor. We assume moreover that Z is normalized in the sense of Lemma 4.19; in other words, Z(γ, a) = Z(γ(a), 0) for all paths γ and all a ∈ R, and Z([F ] a ) = id for all diffeomorphisms F on R. In particular, this implies that hence the W i are already determined by V . It remains to determine the vector bundle maps f i ; we will use Proposition 4.3 for this. To obtain a section P as in the proposition, we use modification functions, which are defined as follows.    Later, in Definition 5.5, we will introduce also left and right modification functions, as well as their family versions. For now, we drop the adjective "two-sided". If χ is a modification function and γ ∈ C ∞ ([0, 1], M ), then γ • χ is a path that is defined on all of R and which is constant on (−∞, ε] ∪ [1 − ε, ∞) for some ε > 0. Hence is well-defined for each γ ∈ C ∞ ([0, 1], X). Note that P (γ) maps Z(γ • χ, 0) = Z(γ(0); 0) = V γ(0) to Z(γ • χ; 1) = Z(γ(1); 1) = V γ(1) . This construction works in families and therefore we get a smooth section P of the bundle ev * 0 V ∨ ⊗ ev * 1 V , as required. The crucial result, which will be shown in Lemma 4.23 below, is then that P (γ) is independent of the choice of modification function.

General paths
In this section, we finish the proof of Theorem 4.13. Having Proposition 4.16 at hand, this will be achieved by establishing the following result. such that res • ext = id, together with a natural isomorphism η : id → ext • res.
We need several lemmas, for which we fix a smooth functor Z : Path(M ) → Vect. We will assume that Z is a strict functor (which is possible by Lemma 3.37), and we will also make the simplifying assumptions discussed in Remark 4.15. We remark that if we choose F to be constant somewhere in between a and b (as allowed by the lemma), the path γ • F will have a sitting instant somewhere in between a and b.
Proof . If F (x) = 0 for all x ∈ [a, b], so that F is a diffeomorphism onto its image in a neighborhood of [a, b], the simplicial structure yields the commutative diagram This, together with the fact that F = id near a and b, so that Z(γ • F ; a) = Z(γ; a) and Z([F ] a ) = id, and similarly for b, proves the result in this case. The general case now follows from the fact that is a one-parameter family of maps such that F 0 = F and such that F ε is a diffeomorphism whenever ε ∈ (0, 1]. Therefore, by the observations above, we have Z(γ; a, b) = Z(γ • F ε ; a, b) for all ε ∈ (0, 1], and, by continuity, the equality persists for ε = 0. We are now able to prove the following essential lemma, which proves the independence of the choice of modification function (see Definition 4.20).  •χ a,b , a, b) as morphisms from Z(γ(a), a) to Z (γ(b), b).
Proof . Clearly, we may assume for simplicity that a = 0 and b = 1. Now, we first argue that we may furthermore assume that χ(t) =χ(t) = t for t in a neighborhood of 1 2 . Since χ is not constant, there exists some t 0 ∈ (0, 1) such that χ (t 0 ) = 0. We may arrange that t 0 = 1 2 : Choose some diffeomorphism F that is the identity near t = 0, 1 and sends 1 2 to t 0 . Then ξ := χ • F is again a modification function, now satisfying ξ ( 1 2 ) = 0. We get that there exists a neighborhood of 1 2 , where ξ is invertible. Hence there exists some small ε > 0 such that we can find a diffeomorphism G of R with Then ξ • G is a modification function that is the identity near t = 1 2 , and we have since F • G is the identity near t = 0, 1. By the above, after replacing χ andχ with equivalent modification functions, we may assume that χ(t) =χ(t) = t near t = 1 2 . Now let F andF be monotonic functions on R such that This is possible since χ andχ are both constant near zero. Now since F ,F are the identity near − 1 2 and 1 2 , Lemma 4.22 gives On the other hand, by the construction of F andF . As all morphisms involved are invertible in view of Lemma 4.17, combining (4.10) with (4.11) implies that Z γ • χ; 0, 1 2 = Z γ •χ; 0, 1 2 . A similar argument shows that Z γ • χ; 1 2 , 1 = Z γ •χ; 1 2 , 1 ; combining these observations finishes the proof.
We are now ready to give the proof of the main result of this section.
for some modification function χ. By Lemma 4.23, this is independent of the choice of modification function. Let F be a diffeomorphism of R, which defines a morphism Using the simplicial structure and Remark 4.11, this determines the functor ext Z completely.
In order to show that ext Z is well defined, we need to check functoriality. Let F be a diffeomorphism of R, which defines an automorphism in Path(M ) q . We need to show that the square commutes. We have Now first notice that F • χ a,b • F −1 =χ a ,b for some modification functionχ, hence the middle term equals Z(γ; a , b ) (here, of course, we use Lemma 4.23 again). Secondly, which was the claim. It is straightforward to check that this gives a natural transformation ext η : ext Z → ext Z and that the assignment η → ext η is functorial in η. This finishes the definition of ext.
We claim that this definition is independent of the modification function χ and the choice of δ and . For notational simplicity, let a = 0 and suppose = 1; the case a = 0 is similar. Now, letχ be another modification function also satisfyingχ(t) = t for t near δ. Since the values of a modification function on [δ, 1] are irrelevant for the definition of η, we may as well assume that χ = χ on [δ, 1] (which then implies that they in fact agree on a neighborhood of [δ, 1]). Now, by Lemma 4.23, we have Z(γ • χ; 0, 1) = Z(γ •χ; 0, 1). On the other hand since χ andχ agree in a neighborhood of δ. But this equals Z(γ •χ; 0, 1) = Z(γ •χ; δ, 1) • Z(γ •χ; 0, δ), from which we obtain the desired equality Z(γ •χ; 0, δ) = Z(γ •χ; 0, δ), by virtue of Lemma 4.17. To see the independence from δ, letχ be a modification function withχ(t) = t nearδ. Without loss of generality, suppose thatδ < δ. By the first step, we are free to choose the modification function χ any way we like, under the constraint that χ(t) = t near δ. We now choose it in such a way that in fact χ(t) = t in a neighborhood of the interval δ , δ . Then However, on δ,δ , we have γ •χ = γ, and by the first step, we have Z γ •χ; 0,δ = Z γ •χ; 0,δ , so that This finishes the argument that η γ,a does not depend on δ. The independence of the choice of now follows immediately.
In order to show that η indeed gives rise to a natural transformation, we have to show that for any diffeomorphism F of R, we have the equivariance property (4.12) First notice that for any path ξ and any diffeomorphism F , we have a commuting square Z(ξ; F (a)) Z(ξ; F (b)).
The last thing to show is that if µ : Z → Y is a smooth natural transformation of smooth functors Y, Z : Path c (M ) → Vect, then η Y • µ = ext µ • η Z . But this is trivial, since η Z and η Y act as the identity on Path c (M ) and ext µ| Pathc(M ) = µ.

Proof of the classification theorem
We are now in a position to prove Theorem 4.2; more precisely, we will prove that the functor Φ from Proposition 4.4 is essentially surjective.
Proof of Theorem 4.2. Let Z : 1-Bord(M ) → Vect be a field theory. By the results of the previous section, we may assume that Z| Path(M ) = Φ(V, ∇), where is the equivalence constructed in Section 4.2. This means that Z(γ, a) = V γ(a) for all paths γ in M and all a, and that Z(γ; a, b) is given by parallel transport along γ from a to b, with respect to the connection ∇.
To get a bilinear form β on V , consider the constant right elbow, which is the bordism R = ((R × M )/M ; ρ 0 , ρ 1 ; γ const ), where γ const (t, p) = p, ρ 1 ≡ −1 and ρ 0 (t, p) := t(1 − t). Then, canonically, Hence Z(R) is a linear map from Z(γ const , 0) ⊗ Z(γ const , 1) to K. However, Z(γ const , 0) = Z(γ const , 1) = V , so we get a bilinear form β := Z(R) on V . It is symmetric because the diffeomorphism F : R → R, determines an automorphism of R in 1-Bord(M ), the restriction of which to R 0 0 gets mapped to the symmetry isomorphism of V ⊗ V under Z. This is because F swaps the two components of R 0 0 = ({0, 1} × M )/M . To see that β is nondegenerate, let L be the constant left elbow, which is the bordism given by L = (R × M/M ; ϑ 0 , ϑ 1 ; γ const ), where ϑ 0 (t) ≡ 1 and ϑ 1 (t) = −ρ 0 (t). Set τ := Z(L), which is a section of the bundle Hom(K, V ⊗ V ), i.e., a section of V ⊗ V . The "snake identity" is satisfied. (To be precise, the left hand side is in fact a map from K ⊗ V to V ⊗ K, but this can be canonically be identified with an endomorphism of V ; the requirement is that this endomorphism be the identity.) We now analyze this identity on each fiber; to this end, write τ p = ij v i ⊗ w j for some elements v i , w j ∈ V p . Then for any u ∈ V p , we have The requirement that this be equal to u ⊗ 1 implies that β must be nondegenerate, as claimed. Note that it also implies that τ = j ε j b j ⊗ b j , where b 1 , . . . , b n is a generalized orthonormal basis for β; thus, is determined by β.
We now want to show that Z = Z V,∇,β = Φ(V, ∇, β), where Φ denotes the functor constructed in Proposition 4.4. We already know that Z = Z V,∇,β on simplicial level zero and Z(B) = Z V,∇,β (B) on all bordisms B that are (tensor products of) intervals and/or constant elbows. Using the techniques from the previous section, we can introduce sitting instants into any nonconstant elbow B and then express B as the composition of two intervals and a constant elbow. This determines the field theory on all intervals and elbows, as well as on circles, since any circle can be decomposed into two elbows. Hence we have Z = Φ(V, ∇, β).

The oriented case
We now briefly comment on the oriented case. The geometry considered here is the one considered in Example 3.35, where manifolds are endowed with orientations. In dimension one, the resulting bordism category will be denoted 1-Bord or (M ) and we write 1-TFT or (M ) for the corresponding groupoid of field theories.
The main difference of 1-Bord or (M ) to the unoriented bordism category is that there are now two different kinds of points: Remember that a point is given as the zero set of the cut function ρ 0 on a one-dimensional manifold (respectively a family of such). Now because ρ 0 (x) = 0 for x ∈ X, the orientation allows to ask whether dρ 0 (x) (which is non-zero and hence a basis of T x X) is positively oriented or negatively oriented. This leads to positive respectively negative points. Correspondingly, we have one more elementary bordism, as displayed in Figure 5. Figure 5. All possible connected oriented bordisms. We call them positively and negatively oriented intervals, left elbow, right elbow, and circle, respectively. The cut functions are ρ i = t − τ i , so these pictures are read from top to bottom.
A field theory Z ∈ 1-TFT or will assign vector spaces V ± to the two different points. The elbow then provides a pairing V + ⊗ V − → C, which must be non-degenerate due to the snake identity; hence V − can be identified with the dual space of V + and vice versa. In particular, we do not obtain the additional datum of a nondegenerate bilinear form on our vector spaces.
Passing from ordinary field theories to oriented field theories over a target M , the result is the following. Here Vect ∼ ∇ (M ) denotes the groupoid of vector bundles with connection on M , with morphisms given by connection-preserving bundle isomorphisms. The proof is analogous to that of Theorem 4.2.

Field theories with values in sheaves
In this section, we consider notions of "families of vector spaces" more general than vector bundles. Let V denote some stack of C ∞ -modules. This means that for each manifold S, objects of V(S) are modules over C ∞ (S), possibly with a bornology or topology of a particular kind. Below, we will fix suitable conditions V should satisfy, but we would like, at a minimum, that the operation of taking global sections determines an embedding Vect → V, and a symmetric monoidal structure on V compatible with this embedding. In line with Definition 3.36, we write for the groupoid of 1-dimensional field theories over M taking values in V, and similarly for the oriented variant. In the TFT case, it is expected that this introduces no new examples, that is, a field theory with values in V automatically takes values in Vect. In this section, we verify that this is indeed the case.
Definition 5.1. We will call a symmetric monoidal smooth stack V an admissible stack of C ∞ -modules if the following holds.
(C1) For each S, V(S) is an additive category with kernels, naturally in S, such that ⊗ is additive in both variables.
(C2) There is a linear symmetric monoidal fibered functor Vect → V which determines, for each S, an equivalence between Vect(S) and the dualizable objects of V(S).
Condition (C3) is a kind of separation axiom. It is often useful to regard a morphism W → W in V(S) as a generalized element of W . Then a morphism σ : V → pr * W as above can be interpreted as a "generalized path of elements" of W , and the axiom says that a generalized path vanishes altogether provided it vanishes on R \ 0.
In Section 5.2, we give examples of admissible stacks of C ∞ -modules. Before doing this, we state and prove the following theorem, which contains the previously mentioned result that the target does not matter for topological field theories. for any admissible V. This is the exact meaning we intend for Theorems 1.1 and 1.2, stated in less detail in the introduction.

Proof of Theorem 5.2
We will focus on the unoriented case, the oriented one being similar. Throughout, we fix a field theory Z : 1-Bord(M ) → V, which we assume to be strict as a morphism of simplicial stacks. Proof . For γ : S × R → M and a < b, we denote by L γ;a,b the left elbow given by this data, that is, the S-family of bordisms with underlying map γ and cut functions ρ 0 (s, t) = (t − a)(t − b) and ρ 1 (s, t) = − 1 4 (b − a) 2 , where t is the coordinate of R (here any ρ 1 can be chosen such that it has no zeroes, and such that ρ 1 ≤ ρ 0 everywhere). We write similarly R γ;a,b for the right elbow. To prove the lemma, we consider in particular S = M and γ = c; then for each δ > 0, the images of L c;0,δ and R c;0,δ are canonically identified as maps where M is the monoidal unit of V(M ). We claim that these morphisms are independent of δ. To see this, pick any diffeomorphism χ : R → R with χ(0) = 0, χ(1) = δ which is affine of slope 1 near 0 and 1. Then χ determines a morphism in 1-Bord(M ) 1 between L c;0,δ and L c•χ;0,1 = L c;0,1 , compatible with the usual identifications of their boundary components with (c; 0). This proves the claim.
Write L = Z(L c;0,δ ), R = Z(R c;0,δ ) for this common value. Then, expressing the interval bordism (c, 0, δ) as a suitable composition of L c;0,δ and R c;0,δ , we get for any δ > 0. Now consider, for instance, (c, 0, δ 2 ) as an (M × R)-family of intervals, where δ now denotes the coordinate on the factor R. Its image under Z is a family of endomorphisms of Z(c; 0) that is constant away from δ = 0, and equal to the identity at δ = 0. Thus, from the separation axiom of V, Z(c; 0, δ) = Z(c; 0, 0) = id Z(c;0) for all δ. This proves that L and R provide the desired evaluation and coevaluation maps. of smooth functors, where Z V,∇ denotes the field theory determined by parallel transport on V . We will eventually show that Z ∼ = Z V,∇,β for some pairing β.
Lemma 4.22 is easily restated for families of bordisms and reproved for V-valued field theories, using the separation axiom. We now adapt some other definitions and lemmas from Section 4 for which special care with the family aspect is crucial. Let γ : S × R → M be an S-family of paths and let a : S → R be any smooth function, so that we can talk about the S-family of objects (γ; a) (where its cut function is ρ a (s, t) = t − a(s)). Note that by the implicit function theorem, any cut function ρ on S × R is equivalent to some ρ a : Just take a defined by ρ(s, a(s)) = 0. Similarly, two functions a ≤ b : S → R determine a family of intervals (γ; a, b). These examples exhaust all possible isomorphism classes of objects, respectively bordisms. We define right modification functions by swapping the roles of a and b, and two-sided modification functions by requiring fiberwise constancy near both ends. is a two-sided modification function. Next, write Due to the bounds on g, this is nonnegative. Finally, set Clearly, h s is nondecreasing, constant near t = a(s), and affine with slope 1 near t = b(s). Moreover, by the choice of κ(s), we have h s (a(s)) = a(s) and h s (b(s)) = b(s). Thus, defines a left modification function. Right modification functions can be constructed similarly. Note that all modification functions agree in a neighborhood of the boundaries, so the domain and codomain of the maps Z(γ • χ, a, b) are canonically identified, as in Remark 4.15.
Proof . The case of two-sided modifications follows immediately from Lemma 4.23, since Vect is full in V, so that the maps Z(γ • χ i , a, b) are identified with maps of vector bundles. Suppose now that χ 0 , χ 1 are left modification functions. Let U be a neighborhood of {t = b}, where χ 0 and χ 1 both agree with the identity. Then we can find a function a with a > a > b and {t = a } ⊂ U , as well as a map F : S × R → S × R over S, fiberwise nondecreasing, which is the identity away from U and near {b = 0}, and fiberwise constant on {t = a }. Thus, using Lemma 4.22, we get for both i = 0, 1. Finally, notice that by the two-sided version of this lemma. This finishes the proof of the left-sided case; the rightsided case is similar. Above, we used the suggestive notation γ(a) for the map s → γ(s, a(s)). Note that V γ(a) ∼ = Z(γ(a), a) is what Z assigns to the family of constant germs with the same value as γ on {t = a}.
Proof . Choose smooth functions a , a : S → R with a < a < a and smooth, fiberwise nondecreasing functions χ 1 , χ 2 : S × R → S × R such that χ 1 is fiberwise constant near a and a and equal to the identity near a, while χ 2 is fiberwise constant near a , a and a . In other words, χ 1 is a left modification function for (γ; a , a) and a right modification function for (γ; a, a ), while χ 2 is a two-sided modification function for both (γ; a , a) and (γ; a, a ). In particular, both χ 1 and χ 2 are two-sided modification functions for (γ; a , a ). The existence of such functions is clear from Lemma 5.6 and the fact that modification functions can be "patched together" in the obvious way for neighboring intervals. We obtain the following diagram of bordisms: obtained by patching together the corresponding diagrams (5.1) in the obvious way. In particular, all bottom maps are parallel translations along segments of γ.
To prove that the second columns of the matrix representing Z(γ; a, b) is zero, we need to show that this map, restricted to the kernel of p a , is zero. This follows from the diagram, since it shows that Z(γ; a, b) factors through p a .
A diagram chase shows that the square composed by Z(γ; a, b), i a , i b , and the parallel translation P : V γ(a) → V γ(b) commutes. This implies that the first column of Z(γ; a, b) is as claimed.
The proof of Theorem 5.2 concludes with the next lemma. Proof . Let pr : S × R → S be the projection and let δ : S × R → R be the coordinate function on the R-factor. Consider the (S × R)-family of bordisms B = (pr × id) * γ; pr * a − δ 2 , pr * a . Then vanishes away from S × 0 by the previous lemma, and therefore is identically 0 by the separation axiom. But this implies that id Z (γ;a) = Z (B)| S×0 = 0, so Z (γ; a) is the zero object of V(S).

Examples of admissible stacks of C ∞ -modules
In this section, we give two examples of suitable target stacks V for TFTs: 1. The stack V alg of sheaves of C ∞ -modules with the algebraic tensor product.
2. The stack V vN of sheaves of complete bornological C ∞ -modules with the completed bornological tensor product, constructed in Appendix A.5 and briefly reviewed below. Here, C ∞ (S) is endowed with its von Neumann bornology (see Example A.2), hence the notation.
Recall that a bornology on a vector space is a collection of subsets deemed to be bounded and satisfying appropriate axioms. Bornological vector spaces and bounded linear maps form a category Born. There is an appropriate notion of completeness, and thus a full subcategory CBorn ⊂ Born of complete bornological vector spaces. We are interested in CBorn for its pleasant categorical properties. In particular, there is a completed tensor product⊗ which makes CBorn into a closed symmetric monoidal category. As is well known, none of the many possible tensor products on the various categories of topological vector spaces have this property. See Appendices A.1 to A.3 for a quick overview, or Meyer [8, Chapter 1] for a comprehensive introduction to the theory of bornological vector spaces.
The goal of this subsection, which will follow immediately by combining Propositions 5.17 and 5.19 and Corollaries 5.18 and 5.20 below, is to prove the following.
Theorem 5.11. V alg and V vN are admissible stacks of C ∞ -modules.
Remark 5.12. The purely algebraic V alg is familiar and attractive in its simplicity, but it has the drawback of not admitting a reasonable "sheaf of sections" functor from any category of infinite-dimensional vector bundles. Moreover, our proof that it satisfies the separation axiom requires the corresponding fact for V vN .
Before proceeding, let us briefly review the definitions of V alg and V vN . It will be convenient to use the sheaf-of-categories approach to stacks.
We define V alg : Man op → Cat ⊗ to be the functor assigning to each manifold S the symmetric monoidal category of sheaves of C ∞ S -modules and C ∞ S -linear maps, with the algebraic tensor product. To a map f : T → S is associated the pullback functor f * defined by That this indeed defines a symmetric monoidal stack on the site of manifolds and satisfies axiom (C1) is a standard fact. Similar constructions can be carried out in the setting of (complete) bornological vector spaces. This is outlined in Appendices A.4 and A.5 and is mostly a formality, except for the fact that sheafification of presheaves with values in CBorn requires additional care. In a nutshell, we promote the sheaf of smooth functions on a manifold S to a sheaf of complete bornological algebras, denoted by C ∞ vN,S , and then define V vN (S) to be the category of sheaves of complete bornological C ∞ vN,S -modules. We have a completed tensor product of bornological sheaves, and the pullback functors f * : V vN (S) → V vN (T ) are defined similarly to (5.2), using this completed tensor product. Thus V vN is a symmetric monoidal stack satisfying axiom (C1).
Next, we construct a functor i : V alg → V vN . Let S be a manifold and V ∈ V alg (S). Given U ⊂ S open, we say that a subset B ⊂ V (U ) is vN-bounded if each point x ∈ U has an open neighborhood W ⊂ U such that B| W ⊂ C 1 b 1 + · · · + C n b n for vN-bounded subsets C 1 , . . . , C n ⊂ C ∞ (U ) and b 1 , . . . , b n ∈ V (W ). It is routine to check that this indeed satisfies the axioms of a (vector) bornology on V (U ). We denote the resulting presheaf of bornological vector spaces by V vN .
Example 5.13. If V = Γ E is the sheaf of sections of a finite-dimensional vector bundle over E → S, then the bornology constructed above coincides with the bornology associated to its Fréchet topology, where a set of sections is bounded if all derivatives of a given order are uniformly bounded on compact sets. Proposition 5.14. For any sheaf V of C ∞ S -modules, V vN is a sheaf of complete bornological C ∞ vN,S -modules. Proof . It follows directly from the definition of the vN-bornology that for each open U ⊆ S, the bornological vector space V vN (U ) is a bornological C ∞ vN (U )-module (i.e., the module action map is bounded), and that restriction maps are bounded. We have to show that V vN satisfies descent. Let U ⊂ S be open and let {U i } i∈I be an open cover of U . Since V is an (algebraic) sheaf, we know that the map is a vector space isomorphism. We have to show that it is an isomorphism of bornological vector spaces. Since we already know that restriction maps are bounded, we have to show that the inverse image under r of bounded sets in i∈I V vN (U i ) is bounded. It suffices to show this for sets of the form i∈I B i with B i ⊂ V vN (U i ) bounded, as these generate the product bornology. Here we have That this is vN-bounded follows again directly from the definition of the bornology, as boundedness can be checked locally. This shows that V vN is a sheaf of bornological C ∞ vN,S -modules. We now show that V vN (U ) is complete for each open U ⊆ S. To this end, we have to show that each bounded subset of V vN (U ) is contained in a complete bounded disc. Here a bounded disc is complete if the normed space V (U ) B = span(B) ⊂ V (U ) is complete, where the norm is the Minkowski functional defined by B. Let {W i } i∈I be an open cover of U , let b i 1 , . . . , b i n i ∈ V (W i ) and C i 1 , . . . , C i n i ⊂ C ∞ (W i ) be complete vN-bounded discs. Then the sets sending (f 1 , . . . , f n i ) to f 1 b 1 + · · · + f n i b n i . It is surjective by definition of B i , and it is bounded since any bounded set C ⊂ X is contained in λ(C i 1 × · · · × C i n i ) for some λ > 0, hence π(C) ⊂ λπ(C i 1 × · · · × C i n i ) = λB. Notice that X is a Banach space as the discs C i k are complete. Because π is bounded, its kernel is closed, so that the quotient X/ Ker(π) is a Banach space. The quotient mapπ : X/ Ker(π) → V (W i ) is a bounded vector space isomorphism. To see that it is a homeomorphism, it remains to show that the inverse is bounded. To this end, let v ∈ V (W i ) B i . Let λ > 0 be such that v ∈ λB i . Because π C i 1 × · · · × C i n i = B i , we can choose a preimage v ∈ λ(C i 1 × · · · × C i n i ) ⊂ X. Then This holds for any λ such that v ∈ λB i . Therefore, we get This shows thatπ −1 is bounded, henceπ : where the right hand side is a product of Banach spaces, endowed with the norm obtained by taking the supremum over all the norms of the V (W i ) B i . In other words, the unit ball of the product is i∈I B i . Observe that this map is well-defined since if v ∈ λB, then v| W i ∈ λB i and (v| W i ) ∈ λ i∈I B i . The map is bornologically proper, since r −1 i∈I B i = B. On the other hand, its image is the kernel of the bounded map This shows that V (U ) B is isomorphic to a closed subspace of a Banach space, hence complete.
To promote the assignment V → V vN to a functor, we have to deal with morphisms. Here we have the following lemma.
Lemma 5.15. Let V and V be (algebraic) C ∞ S -modules and let Φ : V → V be a morphism of sheaves. Then, for each U ⊆ S, Proof . Let B ⊂ V (U ) be vN-bounded. We have to show that also Φ U (B) is vN-bounded. To this end let x ∈ U and choose W such that B| W ⊆ C 1 b 1 + · · · + C n b n for elements b 1 , . . . , b n ∈ V (W ) and vN-bounded subsets C 1 , . . . , C n ⊂ C ∞ (W ). Then since Φ is a morphism of sheaves and C ∞ S -linear, We therefore obtain a functor i : V alg → V vN sending V to V vN and which sends morphisms V → V of sheaves to the corresponding morphism V vN → V vN , which exists by the previous lemma. Since conversely, each morphism of sheaves V vN → V vN gives a morphism of the underlying algebraic sheaves of C ∞ S -modules, we obtain the following corollary.
Corollary 5.16. The functor i : V alg → V vN described above is fully faithful.
Proof . For each S, there is a "sheaf of sections" functor Γ : Vect(S) → V alg (S). To see that these fit together to a map of stacks, we need to show that for each vector bundle V ∈ Vect(S) and each map f : T → S, the natural maps f * (Γ V ) → Γ f * V are isomorphisms. On the stalk at x ∈ T , this yields a map Here, V f (x) is the fiber of the vector bundle V ; all other subscripts denote stalks at the given point. Thus, the above is an isomorphism on stalks and hence an isomorphism of sheaves. That Γ has all dualizable objects as essential image is the content of the Serre-Swan theorem. Proof . Composition with i gives the map Γ : Vect → V vN . Since every dualizable C ∞ vN,S -module has finite rank, any dualizable object in V vN (S) is already dualizable in V alg (S).
Proof . Fix a C ∞ vN,S -module W , and denote by V its pullback to S × R. It suffices to show that if a section σ ∈ V (U ) is such that σ| U \(S×0) vanishes, then in fact σ = 0. By definition, V is the sheafification of the presheaf so we know that σ is determined by a coherent collection of sections σ i ∈ V (S i × T i ) for some collection of opens S i ⊂ S, T i ⊂ R, since products S i × T i form a basis for the topology of U ⊂ S × R. Our question is reduced to showing that if σ i vanishes on S i × (T i \ 0), then it vanishes on S i × T i . Now, The first identification is by definition, the second follows from Proposition A.14, the third from the fact that C ∞ (T i )⊗ -commutes with the colimit defining the tensor product over C ∞ (S i ), and the fourth from Proposition A.13. Thus, σ i gets identified with a smooth function f : T i → V (S i ) which, by definition, is a smooth function with values in some Banach space Hence it vanishes identically if it vanishes away from 0.
Remark 5.21. One might wish for a more elementary proof of the corollary, and we would like to note that the bornological tensor product is handy in this case as well. Arguing as in the proof of Proposition 5.19, one reduces property (C3) for V alg to the claim that if V is a C ∞ (S)-module and V , then σ = 0. This is not immediately clear. Notice, however, that if we give V the fine bornology, Proposition A.11 allows us to regard σ as an element of and therefore as a smooth function from R to some Banach space. The claim follows.

A Bornological sheaves
In this appendix, we construct the symmetric monoidal stack V vN of sheaves of complete bornological C ∞ -modules on the site Man of smooth manifolds. Formally, this construction is very similar to that of its well-known algebraic counterpart V alg , and our main goal here is to highlight the differences. We start recalling several basic facts about bornological vector spaces, providing proofs for the bits that are not easily located in the literature. For this, our main reference is Meyer [8,Chapter 1]. The basic facts about sheaves of (complete) bornological vector spaces and modules are quoted from Houzel [5].

A.1 Bornological vector spaces
A (convex) bornology on a vector space V over K = R or C is a collection S of subsets of V deemed to be bounded. These have to satisfy appropriate axioms which we will not repeat here. For any collection S of subsets of V , there is a smallest bornology containing S , the bornology generated by S . A linear map f : V → W is bounded if it sends bounded subsets to bounded subsets.
A disk in a vector space V is a convex, balanced subset B ⊂ V . We denote by V B = R·B ⊂ V the subspace spanned by B. The closed ball of a seminorm on V B is an absorbing disk, and conversely a disk B ⊂ V determines a unique seminorm on V B . We say that B is norming respectively complete if V B is a normed respectively a Banach space. A bornological vector space is separated if every bounded disk is norming, and complete if every bounded subset is contained in a complete bounded disk.
Example A.1. The smallest possible bornology on a vector space V is the one generated by convex hulls of finite subsets. This is called the fine bornology. It is always complete. We denote by Fin(V ) this bornological vector space. With this bornology, we have the relation V ∼ = colim V α , where V α runs through all finite-dimensional subspaces of V , endowed with their fine bornology.
Example A.2. Let V be a locally convex topological vector space. Traditionally, a subset of B ⊂ V is called bounded if it is absorbed by any neighborhood of the origin. This defines a bornology on V , called the von Neumann bornology. It is complete if V is complete as a topological vector space. We denote by vN(V ) this bornological vector space.
We are interested in bornological vector spaces for their convenient categorical properties. We denote by The forgetful functor Born → Vect preserves limits and colimits; in other words, limits and colimits in Born are obtained by putting appropriate bornologies on the corresponding constructions with plain vector spaces. The bornology on a direct sum i∈I V i is generated by images of bounded subsets of each V i via the standard inclusions; a subset B ⊂ i∈I V i is bounded if and only if each projection p i (B) ⊂ V i is bounded; kernels and cokernels are endowed with the subspace and quotient bornology, respectively.
The inclusion i : CBorn → Born has a left adjoint, the completion functor Born → CBorn, V → V c . This means that there are natural isomorphisms Hom(V, W ) ∼ = Hom V c , W for W complete. In particular, limits in CBorn are calculated as limits in Born. This is also true for direct sums. However, completion does not preserve colimits in general: the cokernel of a map f : V → W in CBorn is obtained by modding out the bornological closure of the image, Coker f = W/f (V ) [8,Section 1.3.3]. We may write sep colim, sep Coker, etc. to emphasize that a construction is taken inside CBorn.
The space Hom(V, W ) of all bounded linear maps has a natural bornology generated by the uniformly bounded subsets, i.e., those L ⊂ Hom(V, W ) such that We equip the algebraic tensor product V 1 ⊗ V 2 with the bornology generated by subsets of the form B 1 ⊗ B 2 , with B 1 ⊂ V 1 , B 2 ⊂ V 2 bounded. We define the completed bornological tensor product by Theorem A.4 ([8, Proposition 1.111]). With the completed tensor product and the bornology on hom-sets defined above, CBorn becomes a closed symmetric monoidal category. This means that⊗ is unital, commutative and associative in the appropriate sense, and there are natural isomorphisms for all V 1 , V 2 , W ∈ CBorn. Likewise, Born is closed symmetric monoidal with its uncompleted tensor product ⊗. Remark A.6. Complete bornological vector spaces are directly related to the category Ind(Ban) of inductive systems of Banach spaces. The assignment of the directed system {V B } B∈Sc(V ) to V ∈ CBorn defines the dissection functor diss : Born → Ind(Ban). There are also a versions of dissection for non-complete and non-separated spaces.
In sheaf theory, filtered colimits and their commutation properties with certain kinds of limits play an important role. Thus, we analyze now such colimits. Recall that the underlying set of a filtered colimit of vector spaces (and hence also of bornological vector spaces) is the filtered colimit of the underlying diagram of sets.
Lemma A.7. Let V = colim i∈I V i be a filtered colimit in Born, and denote by j i : V i → V the standard map. Then a subset of V is a bounded disk if and only if it is of the form j i (B) for some i ∈ I and bounded disk B ⊂ V i .
Proof . Expressing V in terms of coproducts and a coequalizer, we find a map p : such that bounded subsets of V are precisely those of the form p(B) with B bounded. A bounded disk B in the direct sum is given by the convex hull of a finite collection B i 1 , . . . , B in of bounded disks B i j ⊂ V i j . Pick an upper bound i for the collection {i 1 , . . . , i n } ⊂ I, and let B i ⊂ V i be the smallest bounded disk containing the images in V i of each B i j ⊂ V i j . Then clearly p(B) = p(B i ), which finishes the proof.
Proposition A.8. Small filtered colimits commute with finite limits in Born.
Proof . Let I be a finite indexing category, and J filtered. Fix a diagram V : I × J → Born. We want to show that the natural bounded linear map φ : colim is an isomorphism. Since filtered colimits of vector spaces commute with finite limits, we know that φ is a linear isomorphism. It remains to show φ that has a bounded inverse, that is, every bounded disk B in the codomain is contained in the image of a bounded disk.
By Lemma A.7, each of the bounded disks B i = p i (B) ⊂ colim j∈J V (i, j) is the image, through the standard map ι i,k i : V (i, k i ) → colim j∈J V (i, j), of a bounded disk B i,k i ⊂ V (i, k i ), for some k i ∈ J. Since I is finite, we can pick an upper bound k for the collection {k i } i∈I , and we can write B i = j k (B i,k ) for suitable bounded disks B i,k ⊂ V (i, k). Now, lim i∈I V (i, k) is a subspace of the direct product i∈I V (i, k), which, again by finiteness, is also a direct sum. The bounded disks B i,k ⊂ V (i, k) span a bounded disk in that direct sum, and by restriction, a bounded disk B k ⊂ lim i∈I V (i, k). This in turns determines a bounded disk B = ι k (B k ) in the domain of φ, and φ(B ) = B.
In general, a filtered colimit of separated, or even complete, bornological spaces need not be separated, as the following example shows.
Example A.9. Let X be a manifold and x ∈ X. Consider the space of germs of smooth functions around x, Endowing C ∞ (U ) with its usual von Neumann bornology and taking the colimit in Born gives C ∞ X,x a bornology. Then the germ of a function f ∈ C ∞ (X) which vanishes to infinite order at 0 is in the closure of {0} in C ∞ X,x . In fact, we can find a sequence of functions f n ∈ C ∞ (X) with trivial germ at 0 converging to f in the Fréchet sense, so (equivalently) in the bornological sense. Therefore, the same holds for their images in C ∞ X,x .
On the other hand, filtered colimits V = colim V i of separated or complete bornological spaces remain separated or complete for diagrams with only injective morphisms or, more generally, if every bounded B i ⊂ V i with j i (B i ) = 0 ⊂ V is already in the kernel of some morphism V i → V j of the diagram. Such inductive systems are called stable by Houzel [5].
Proposition A.10. Let {V i } i∈I be a stable filtered diagram of separated or complete bornological spaces. Then V = colim i∈I V i is separated or complete, respectively.
Proof . Let B ⊂ V be a bounded disk. Then there exists i ∈ I and a bounded disk B i ⊂ V i such that j i (B i ) = B. Now, B i ∩ Ker j i is a bounded disk with trivial image in V , so it has trivial image already in V k for some k ∈ I. Let B k be the image of B i in V k , so that B = j k (B k ). Then (V k ) B k → V B is injective, so V B is a normed space.
Proposition A.11. Let V, W ∈ CBorn and assume V has the fine bornology. Then V ⊗ W is already complete. If both V and W have the fine bornology, the same is true of their tensor product.
Proof . The first assertion is clear if V is finite-dimensional. Otherwise, we have, by Proposition A.5, V ∼ = colim V α , where V α ranges through the collection of finite-dimensional subspaces of V , endowed with their fine bornologies and partially ordered by inclusion. Since, like any left adjoint, tensor product commutes with colimits, we have This is a stable filtered colimit of complete bornological spaces, hence complete. If also W is fine, then where V α ⊂ V and W β ⊂ W range through all finite-dimensional subspaces. This is a cofinal subcollection of the finite-dimensional subspaces of V ⊗ W , and therefore induces the fine bornology.

A.2 Bornological algebras and modules
It is straightforward to define the notion of bounded bilinear maps V 1 × V 2 → W between complete bornological vector spaces, and a bornology on the space Hom (2) (V 1 , V 2 ; W ) of such. Moreover, there is a natural isomorphism A bornological algebra is a bornological vector space A with a bounded, associative bilinear product A × A → A. A bornological module is a bornological vector space M with a bounded, associative, bilinear action map A × M → M . Any of those is called complete if the underlying bornology is complete.
In this paper we only consider unital, commutative bornological algebras and unital modules, so we drop the extra adjectives. Moreover, we focus on the complete case. Thus we get a category CAlg of complete bornological algebras and, for each A ∈ CAlg, a category CMod(A) of complete bornological modules. Module categories are enriched in themselves, that is, for M, N ∈ CMod(A), the space of A-linear maps Hom A (M, N ), with its subspace bornology, is naturally a complete A-module.
By Proposition A.11, the functor Fin : Vect → CBorn sends algebras and modules over them to complete bornological algebras and modules. Similarly, a Fréchet algebra A induces a complete bornological algebra structure on vN(A), and a Fréchet A-module M gives a complete vN(A)-module structure on vN(M ) [8,Theorem 1.29].
Given modules M and N over the bornological algebra A, we set where the map is specified by (m, a, n) → am ⊗ n − m ⊗ an. The usual adjunction between extension and restriction of scalars carries over to the bornological setting. Below, we use j to regard B and N as A-modules. for all a ∈ A, b, b ∈ B and m ∈ M . With this observation, the usual algebraic manipulations give a linear bijection between the hom spaces in question, and it is easy to check that everything is compatible with the bornologies. Letting j = id : A → A, we see that A⊗ A M ∼ = M . From this, and the fact that tensor products over A and B are associative, one deduces that B⊗ A -is monoidal.

A.3 Smooth functions
Given a smooth manifold S, we denote by C ∞ vN (S) the space of smooth functions with the von Neumann bornology associated to its usual Fréchet topology. This can be described as the bornology of uniform boundedness of all derivatives of each given order on compacts. If V is a complete bornological space and f : S → V a function, we say that f is smooth if there exists a complete bounded disk B ⊂ V such that f takes values in V B , and f : S → V B is smooth as a function with values in a Banach space.
Proposition A.13. For any smooth manifold S and complete bornological vector space V , there is a natural isomorphism of vector spaces This defines a bornology on the left-hand side, which we denote C ∞ vN (S; V ).
Proof . Using the dissection isomorphism V = colim B V B , where B ranges over all complete bounded disks in V (Proposition A.5), we get On the other hand, we have C ∞ (S; V ) = colim B C ∞ (S, V B ) by definition. This reduces the proposition to the case V = vN(V ν ) for a Banach space V ν . The rest of the argument is functional analysis. We have C ∞ (S; vN(V ν )) = vN(C ∞ (S; V ν )) by [7,Corollary 3.9]. It is well known [14,Theorem 44.1] that C ∞ (S; V ν ) ∼ = C ∞ (S)⊗ π V ν , where we used the completed projective tensor product. Finally, vN(V )⊗ vN(W ) ∼ = vN(V⊗ π W ) if V is nuclear [8, Theorem 1.91], which finishes the proof, since C ∞ (S) is nuclear for any manifold S (see, e.g., the corollary of Theorem 51.5 in [14]).
Proposition A.14. For any smooth manifolds S and T , we have Proof . This follows from the corresponding statement in the Fréchet setting, using the completed projective tensor product [14,Theorem 51.6], and the fact that vN(V )⊗ vN(W ) ∼ = vN(V⊗ π W ) if V is nuclear.

A.4 Bornological sheaves
In this section, we outline the theory of sheaves of bornological vector spaces. We follow Houzel [5,Section 2]. Let C be a category with limits, and denote by PSh C (X) and Sh C (X) the categories of presheaves and sheaves with values in C on a topological space (or, more generally, a Grothendieck site) X. We are interested in the cases C = CBorn and, as a preliminary step, C = Born. We will say that a (pre)sheaf with values in Born is complete if it takes values in CBorn.
The first item in our wish list is to have a left adjoint to the inclusion Sh C (X) → PSh C (X), which we call the sheafification functor. Let F ∈ PSh C (X), U ⊂ X open, and U = {U i ⊂ U } i∈I be an open cover. As usual, we write and say that F is a sheaf if the canonical map F (U ) → F (U) an isomorphism for all U. We will call F a semisheaf if the maps F (U ) → F (U) are always monomorphisms. Since the inclusion CBorn ⊂ Born preserves limits, the condition of being a (semi)sheaf in one of these categories is equivalent to being a (semi)sheaf in the other. Set where U runs through all open covers of X. For the traditional choices C = Set, Vect, etc., F + is a semisheaf, and is a sheaf if F is already a semisheaf. This defines the sheafification functor F → F ++ . When C = Born, F ++ is obviously a sheaf when regarded as taking values in Vect, and it can be checked directly that it is in fact a bornological sheaf.
If F ∈ PSh Born (X) is a complete semisheaf, then the inductive system defining F + is stable, so it follows, from Proposition A.10, that F + is a complete sheaf. However, if F fails to be a semisheaf, then F + may not be complete even if F is. This issue is resolved in Houzel [5, p. 32] as follows. Let F ∈ PSh Born be arbitrary and let E be the smallest sub-presheaf of U → F (U ) such that F : U → F (U )/E(U ) is a complete semisheaf. Then (F ) + is a complete sheaf, and this construction gives a left adjoint to the inclusion Sh CBorn (X) → PSh Born (X). By restriction, this left adjoint gives us both a sheafification functor PSh CBorn (X) → Sh CBorn and a completion functor Sh Born → Sh CBorn .
The second item in our wish list is to get tensor products and internal homs. If E, F ∈ PSh Born (X), then Hom(E, F ) gets a bornology as a subspace of the product U ⊂X Hom(E(U ), F (U )), U ⊂ X open, and we get also a Born-valued presheaf Hom(E, F ). It is a sheaf if so is F , and it is complete if so is F . We define E ⊗ F as the sheafification of the presheaf U → E(U ) ⊗ F (U ), and E⊗ F as the associated complete sheaf.
The following statement summarizes this discussion.

A.5 The stack V vN
The assignment U → C ∞ vN (U ) defines a sheaf of complete bornological algebras on the site Man of smooth manifolds. Its restriction to the small site of a manifold S will be denoted C ∞ vN,S . We denote by V vN (S) = CMod(C ∞ vN,S ) the category of sheaves of complete C ∞ vN,S -modules. As observed in the previous subsection, this is a closed symmetric monoidal category.
Any smooth map f : T → S induces a homomorphism C ∞ vN,S → f * C ∞ vN,T of sheaves of complete bornological algebras. We denote by f : f −1 (C ∞ vN,S ) → C ∞ vN,T its adjoint. This makes C ∞ Finally, we need to show that V vN is in fact a stack. This is the case because the pretopology of all open covers of a manifold S and the pretopology of open covers subordinated to a given open cover generate the same Grothendieck site. More concretely, if U = {U i ⊂ S} is an open cover and V i ∈ V vN (U i ) is a collection of objects with coherent isomorphisms between their restrictions to overlaps U i ∩ U j , we can construct a presheaf of C ∞ vN,S -modules V which agrees with V i on U i and assigns 0 to any open not contained in some U i . Sheafification then gives the desired "glued together" object V ∈ V vN (S).