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Extended field theories are local and have classifying spaces

We show that all extended functorial field theories, both topological and nontopological, are local. We define the smooth (infinity,d)-category of bordisms with geometric data, such as Riemannian metrics or geometric string structures, and prove that it satisfies codescent with respect to the target S, which implies the locality property. We apply this result to construct a classifying space for concordance classes of functorial field theories with geometric data, solving a conjecture of Stolz and Teichner about the existence of such a space. We use our classifying space construction to develop a geometric theory of power operations, following the recent work of Barthel, Berwick-Evans, and Stapleton.


Introduction
A key feature of functorial field theory (FFT) that is usually expected, or demanded, is that it is local in the sense that global phenomena are entirely determined by local phenomena. All influences should propagate through space (or spacetime) at some finite speed and there is no 'action at a distance'. This is mostly relevant for classical field theories, but there is still nontrivial locality even for quantum field theories (where one has nontrivial moduli for bordisms).
Although the axiomatization of field theories via functorial field theories (see the original paper by Segal [2004.a] or Atiyah's paper [1988.b] for the topological case) provides a beautiful framework that captures many necessary ingredients, it has the unfortunate side effect of allowing field theories that are not local. This is almost evident from the axioms, for suppose we are given a bordism Σ (i.e., a d-dimensional smooth manifold with boundary) between two (d − 1)-manifolds Σ 0 and Σ 1 and suppose we have a smooth map f : Σ → X to some target manifold X. To such data, a functorial field theory associates a linear map between Hilbert spaces Z (Σ,f ) : Now in this setting 'local' means local with respect to the target manifold X. In applications, Σ should represent the 'worldline' of a particle, or 'worldvolume' of a string and X is the ambient space in which it propagates. If this field theory were local, we should be able to restrict it to some small region U ⊂ X. Moreover, we should be able to reconstruct all of the field theory from such restrictions, to elements of an open cover of X. Here we have a problem. If the bounding manifolds Σ 0 and Σ 1 have images in X (under f ) that cover a large region in X, we have no hope for being able to assemble the field theory from its restriction to a cover {U α } of X. Indeed, there is no reason to expect that the maps f i : Σ i → X, i = 0, 1, factor through some U α and we are unable to cut Σ i into smaller pieces so that it factors. Fortunately, all hope is not lost. This defect of functorial field theories can be resolved by allowing ourselves to further cut Σ i into smaller pieces by codimension 1 submanifolds. This naturally leads to the fully-extended setting of Freed [1992], Baez-Dolan [1995], and Lurie [2009.a], where we have a d-category of bordisms, starting with points as objects, 1-manifolds as morphisms, 2-manifolds as morphisms between morphisms etc. It has long been expected that fully-extended field theories ought to be local, but no actual proof has emerged in the literature. One part of the issue for this is that a rigorous definition (in the topological case) of the fully-extended bordism category has not appeared until Lurie [2009.a] and Calaque-Scheimbauer [2015]. Moreover, to talk about the nontopological case, one really needs to parameterize the bordism category over smooth manifolds, and this adds another layer of complexity that has not yet been explored in the literature in the fully extended case (see Teichner [2004.b, 2011.a] for the nonextended case).
The main goal of this paper is to provide a rigorous proof that all fully-extended field theories are local. To incorporate a variety of geometric structures on smooth manifolds, such as Riemannian metrics, orientations, spin and string structures, as well as more complicated higher geometric structures such as geometric string structures, we generalize the usual definition of tangential structures, i.e., the space of lifts where τ M denotes the tangent bundle and Y is a space encoding the tangential structure. Our motivation is that, in the geometric setting, one is interested in a variety of structures that do not fit into the above formalism. For example, we can consider Riemannian metrics with constraints on the sectional curvature (e.g., positive, negative, nonpositive, nonnegative). We adopt an alternative perspective. Tangential structures can be pulled back along open embeddings. They are also local in that structures can be glued from local data. Thus, a tangential structure naturally gives rise to an ∞-sheaf on the site of d-dimensional manifolds, with open embeddings between them, and we might as well consider any ∞-sheaf on this site as a legitimate geometric structure. This perspective is explained in detail in Section 3. The usual notion of a target in physics, for example, in the sigma model, can be encoded as a geometric structure of this type by taking S(M ) = C ∞ (M, X), where X is the target manifold.
For a geometric structure S, a T -valued, fully-extended field theory F with S-structure is a symmetric monoidal functor F : Bord S d → T , where T is some smooth symmetric monoidal (∞, d)-category. To say that F is local roughly means that for any open cover of the target S we can reconstruct F from its restriction to bordisms M → S that factor through some element of the open cover.
Thus, a natural formulation of locality involves descent on a general smooth stack S ∈ PSh ∆ (FEmb d ), as defined in Definition 3.1.2. This leads us to the main result of the paper. Roughly, the strategy for our proof is to decompose bordisms into pieces whose map to the target S factors through some element of the open cover. Decomposing bordisms into such pieces turns out to be quite delicate. Furthermore, the entire space of such decompositions must be contractible (see Proposition 6.6.6), which is the most difficult part of the proof.
The results of Berwick-Evans, Boavida de Brito, and the second author [2019], see also [2022] can be used to construct a classifying space for field theories. More precisely, we can construct (rather explicitly) a space B ∫ FFT d,T,S such that the following holds. where on the right we take homotopy classes of maps.
This resolves a long-standing conjecture of Stolz and Teichner [2011.a] that postulates the existence of such a classifying space for extended functorial field theories with any geometry. For a more general version, which allows for twisted structures on X, see Theorem 7.2.9. We also use this classifying space construction to provide a geometric model for power operations, building on the work of Barthel, Berwick-Evans, and Stapleton [2020.b].
In Section 2, we introduce the notion of a smooth, symmetric monoidal (∞, d)-category. In Section 3, we define geometric structures. In Section 4, we introduce two smooth variants of the bordism category, analogous to the distinction between bicategories and double categories. In Section 5, we develop necessary tools from homotopy theory that will be used in the proof of Theorem 1.0.1. In Section 6, we prove that the various bordism categories we have introduced satisfy codescent, which is the main theorem of the paper. In Section 7 we establish several applications. After reviewing background in Section 7.1, we construct an explicit classifying space of field theories (in the sense of Theorem 1.0.2) in Section 7.2. In Section 7.3, we construct a geometric model for power operations.

Smooth symmetric monoidal (∞, n)-categories
In this section we develop a model for smooth symmetric monoidal (∞, d)-categories. The encoding of symmetric monoidal structures follows Segal [1974]. The encoding of categorical composition follows the same idea of Segal, as developed by Rezk [1998.b] (in the (∞, 1)-categorical case) and Barwick [2005] (in the (∞, n)-categorical case). The encoding of smoothness follows Stolz-Teichner [2004.b, 2011. The use of the cartesian site in this context was proposed by Urs Schreiber (see, e.g., Fiorenza-Schreiber-Stasheff [2010]).

Homotopy theory and higher categories
Recall the simplex category ∆, whose objects are nonempty, totally ordered sets [n] = {0, . . . , n}, and whose morphisms are order-preserving maps. Recall also Segal's category Γ [1974] (see also Bousfield-Friedlander [1978] and Schwede [1999.b]), whose opposite category has objects which are finite pointed sets ℓ = { * , 1, . . . , ℓ} and basepoint-preserving functions as morphisms. Finally, consider the category Cart whose objects are cartesian spaces R n and morphisms are smooth functions between them. The basic building blocks for symmetric monoidal, smooth (∞, n)-categories lie in the threefold product category ∆ ×n × Γ × Cart.
One should think of each category as capturing an independent nature of the objects we are constructing. We provide some indications of how to think of each piece conceptually.
The category ∆ ×n : Here an object is a multisimplex k = ([k 1 ], [k 2 ], . . . , [k n ]) ∈ ∆ ×n , which we can think of as indexing a composable chain of morphisms, with k i composed morphisms in the ith direction. We will generally think of such composable chains as a grid. For example, for n = 2, we have two composition directions. If k 1 = k 2 = 4, this gives a grid because any ℓ-tuple can be deformed to an ℓ-tuple that can be multiplied. The elegance of Segal's method is that it circumvents the need for explicitly keeping track of coherent homotopies in the symmetric monoidal structure, although these are of course still hidden in the above equivalence. For more information, see Segal [1974] The category Cart: This category captures the smooth structure. An object of Cart is an open subset of R n that is diffeomorphic to R n and morphisms are smooth maps. This category is turned into a site by equipping it with the coverage of good open covers, i.e., open covers for which every finite intersection is either empty or diffeomorphic to R n . Objects in the functor category Fun(Cart op ; D) can be thought of as the category of smoothly parametrized objects of D over cartesian spaces. As a basic example, one can consider a smooth manifold M as a functor M : Cart op → Set, by assigning to each cartesian space U the set of smooth maps C ∞ (U, M ). For us, this category will allow us to form smooth families of symmetric monoidal (∞, n)-categories. For more information about sheaves on this site, see Fiorenza-Schreiber-Stasheff [2010] or Sati-Schreiber [2020.c].
Notation 2.1.2. Throughout the remainder of the paper, we will denote simplicial presheaves on some category C as PSh ∆ (C), i.e., PSh ∆ (C) := Fun(C op , sSet). Ordinary presheaves with values in another category D will be denoted If C is equipped with a (symmetric) monoidal structure, then PSh ∆ (C) is equipped with a (symmetric) monoidal structure given by the Day convolution.
Throughout this paper, we will make use of internal homs and simplicial enrichments of presheaves on monoidal categories (like Cart or ∆ with its cartesian structure or Γ with its smash product structure). We introduce the following systematic notation.
Notation 2.1.3. Given simplicial presheaves X, Y ∈ PSh ∆ (R) on a (symmetric) monoidal category R, we denote by Map(X, Y ) ∈ sSet the simplicial enrichment and by Hom(X, Y ) ∈ PSh ∆ (R) the internal hom. We also use the notation Fun ⊗ for internal homs in the case R = Cart × Γ × ∆ ×d .

Smooth multiple (∞, n)-categories
The target category for our functors will be the category of simplicial sets sSet and we will present an (∞, 1)category of sheaves of symmetric monoidal (∞, n)-categories by a particular model structure (following Barwick [2005] in the case of ∆ ×n ). We begin by taking the injective model structure on PSh ∆ (∆ ×n × Γ × Cart). To encode the Segal conditions for ∆ and Γ, completeness conditions for ∆, and descent conditions for Cart on these objects, we perform a left Bousfield localization of this model category.
Notation 2.2.1. Fix some n ≥ 0, which corresponds to n in (∞, n)-categories. The functor is defined as the composition where the first functor is the Yoneda embedding, the second functor is the inclusion of presheaves of sets into simplicial presheaves, and the third functor is induced by the projection functor ∆ ×n × Γ × Cart → Γ. Analogously, we define functors where the latter uses the projection ∆ ×n × Γ × Cart → ∆ onto the kth copy of ∆, where 1 ≤ k ≤ n.
Definition 2.2.2. Fix n ≥ 0. We define the following maps, with respect to which we will perform a left Bousfield localization.
Local objects are precisely Segal's special ∆-objects X, defined by the condition that the map X a+b → X a × X0 X b is a weak equivalence.
(ii) (Completeness condition.) For 1 ≤ k ≤ n, the map where E is obtained by evaluating the composition of functors on the nerve of the groupoid with two objects x and y and two nonidentity morphisms x → y and y → x, where the first functor converts presheaves of sets into presheaves of discrete simplicial sets and the second functor is induced by the projection ∆ n × Γ × Cart → ∆ onto the kth copy of ∆.
Local objects with respect to (i) and (ii) are Rezk's complete Segal objects, defined by Segal's special ∆-condition and the completeness condition, which says that the map X 0 → X 1 that sends objects to their identity morphisms is a weak equivalence onto the subobject of invertible 1-morphisms.
(iv) (Sheaf condition.) For {U α } α∈I a good open cover of V in Cart, the Čech nerve c U ∈ PSh ∆ (Cart) has as its m-simplices the presheaf of sets α0,...,αm: The kth face map deletes U α k , the kth degeneracy map duplicates U α k . We have a canonical map that in degree m is induced by the inclusions U α0 ∩ · · · ∩ U αm → V . The local objects are ∞-sheaves (alias ∞-stacks), for which the restriction map is a weak equivalence, where we keep all arguments but Cart fixed and arbitrary.
Remark 2.2.8. Although it may seem as if the nature of the two maps in (iii) are different, both of them arise from inverting maps of the form Taking m = 0 produces the map τ . Taking m = 2 produces a map weakly equivalent to the map t s,ℓ (note that j Γ 0 in the pushout was already forced to be contractible by the map τ ). In fact, it is clear that we can alternatively localize at maps of the above form, which gives a somewhat more uniform description.
The above conditions should be satisfied for an arbitrary fixed choice of all objects in factors other than the one under consideration. For instance, fixing an arbitrary object in Γ × ∆ ×n should yield an ∞-sheaf of simplicial sets on Cart. This idea can be precisely expressed using the following definition.
Definition 2.2.9 (The multiple model structure). We define a model structure C ∞ Cat ⊗,uple ∞,n := PSh ∆ (∆ ×n × Γ × Cart) uple by performing the ordinary left Bousfield localization of the model category PSh ∆ (∆ ×n × Γ × Cart) inj at the morphisms given by taking the product of one of the maps (i), (ii), (iii), or (iv) of Definition 2.2.2 with the identity map of an object in the image of the composition where R denotes the product ∆ ×n × Γ × Cart with the corresponding factor omitted. For example, one such local morphism for n = 1 is Locality with respect to such a map forces the simplicial object obtained by fixing the factors corresponding to Γ and Cart to be a special Segal object.
The existence of this localization follows from the Smith recognition theorem (see Barwick [2007, Theorem 4.7]).

Smooth globular (∞, n)-categories
The previous section gives the correct smooth variant of n-fold complete Segal spaces. However, these are not quite a model for (∞, n)-categories until we impose the globular condition of Barwick-Schommer-Pries [2011.b, Notation 12.1]. For n = 2 this amounts to passing from double categories to bicategories. Recall that double categories have two distinct notions of 1-morphisms: horizontal and vertical. Both can be composed, and 2-cells are squares involving two vertical and two horizontal morphisms. In other words, our model so far describes the n-fold analog of double categories. In order to get rid of the extra 1-morphisms, we further localize the functor category PSh ∆ (∆ ×n × Γ × Cart) uple at the morphisms given by taking the product of a map (v) with the identity map of an object in the image of the composition Definition 2.3.1. Fix n ≥ 0. We define the following maps, which we include in our left Bousfield localization for the globular model structure.
(v) (Globular maps.) For an object k = ([k 1 ], . . . , [k n ]) ∈ ∆ ×n , letk be the object with jth component given by There is a canonical map from k tok given by identities or unique maps to [0] for each index j. The globular maps are defined as Here j ∆ ×n is the composition The local objects are multisimplicial spaces X such that X 0 , which we interpret as an object in (n − 1)fold simplicial spaces, is constant and X k is a local object in (n − 1)-fold simplicial spaces. For n = 2, the locality condition boils down to forcing the maps X 0,b → X 0,0 to be equivalences. This makes all vertical morphisms identities.
Definition 2.3.3 (The globular model structure). We define the model structure in the same way as in Definition 2.2.9, but with the maps (2.3.2) thrown into the list, with R = Γ × Cart.
The existence of the localization is established in Barwick [2007, Theorem 4.7].
Remark 2.3.4. Our results apply to bordism categories which are fibrant objects in the multiple or globular model structure. We use the shorthand notation C ∞ Cat ⊗ ∞,n to refer to the model category C ∞ Cat ⊗,glob ∞,n or C ∞ Cat ⊗,uple ∞,n , depending on the context. We also write PSh ∆ (∆ ×n × Γ × Cart) local when referring to either model structure.
Definition 2.3.5. We have a globularization functor given by the homotopy right adjoint to the inclusion functor going in the opposite direction.
Remark 2.3.6. Concretely, Glob extracts those connected components of a given object in C ∞ Cat ⊗,uple ∞,n that satisfy the globularity condition, i.e., extracting any vertex with respect to the ith factor of ∆ produces a point in the connected component in the image of the simplicial degeneracy map with respect to the (i + 1)st, . . . , dth factors of ∆, applied to some connected component over the multisimplex 0 ∈ ∆ ×d−i+1 .

Functor categories
Definition 2.4.1. We equip the category Γ with the symmetric monoidal structure given by the smash product of pointed finite sets. That is, if A and B are pointed finite sets, we set after which we set κ ∧ λ := κλ for any integers κ, λ ≥ 0, with the formula for A∧B yielding functoriality in both arguments. We equip PSh ∆ (Γ) with the induced Day convolution monoidal product. If R = Cart× ∆ ×d (or a factor of this product), we equip PSh ∆ (Γ × R) with the Day convolution monoidal product, where the monoidal structure on Γ × R is given by the smash product on Γ and cartesian product on R.
Definition 2.4.2. For X, Y ∈ C ∞ Cat ⊗,uple ∞,n , the internal ∞-hom Fun ⊗ (X, Y ) can be computed by deriving the internal hom functor, which is a right Quillen functor because C ∞ Cat ⊗,uple ∞,n is a monoidal model category. For X, Y ∈ C ∞ Cat ⊗,glob ∞,n , the internal ∞-hom Fun ⊗ (X, Y ) can be computed by deriving the internal hom functor in C ∞ Cat ⊗,uple ∞,n and then applying the functor Glob of Definition 2.3.5.
Remark 2.4.3. One can easily see that including the globular condition yields a model structure that does not satisfy the pushout product axiom, so internal ∞-homs in the underlying ∞-category can no longer be computed by deriving the internal hom, which is why we had to apply the functor Glob.

Geometric structures
We now describe how to encode geometric structures on bordisms. As pointed out in the introduction, our treatment generalizes the traditional treatment of tangential structures and is more general. Roughly, the passage from the traditional approach to our approach is given by taking the sheaf of sections of the homotopy pullback where τ : M → BGL(d) is a classifying map for the tangent bundle. There are two subtleties which need to be accounted for. First, we need to encode geometric structures. Hence, we need to replace the classifying space BGL(d) with the classifying stack of vector bundles or rank d. Second, the sheaf must live on a larger site than the open subsets of M , since we will need to pullback the structure along all open embeddings. Moreover, we will need all geometric structures to vary in families, parametrized over cartesian spaces. We resolve these issues by working with sheaves on the site of submersions (with d-dimensional fibers), with fiberwise open embeddings between them.
Remark 3.0.1. To make all sites small and to make presheaves valued in sets, rather than proper classes, we require that the underlying set of any manifold is a subset of R. We do not require any compatibility with the smooth structure on R.

Sheaves on FEmb d
Our definition resembles that of Ayala-Francis [2012.a], except that we do everything in smooth families.  Our definition of geometric structures is extremely versatile and captures all significant geometric structures we can think of, including metrics, topological structures, tangential structures, etc. We begin with some basic examples that are non-tangential in nature.
Example 3.1.5 (manifolds). Let X be a smooth manifold of any dimension. We can regard X as a geometric structure via the sheaf which assigns to all submersion M → U . This is clearly a sheaf on FEmb d since the total space functor FEmb d → Man, which maps M → U to M , sends covering families to covering families. This sheaf is not representable.
Example 3.1.6. The previous example can be generalized easily to all simplicial presheaves on the site of smooth manifolds. Indeed, the total space functor T : FEmb d → Man induces a restriction which manifestly preserves the homotopy descent property.
Example 3.1.7. An example of a geometric structure that does not come from a simplicial presheaf on smooth manifolds and smooth maps is given by the presheaf of Riemannian metrics. Riemannian metrics restrict along immersions (in particular, étale maps), but not along arbitrary smooth maps. This presheaf allows us to encode Riemannian metrics as nontangential structures. We can also consider Riemannian metrics with restrictions on sectional curvature (e.g., positive, negative, nonpositive, nonnegative), since these properties are preserved by pullbacks along étale maps.

Tangential structures
We explain how to treat traditional tangential structures in this setting (see, e.g., Lurie [2009.a]). Since we have chosen to work with model categories, we will need a strict presentation of the fiberwise tangent bundle functor on FEmb d . We start with some preparatory definitions and lemmas. The following definition describes the stack of vector bundles on the site FEmb d .
The remaining structure of a pseudofunctor is given by the natural isomorphisms and The definition of Vect(d) as a pseudofunctor is necessary since pullbacks (implemented as tensor products here) do not compose strictly. Since we will need a strict simplicial presheaf in order to describe tangential structures, we strictify this pseudofunctor as follows.
denotes the inclusion of strict functors into pseudofunctors, denoted by ΨPSh. The left adjoint F formally adds pullbacks. We define the strict presheaf of groupoids BGL(d) := F Vect(d). We have a canonical pseudonatural transformation given by the unit of the adjunction.
We now define the category of vector bundles over base spaces that are objects in FEmb d . More precisely, the vector bundles live over the total space of the object.
where f is a morphism in FEmb d and f * = BGL(d)(f ).
We now define the fiberwise tangent bundle functor on FEmb d .
The following construction takes as an input a vector bundle over a stack Y , specified by a map Y → BGL(d), and sends it to the sheaf, on the site FEmb d , whose value on M → U is the simplicial set of lifts of the fiberwise tangent bundle map M → BGL(d) through the map Y → BGL(d). The simplicial set of lifts is modeled by the simplicial set of sections of the base change of Y → BGL(d) along M → BGL(d).
Definition 3.2.5. Given an integer d ≥ 0, we define a functor This amounts to pulling back the bundle Y → BGL(d) to M along the fiberwise tangent bundle map M → BGL(d) and taking the simplicial set of sections.
This construction defines a (strict) presheaf, since the resulting simplicial set of sections can be expressed as the simplicial subset of Y (M → U ) comprising those n-simplices whose image in BGL(d)(M → U ) n is the degeneration of the given element in In other words, a morphism M → M ′ in FEmb d is sent to the simplicial map This defines a (strict) functor because τ is a strict functor.
Proposition 3.2.6. The functor τ * is a right Quillen functor, where both sides are equipped with the local injective model structure.
Proof. (Acyclic) cofibrations in the slice model structure are created by the forgetful functor

2.7)
and (acyclic) cofibrations in the injective model structure are defined objectwise.
By construction, the functor τ * preserves small limits and filtered colimits. The domain and codomain of τ * are locally presentable categories, so τ * is a right adjoint functor.
Its left adjoint functor is the unique cocontinuous functor where the first map projects ∆ n → ∆ 0 and the second map is the tangent bundle map τ (M → U ). The composition of the left adjoint with the forgetful functor (3.2.7) is identity. Therefore, the left adjoint functor preserves injective (acyclic) cofibrations, so it is a left Quillen functor for the injective model structure and its right adjoint τ * is a right Quillen functor. Čech covers are sent to Čech covers, equipped with a map to BGL(d). Hence, this Quillen adjunction descends to the local injective model structure.
Definition 3.2.8. A tangential structure is a map Y → BGL(d). Its underlying geometric structure, in the sense of Definition 3.1.2, is given by applying the right derived functor of τ * .

Examples of geometric structures
We now discuss some examples of geometric structures.
Example 3.3.1 (families of differential string structures). Let BSpin(d) ∇ denote the simplicial presheaf on FEmb d which is the homotopy sheafification of the simplicial presheaf (d)) denotes fiberwise Lie-algebra valued 1-forms. This simplicial presheaf assigns to a smooth manifold M the nerve of the action groupoid given by the action of C ∞ (M, Spin(d)) on Ω 1 U (M ; so(d)) by gauge transformations A → g −1 Ag + g −1 dg (where gAg −1 denotes the adjoint action and g −1 dg denotes the Maurer-Cartan form). This sheaf classifies smooth principal Spin(d)-bundles with connection, which live on the total space M . Then we have a morphism of simplicial presheaves which forgets the connection (projects out forms) and maps via the composition where the first map is the double cover and the second is the inclusion. Now consider the simplicial presheaf arising as the pullback in PSh ∆ (Man): The maps in the diagram are defined as follows. The map a: Ω 3 → B 3 U(1) ∇ is the canonical map that includes globally defined 3-connections into circle 3-bundles with connections. It can be obtained (for example) via the morphism of sheaves of positively graded chain complexes via the Dold-Kan correspondence. All forms appearing in the diagram are fiberwise differential forms with respect to the projection M × U → U . The maps e 0 , e 1 : B 3 U(1) → BU(1) ∇ are evaluations at the endpoints. The map 1 2 p 1 is the fractional differential refinement of the first Pontrjagin class, which can be obtained via Chern-Weil theory (see, e.g., Bunke [2012.b]) from the invariant polynomial p 1 in the expansion where x ∈ so(n). The map BString(d) ∇ → BGL(d) exhibits a geometric structure. Let us unwind the space of differential string structures. We fix a fiberwise Riemannian metric and a fiberwise orientation. Since BString(d) ∇ → BSO(d) is a projective fibration, the (derived) space of sections is a Kan complex (this can also be computed as the fiber of the map Using the universal property of the pullback describing BString(d) ∇ , a vertex in this Kan complex is uniquely determined by the following data: S1 A fiberwise spin structure on the fiberwise oriented frame bundle and a spin connection ∇ on the resulting spin principal bundle.
S2 A fiberwise differential 3-form ω corresponding to a connection 3-form on the trivial circle 3-bundle.
S3 A fiberwise isomorphism of circle 3-bundles with connection λ: 1 2 p 1 (∇) ∼ = a(ω). The sheaf of sections sending thus has vertices which are precisely fiberwise differential string structures (including a choice of a fiberwise oriented Riemannian metric) in the sense of Fiorenza-Schreiber-Stasheff [2010] (an equivalent definition was first given by Waldorf [2009.b]). The superscript h denotes the homotopy pullback presented by the path space construction like the one used to define BString(d) ∇ .
Example 3.3.3. The geometric spin and spin c structures are encoded in exactly the same manner as differential string structures, but with the homotopy pullback diagrams Here I: BU(1) ∇ → B 2 Z is the canonical map which is part of the differential refinement (see Bunke [2012.b], Fiorenza-Sati-Schreiber [2013.a], Schreiber [2017.b]) and ρ 2 is the mod 2 reduction. These homotopy pullbacks are again presented by the path space construction.
In the same manner (by modifying the obstruction class), we can also consider higher categorical structures such as fivebrane or differential fivebrane structures of Sati-Schreiber-Stasheff [2008 and ninebrane structures of Sati [2014].
Example 3.3.4. The rigid geometries of Stolz-Teichner [2011.a] (see also Ludewig-Stoffel [2020.a]) provide an example of geometric structures. Let G be a Lie group acting on a d-dimensional manifold M. Recall that a (G, M)-structure on a manifold Y is a maximal atlas {(U α , φ α )} consisting of charts that are diffeomorphisms onto open subsets of M, and a collection of constant transition functions g αβ ∈ G such that the diagram Such rigid geometries can be encoded in our language as follows. We can view M as a sheaf on FEmb d by assigning a submersion M → U to the set of fiberwise open embeddings M → M. Let G δ denote the constant sheaf associated to the G. Taking a Čech cofibrant replacement by a maximal atlas, one sees that a (G, M)-structure on Y is precisely a (projectively) derived map where Y is the representable presheaf Y → * and M//G δ is the homotopy quotient by the G δ action. More generally, for a submersion Y → U , a morphism (Y → U ) → M//G δ encodes fiberwise rigid geometries on Y over U .
Remark 3.3.5. The following table provides a list of a few examples of geometric structures which can be encoded using our machinery. In the table, the functor τ * is derived, see Definition 3.2.5. We regard simplicial sets as ∞-sheaves on FEmb d via the constant sheaf functor, which we leave implicit.
Stack Description

Smooth bordism categories
In this section, we give a precise definition of the smooth bordism category as a symmetric monoidal smooth (∞, d)-category.

The smooth d-uple bordism category
In this section we define the d-uple and d-globular bordism categories, in the terminology of Calaque-Scheimbauer [2015]. The d-uple bordism category is not local with respect to the globular morphisms (2.3.2) and hence there are distinguished composition directions, which need to be accounted for, just as a double category has distinguished composition directions that are put on equal footing. The formalization of smoothness for bordism categories is due to Stolz-Teichner [2004.b, 2011.
There is a functor Cut: (FEmb d ) op → Set that associates to an object p: M → U its set of cuts, and to a morphism the induced map of sets that takes preimages of cuts. We equip the set of cuts with a natural ordering There is a functor CutTuple: The functor associates to a morphism the induced map of sets that takes preimages of the cuts and reindexes them according to the map of simplices. Thus, a face map removes a cut and a degeneracy map duplicates a cut.
We remark that the above definition implies that as well as the analogous chains for C ≤ and C ≥ .
Definition 4.1.3 (d-uple bordisms). Given d ≥ 0, we specify an object Bord d,uple in the category as follows. For an object (m, ℓ , U ) ∈ ∆ ×d × Γ × Cart, the simplicial set Bord d,uple (m, ℓ , U ) is the nerve of the following groupoid, which is small by Remark 3.0.1.
Objects: An object of the groupoid is a bordism given by the following data.
3. A choice of map P : M × U → ℓ , which gives a partition of the set of connected components of M × U into ℓ disjoint subsets and another subset corresponding to the basepoint (the "trash bin"), which satisfy the transversality property: ⋔. For every subset S ⊂ {1, . . . , d} and for any j: where π i : R S → R is the ith projection, yields the j i -th cut C i ji in the cut tuple C i , as in Definition 4.1.1. We require that the fiberwise-regular values of (h S , p) form an open neighborhood of {0}×U ⊂ R S ×U .
To simplify notation, we define and we require it to be compact, for all choices of j, j ′ . We will omit P in the notation when it is clear from the context. Morphisms: A morphism of the groupoid is a cut-respecting map given by an equivalence class whose representatives are morphisms ψ: V →Ṽ in FEmb d that commute with the maps to ℓ , where V is an open subset of M × U that contains core([0, m], P ), and likewise forṼ . Moreover, ψ satisfies the property: That is, ψ restricts to a diffeomorphism on germs of all grid faces of all codimensions, throwing away the trash bin.
Two such maps ψ: V →Ṽ and ψ ′ : That is, we take germs of such maps around the grid.
Presheaf structure maps: The structure maps corresponding to morphisms in ∆, Γ, and Cart are given by nerves of functors of groupoids specified as follows. For ∆, the structure maps are induced by the maps in Definition 4.1.2. A face map removes the corresponding cut and the outer face maps shrink the core and restrict germs appropriately. A degeneracy map duplicates a cut. The properties defining ψ are preserved by these operations and hence the above data is natural with respect to maps in ∆. For Γ, a map ℓ → ℓ ′ is simply composed with the given map P : M × U → ℓ , possibly shrinking the core and restricting germs. For a given map U ′ → U in Cart, we pull back cut tuples and P via this map. Likewise for ψ.
Remark 4.1.5. We have chosen to work exclusively with trivial bundles M × U → U in Definition 4.1.3 in order to ensure functoriality of pullbacks of bundles. Since we are working over the cartesian site, all bundles are trivializable and our presheaf of (∞, d)-categories satisfies descent on cartesian spaces. Alternatively, we could include all bundles, but then we would need to work with Grothendieck fibrations instead of presheaves.
Remark 4.1.6. The multisimplicial presheaf, given by taking the nerve of the groupoid, is already local with respect to the Segal morphisms (2.2.3). Indeed, we can glue two bordisms together along a given diffeomorphism of tubular neighborhoods of the respective boundaries.

Bordisms with geometric structures
In the above definition, the bordisms are equipped with no additional structure. We would like to equip bordisms with geometric structure, or tangential structure, in the sense of Section 3. Recall that these are given simply by simplicial presheaves on the site FEmb d . Although it is relatively clear how to modify Definition 4.1.3 so that M is equipped with a geometric structure, we still need to say what happens when we pass to isomorphisms.
Let S be a simplicial presheaf on FEmb d . Consider a bordism M × U → U (Definition 4.1.3). We have a well-defined germ of geometric structures Figure 1: A bordism defined by a cut tuples C 1 and C 2 . Everything is parametrized by elements x ∈ U in Thus, the map ϕ * fiberwise pulls back geometric data on a germ of the core.
Example 4.2.3. Let X be a smooth manifold, viewed as a sheaf on FEmb d via For N = M × U → U , the corresponding geometric structure on bordisms has elements given by the germ of a smooth function f : M → X around the core. The pullback ϕ Definition 4.2.4 (d-uple bordisms with structure). Fix d ≥ 0 and S a simplicial presheaf on FEmb d . We specify an object Bord S d,uple in the category by taking the diagonal of the nerve of the following simplicial groupoid.
(GO) The simplicial set of objects is given by where the coproduct ranges over the objects of Definition 4.1.3 and the subscript denotes the germ in (4.2.1). This groupoid is small because of our convention on manifolds (see Remark 3.0.1).
(GM) The simplicial set of morphisms is given by where the coproduct is taken over the morphisms in Definition 4.1.3. The target map of the groupoid structure sends the component indexed by a germ ϕ: (M, C, P ) → (M ,C,P ) to itself by identity. The source map pulls back the structure by ϕ as defined in (4.2.2). Composition is given by functoriality of S.
The above simplicial sets assemble to form a presheaf on Cart × Γ × ∆ ×d in the obvious way, using the fact that S is a presheaf on FEmb d .
We also have the following variant of the bordism category, which incorporates isotopies between cuts, which can be seen as implementing d-thin homotopies between geometric structures, i.e., homotopies implemented by smooth maps of rank at most d. For example, parallel transport maps for a connection on a principal G-bundle are invariant under 1-thin homotopies even though they are not invariant under arbitrary smooth homotopies since not all principal bundles with connection are flat.
Definition 4.2.5 (d-uple bordisms with d-thin homotopies). Fix d ≥ 0 and S a simplicial presheaf on FEmb d . We specify an object Bord S d,uple in the category We define Bord S d,uple ((U, ℓ , m) by taking the diagonal of the nerve of the following simplicial groupoid.
(GO) The simplicial set of objects is given by where the coproduct ranges over pairs (M, P ) as in Definition 4.1.3. The simplicial set Cut(M × U → U, P ) is a Kan complex that has as its l-simplices ∆ l -indexed smooth collections C of cut tuples such that for any t ∈ ∆ l the triple (M, C t , P ) is an object in Bord d,uple (U, ℓ , m), and the subscript denotes the germ in (4.2.1), taken around the union of cores of C t inside M , for all t ∈ ∆ l . We identify two such ∆ l -indexed collections if they have the same germ around the compact part of ∆ l = R l+1 (given by {x ∈ R l+1 ≥0 | i x i = 1}). This simplicial set is a Kan complex because such germs of families can be glued along their common boundary.
(GM) The simplicial set of morphisms is given by where the coproduct is taken over the same pairs as for Ob. The simplicial set DiffCut(M × U → U, P ) has as its l-simplices an l-simplex of Cut(M × U → U, P ) together with a germ of a diffeomorphism M →M taken around the union over all t ∈ ∆ l of cores of C t insideM (that is, the germ is taken for the codomain, not the domain) and that commutes with the maps P andP . The simplicial structure maps are given by the simplicial structure maps for Cut together with the appropriate restriction maps for germs of diffeomorphisms. The target map of the groupoid structure sends the component indexed by (M ,P ) to itself and discards the data of the germ of a diffeomorphism. The source map pulls back the geometric structure (as defined in (4.2.2)) and the cut tuples via the given diffeomorphism M →M , landing in the component indexed by (M, P ). Composition is given by functoriality of S.
The above simplicial sets assemble to form a presheaf on Cart × Γ × ∆ ×d in the obvious way, using the fact that S is a presheaf on FEmb d .

The globular bordism category
We now extract a globular subcategory from the d-uple bordism category.  Geometrically, Glob extracts those bordisms from Bord d (respectively Bord d ) that are diffeomorphic to identity bordisms (respectively cylinders) in all globular directions.
• • Remark 4.3.4. In general, the categories Bord S d,uple , Bord S d,glob , Bord S d,uple , and Bord S d,glob are not fibrant (or even local) objects in the uple and glob model structures (respectively). Fibrancy can be established in some special cases. However, this is not needed for the main theorem. In particular, field theories are defined using derived homs from the bordism category, which does not require any fibrancy or cofibrancy properties.
Remark 4.3.5. Our results work equally well for both the variants Bord S d,uple and Bord S d,glob . Henceforth, we will denote both bordism categories ambiguously as Bord S d . Likewise for Bord S d . When distinctions need to be made between the two, we will make it explicit.
5 Criteria for weak equivalences of smooth symmetric monoidal (∞, n)-categories Our goal in this section is to establish sufficient criteria for weak equivalences in C ∞ Cat ⊗ ∞,n that will be used in the next section to prove the main theorem.

Cartesian spaces and stalks
First, we show how to reduce the problem of showing that a morphism in C ∞ Cat ⊗ ∞,n = PSh ∆ (∆ ×n × Γ × Cart) local is a weak equivalence to the same problem in the model category PSh ∆ (∆ ×n × Γ) local .
Proposition 5.1.1. Weak equivalences in the model category C ∞ Cat ⊗ ∞,n can be detected stalkwise. Specifically, on the site Cart, points are indexed by natural numbers p ∈ N corresponding to the dimension of the cartesian space. For each point p ∈ N, the associated stalk is given by the functor p * : C ∞ Cat ⊗ ∞,d → Cat ∞,d that assigns to a smooth symmetric monoidal (∞, d)-category X the stalk

Reduction of multiple to single
Next, we explain how the problem of showing that a morphism in PSh ∆ (∆ ×n × Γ) local is a weak equivalence can be reduced to the same problem in the model categories PSh ∆ (∆) local or PSh ∆ (Γ) local . The model structure on Γ-spaces is given by taking the injective model structure on PSh ∆ (Γ) and performing left Bousfield localization at the morphisms (2.2.5) and (2.2.6). This model structure has more cofibrations than the Bousfield-Friedlander model structure [1978], which, in turn, has more cofibrations than Schwede's Q-model structure [1999.b]. All three model structures have the same weak equivalences.
Recall also the Rezk model structure [1998.b] on ∆-spaces, given by taking the Reedy model structure on PSh ∆ (∆), which coincides with the injective model structure, and performing left Bousfield localization at the morphisms (2.2.3) and (2.2.4).
Proposition 5.2.1. If f is a morphism in PSh ∆ (∆ ×n × Γ) local such that for each m ∈ ∆ ×n the functor sends f to a weak equivalence, then f is a weak equivalence.
Likewise, if f is a morphism in PSh ∆ (∆ ×n × Γ) local such that for each i ∈ 1, . . . , n, ℓ ∈ Γ, and m ∈ ∆ n−1 the functor sends f to a weak equivalence, then f is a weak equivalence.
Proof. We will prove the second claim, the first claim is analogous. If f : F → G is a morphism with indicated properties, then the induced natural transformation in PSh(∆ ×n−1 × Γ, PSh ∆ (∆) local ) with components f m, ℓ : F m, ℓ → G m, ℓ is objectwise a local weak equivalence in PSh ∆ (∆) local . By the co-Yoneda lemma, the weighted colimit of f m, ℓ , weighted by the Yoneda embedding, is canonically isomorphic to f . The weighted colimit computes the weighted homotopy colimit of the same pair. This weighted homotopy colimit can be rewritten as a homotopy coend of a natural transformation (of bifunctors) whose components are These maps are local weak equivalences in simplicial presheaves on ∆ ×n × Γ, by definition of the multiple model structure in Definition 2.2.9. Since local weak equivalences are closed under homotopy colimits in the injective model structure, f is a local weak equivalence.

Reduction of simplicial presheaves to presheaves of sets
We now explain how the problem of showing that a morphism in PSh ∆ (∆) local or PSh ∆ (Γ) local is a weak equivalence can be reduced to the same problem in the categories PSh(∆) respectively PSh(Γ), where weak equivalences of presheaves of sets are created by the inclusion functor that turns them into presheaves of discrete simplicial sets. sends f to a weak equivalence (here presheaves of sets are promoted to presheaves of discrete simplicial sets), then f is a weak equivalence. Likewise for Γ instead of ∆.
Proof. This proof is analogous to the proof of Proposition 5.2.1. If f : F → G is a morphism with indicated properties, then the induced natural transformation of simplicial diagrams in PSh ∆ (∆) local with components f l : F l → G l is an objectwise local weak equivalence. The objectwise diagonal of this natural transformation is precisely f . The objectwise diagonal functor computes the homotopy colimit of simplicial diagrams in the injective model structure. Since local weak equivalences are closed under homotopy colimits in the injective model structure, f is a local weak equivalence. The case of Γ is analogous.

The simplicial Whitehead theorem
there is a diagonal arrow δ: ∆ n → X and a homotopy H: ∆ 1 × ∆ n → Y from f δ to b, relative boundary.

Lemma 5.4.2. A map f : X → Y of simplicial sets is a simplicial weak equivalence if and only if for any commutative diagram
Sd there is l ≥ k such that after precomposing both horizontal maps with the natural transformation Sd l → Sd k given by the iterated last vertex map, the resulting commutative square with newly constructed horizontal maps a ′ and b admits a diagonal arrow δ: Sd l ∆ n → X and a homotopy H: Sd l (∆ 1 × ∆ n ) → Y from f δ to b ′ , relative boundary.
We will need the following special case of the above criterion, which establishes weak contractibility of simplicial sets.
Lemma 5.4.3. Suppose X is a simplicial set such that for any n ≥ 0, k ≥ 0, and any simplicial map f : Sd k ∂∆ n → X, there is a finite chain of simplicial homotopies in either direction along ∆ 1 of the form H × Sd k ∂∆ n → X that connects f to a simplicial map that factors through ∆ 0 , where H is an arbitrary zigzag of 1-simplices. Then X is a weakly contractible simplicial set, i.e., the map X → ∆ 0 is a simplicial weak equivalence.
Proof. We prove that Ex ∞ X is a contractible Kan complex. To this end, we show that any map g: ∂∆ n → Ex ∞ X extends to a map ∆ n → Ex ∞ X. The map g is adjoint to a map for the form f : Sd k ∂∆ n → X for some k ≥ 0. By assumption, there is a simplicial homotopy h 0 : H × Sd k ∂∆ n → X from f to a constant simplicial map Sd k ∂∆ n → ∆ 0 → X, where H is an arbitrary zigzag of 1-simplices.
The adjoint map to h 0 has the form H → Hom(Sd k ∂∆ n , X) = Hom(∂∆ n , Ex k X), and we postcompose it with the map Hom(∂∆ n , Ex k X) → Hom(∂∆ n , Ex ∞ X) induced by the inclusion Ex k X → Ex ∞ X. The latter composition is adjoint to the map h 1 : C = H × ∂∆ n → Ex ∞ X, which is a simplicial homotopy from g to a constant simplicial map p: ∂∆ n → ∆ 0 → Ex ∞ X. We extend h 1 to a map h ′ 1 : A → X, where A = C ⊔ ∂∆ n ∆ n , and the map h ′ is defined on ∆ n by extending p to ∆ n as a constant map.
The inclusion A → H × ∆ n is an acyclic cofibration, thus the map A → Ex ∞ X can be extended to a map H × ∆ n → Ex ∞ X. The restriction of the latter map to ∆ 0 × ∆ n = ∆ n provides the desired extension of the map g: ∂∆ n → Ex ∞ X to ∆ n .

Dugger and Spivak's rigidification of quasi-categories
The final tool needed for the proof of the main theorem is a rigidification of quasi-categories. The reason we will need this rigidification is the following. In Section 6, we will have a monomorphism of simplicial sets f : X → Y which we claim is a weak equivalence in the Joyal model structure. Thus, f admits a presentation by an inner anodyne map. However, it is not at all clear how to express our given f as a retract of transfinite composition of cobase changes of inner horn inclusions. On the other hand, if we work with simplicial categories, then the induced map on hom spaces turns out to be much easier to understand.
By definition, a morphism of simplicial sets X → Y is an equivalence in the Joyal model structure if it induces a Dwyer-Kan equivalence on corresponding simplicial categories C(X) → C(Y ). Here C is left adjoint to the homotopy coherent nerve functor. Dugger and Spivak have a concrete model [2009.c] for C, where the mapping simplicial sets are described using necklaces. We now review this description.
Definition 5.5.1. A necklace is a simplicial set of the form ∆ n0 ∨ ∆ n1 ∨ · · · ∨ ∆ n k , where each wedge is obtained by gluing the final vertex of ∆ ni to the initial vertex of ∆ ni+1 . Each ∆ ni is called a bead and each the initial/final vertex of each ∆ ni is called a joint.
Morphisms of necklaces are simply morphisms of simplicial sets. Such morphisms are a composition of morphisms induced by the canonical inclusions ∆ ni ∨ ∆ ni+1 ֒→ ∆ ni+ni+1 , and the face and degeneracy maps of each bead. Now let X be a simplicial set. By Dugger-Spivak [2009.c, Theorem 5.2], the simplicial category C(X) is connected by a zig-zag of equivalences to the simplicial category whose objects are vertices in X and whose simplicial set of morphisms are given by the nerve of the following category 1. The objects are morphisms of necklaces ∆ n1 ∨ · · · ∨ ∆ n k → X sending the initial vertex of the first bead to x and the final vertex of the last bead to y.
2. The morphisms are morphisms of necklaces covering X.
Following Dugger-Spivak [2009.c], we call this simplicial category the necklace category and denote it C nec (X). Thus, we have the following.
Proposition 5.5.2. Let f : X → Y be a morphism of simplicial sets which is bijective on vertices. Then f is an equivalence in the Joyal model structure if for all vertices x, y ∈ X, the induced map is an equivalence in the classical model structure on simplicial sets.
Proof. Since f is bijective on vertices, C nec (f ) is essentially surjective. If C nec (f ) is an equivalence on mapping spaces, then it is a Dwyer-Kan equivalence of simplicial categories. By Dugger-Spivak [2009.c, Theorem 5.2], C nec (X) is connected to C(X) by a zig-zag of weak equivalences. Therefore, C(f ) is a weak equivalence (by 2 out of 3).

Γ-spaces
In this section, we give a sufficient condition to detect acyclic cofibrations in the local injective model structure on Γ-spaces. Since we already reduced the problem from the case of simplicial presheaves to presheaves of sets, we work in this simplified setting.
Definition 5.6.1. Let j: Γ ֒→ PSh(Γ) denote the Yoneda embedding. We define the boundary of a representable j ℓ ∈ PSh(Γ) as the subpresheaf ∂ ℓ ֒→ j ℓ , whose value at m consists of all maps of finite pointed sets ℓ → m such that at least one element k ∈ ℓ with k = * is sent to * ∈ m .
Proposition 5.6.2. The inclusion ∂ ℓ ֒→ j ℓ , viewed as a morphism of discrete simplicial presheaves, is an acyclic cofibration in the local injective model structure on Γ-spaces.
Proof. We prove by induction on ℓ that ∂ ℓ → j ℓ is a weak equivalence. The map in PSh(Γ), is an acyclic cofibration for any ℓ , which follows by repeatedly composing cobase changes of maps of the form (2.2.5). The resulting subpresheaf picks out those maps of pointed sets ℓ → m that send all elements to * except for some k ∈ ℓ , k = * .
Suppose ∂ m → j m is a weak equivalence for all m < ℓ. To show that ∂ ℓ → j ℓ is a weak equivalence, we present the inclusion S ℓ → ∂ ℓ as a composition of cobase changes of coproducts of maps of the form ∂ m → j m , and then use the 2-out-of-3 property for the maps S ℓ → ∂ ℓ and S ℓ → j ℓ to conclude that ∂ ℓ → j ℓ is a weak equivalence.
We introduce the following filtration on S ℓ → ∂ ℓ : Here W k is a subpresheaf of j ℓ comprising precisely those maps of pointed finite sets ℓ → m such that the preimage of m \ { * } has k or fewer elements. By construction, S ℓ = W 1 and W ℓ−1 = ∂ ℓ .
It remains to present the inclusion W k−1 → W k for any 1 < k < ℓ as a cobase change of a coproduct of maps of the form ∂ k → j k . The indexing set of the coproduct is the set of all maps f of pointed finite sets ℓ → k such that the preimage of any p ∈ k , p = * has exactly one element (such morphisms are also known as inert, see Lurie [2017.c, Definition 2.1.1.8]) and f is (strictly) increasing when restricted to this preimage. (Here we use the natural order on ℓ = { * , 1, . . . , ℓ}.) The map f : ℓ → k corresponds to a map j k → W k under the (contravariant) Yoneda embedding, and the attaching map ∂ k → W k−1 is the composition ∂ k → j k → W k , which factors through W k−1 by construction.
To show that the resulting commutative square is a pushout square, pick an arbitrary element of W k that is not in W k−1 , i.e., a map of pointed finite sets g: ℓ → m such that the preimage A of m \ { * } has exactly k elements. We have to show that this element comes from a unique element in the coproduct, i.e., there is a unique pair (f, σ), where f : ℓ → k is an element in the indexing set of the coproduct and σ: k → m is a map of pointed finite sets such that σf = g. Indeed, f must be the unique map f : ℓ → k of pointed finite sets such that the preimage of k \ { * } equals A and f is strictly increasing. For such an f there is a unique σ: k → m for which σf = g because f is injective away from the preimage of { * }.

Codescent for bordism categories
In this section, we prove the main theorem for bordism categories. Throughout this section, we will use multi-index notation. Greek letters α and β will denote multi-indices, which are maps of sets α: The simplicial maps induce maps between corresponding intersections, by removing or duplicating an open set in the intersection.
Since bordisms and their cut tuples will be used frequently, we will often denote bordisms simply by M , leaving the cut tuples and map P : M → ℓ implicit. is an equivalence in C ∞ Cat ⊗ ∞,d . We establish this as follows. First, by Proposition 6.1.2, the canonical map from the homotopy colimit in (6.0.2) to the corresponding strict colimit is an equivalence. Then we show that the map out of the strict colimit is a stalkwise weak equivalence (see Proposition 5.1.1) by factoring each stalk as a composition of monomorphisms (see Definition 6.3.1) The map 1 is a weak equivalence by Proposition 6.4.1. The map 2 is a weak equivalence by Proposition 6.5.1. The last equality holds by Lemma 6.6.4. We remark that our constructions respect the globularity of the existing bordisms. Therefore, our proof works in both cases: globular and multiple.

Reduction of homotopy colimits to strict colimits
Recall the following fact, which will be used twice below.
Lemma 6.1.1. If I is a direct category (meaning it contains no infinite descending chains of nonidentity morphisms · · · → X 2 → X 1 → X 0 ), and D: I → C is a diagram valued in a model category C such that the canonical map colim k<i D(k) → D(i) is a cofibration for any i ∈ I, then the canonical map hocolim D → colim D is a weak equivalence. The cofibrancy condition holds if C is a category of simplicial presheaves, cofibrations in C are monomorphisms of simplicial presheaves, and all transition maps D(k) → D(i) are monomorphisms of presheaves.
Proof. This follows immediately if we equip I with a Reedy structure whose arrows are all positive and observe that the cofibrancy condition expresses cofibrancy in the Reedy structure, which coincides with the projective structure. For the case of simplicial presheaves, observe that colim k<i D(k) → D(i) is a monomorphism because the left side is a union of subobjects. Proof. The same proof works for both Bord d and Bord d ; we only mention Bord d below. To prove the claim, we will rewrite our simplicial diagram as a filtered homotopy colimit over homotopy colimits of cubical diagrams. We then use Lemma 6.1.1 to compute these homotopy colimits as strict colimits, proving the claim.
Observe that if A is the indexing set of the cover {U a }, commutativity of (homotopy) colimits and (homotopy) coproducts ensures that we can rewrite the (homotopy) colimit as a diagram over ∆ op /A , with objects morphisms of sets [n] → A and morphisms commutative triangles. Explicitly, we assign Now let ∆ s ⊂ ∆ be the subcategory with the same objects, but only the injective order preserving maps. This inclusion is an initial functor (see, for example, Lurie [2017.a, Lemma 6.5.3.7]). Fix any subset J ⊂ A and consider the category ∆ s /J whose objects are injective maps j: where the first homotopy colimit appearing on the right is taken over inclusions of finite subsets J ⊂ A. The second homotopy colimit, indexed by (∆ s /J) op , is represented by its corresponding strict colimit. This follows from Lemma 6.1.1 because all transition maps are manifestly monomorphisms of simplicial presheaves. Observe that the poset of all J is a direct category, with transition maps being monomorphisms for all inclusions J ⊂ K, so by Lemma 6.1.1 the strict colimit over all J ⊂ A computes the homotopy colimit.

Reduction to representable geometric structures
As explained in the proof of Theorem 6.0.1, we reduce to the case of representable presheaves on FEmb d . Hence, we will be interested in the bordism categories Bord W →U , ℓ , V ) is a discrete groupoid: any two parallel morphisms are equal. As described in Definition 4.2.5, such a pair of morphisms is given by germs of diffeomorphisms of unions of cores, and these germs of diffeomorphisms must commute with the given family of germs of embeddings of cores into R d . By definition, the union of the latter cores is precisely the codomain of the germs of diffeomorphisms. Thus, if the sources of two morphisms are equal, the morphisms themselves must coincide.

Filtration on the bordism category
We now define the filtration on p * Bord W →U d that we used in Theorem 6.0.1. The idea will be to show that all but the first layer in the filtration come from attaching simplices by inner horns, and the corresponding transition map is an acyclic cofibration. The first layer of the filtration will use the Γ-boundary inclusions (Proposition 5.6.2) and we will exhibit it as a transfinite composition of cobase changes of these maps. consisting of connected components whose vertices we now specify. A vertex x ∈ B 0 (m, ℓ ) is given by a germ f : M ⇒ W around core[0, m] such that f maps each connected component of the germ into some W a ⊂ W . We define the vertices of B i (m, ℓ ) to be the subset of vertices x ∈ p * Bord W →U d (m, ℓ ) that admit a cut tuple C with the following properties: • C contains the cut tuple of x in the i-th direction.
• For each 0 ≤ j i ≤ m i − 1, the bordism Υ ji with the same data as x, but with cut tuple in the i-th direction given by two successive cuts C ji and C ji+1 , is in B i−1 .
We also define a filtration on p * Bord W →U d in the same manner, defining n-simplices of B i (m, ℓ ) by taking ∆ n -indexed smooth families of embedded bordisms such that for any t ∈ ∆ n the corresponding bordism over t is a 0-simplex in B i (m, ℓ ). Proof. Fix m ∈ ∆ ×d . We evaluate ρ on m, obtaining a morphism in PSh ∆ (Γ). By Proposition 5.2.1, it suffices to show that this is an acyclic cofibration in PSh ∆ (Γ) local . We will not include m in the notation.

Γ-direction
Step 1 (A filtration on B 0 ): Recall Remark 6.2.1 and Remark 6.2.2. For Bord d , denote by B 0 0 ⊂ B 0 the domain of the map ρ. For Bord d , denote by B 0 0 ⊂ B 0 the simplicial subpresheaf of B 0 comprising n-simplices (n ≥ 0) given by smooth ∆ n -indexed families of bordisms such that for any t ∈ ∆ n the bordism given by the fiber over t belongs to the strict colimit. For k > 0, denote by B k 0 ⊂ B 0 the simplicial subpresheaf given by the union of B k−1 0 and simplices given by bordisms (or smooth families of bordisms in the case of Bord d ) such that the germ of core[0, m] has at most k connected components. Hence, we have a filtration In the case of Bord d , the first inclusion is an equality by definition of B 0 0 . In the case of Bord d , the first inclusion is a levelwise weak equivalence (for every ℓ ∈ Γ) by the simplicial Whitehead theorem (Lemma 5.4.2): any map b: Sd k ∆ n → B 0 0 ( ℓ ) can be further barycentrically subdivided so that in the resulting map b ′ : Sd l ∆ n → B 0 0 ( ℓ ) each constituent simplex belongs to the strict colimit. This is possible because cores of bordisms are compact, the smooth families are indexed by compact manifolds (namely, ∆ n ), and any point in a ∆ n -indexed family belongs to the strict colimit by definition of B 0 0 . We now show that each inclusion B k−1 0 ֒→ B k 0 is a weak equivalence, for k ≥ 2. By Proposition 5.3.1, it suffices to show that for each [l] ∈ ∆, the map (B k−1 0 ) l ֒→ (B k 0 ) l is a weak equivalence of presheaves of sets on Γ. We claim that (B k−1 0 ) l ֒→ (B k 0 ) l is a cobase change of a coproduct of boundary inclusions ∂ k ֒→ j k , which are weak equivalences by Proposition 5.6.2. Fix [l] ∈ ∆, which we will omit from the notation below.
Below, we treat the cases of Bord d and Bord d simultaneously. We will always work with a single bordism, since for a ∆ l -indexed family of bordisms the indexing manifold ∆ l is connected, so doing things in families makes no difference when we talk about connected components.
Step 2 (The proposed pushout): Consider the set of maps where j is the Yoneda embedding, that do not factor through B k−1 0 ֒→ B k 0 , and the bordism (M, C, P ) corresponding to f under the Yoneda lemma has the following property: the map P : M → k , restricted to the germ of core[0, m], is surjective away from the basepoint in k and maps all other connected components of M (the ones that are disjoint from the core) to the basepoint in k . The symmetric group Σ k acts on the set of connected components of the core by permuting the elements in k . This action is free, since the restriction of P : M → k is required to be surjective, away from the basepoint. In fact, the restriction of P is also injective on connected components because the germ of core[0, m] has at most k connected components. Choose a single representative from each Σ k -orbit and denote the resulting set by S.
For f ∈ S, the composition ∂ k → j k , since elements of ∂ k are maps of finite pointed sets k → ℓ that have at most k − 1 elements of k map to non-basepoint elements of ℓ . We have a commutative diagram We show that this is an objectwise pushout square.
Step 3: Fix ℓ ∈ Γ and evaluate at this object. We show that the bottom map in (6.4.2) induces a bijection of sets Elements in the left side are precisely pairs (f, σ), where f ∈ S and σ: k → ℓ satisfies σ −1 { * } = { * }. The displayed map sends (f, σ) → f (σ), i.e., we apply the Γ-structure map σ to f . Observe that the core of f (σ) is the same as the core of f . In particular, the core of f (σ) has k connected components and does not factor through any W a , hence f (σ) ∈ B k−1 0 ( ℓ ).

It remains to show that any element
, then x and f must have exactly the same bordism data (including the core) except for P , for which we have P x = σ • P f . By definition, the restriction of P f to the germ of the core is bijective on π 0 , after removing the basepoint in k . Hence the equation P x = σ • P f has exactly one solution in terms of σ, which establishes uniqueness of (f, σ).
To show the existence of (f, σ), we construct f from x by changing the P -component of the bordism data of x to an arbitrary surjection P f from the germ of core([0, m], P x ) to k \ { * } ⊂ k , which is bijective on π 0 . Among all such surjections, there is a unique one with the corresponding map f ∈ S, since any two surjections have corresponding f 's in the same Σ k -orbit.

∆-direction
We now turn to the most technical part of the proof. Supporting lemmas will be proved subsequently.
Proposition 6.5.1. The monomorphism ρ i : B i−1 ֒→ B i in Definition 6.3.1 is an equivalence. Likewise for Bord d instead of Bord d .
By Proposition 5.2.1, it suffices to show that ρ i (m, ℓ ) is an equivalence in the Rezk model structure on simplicial objects in simplicial sets. For Bord d , recall that both B i−1 and B i are discrete in the target simplicial direction [l] ∈ ∆, i.e., the canonical maps B i → π 0 (B i ) and B i−1 → π 0 (B i−1 ) are objectwise equivalences (see Remark 6.2.1).
This amounts to working with the images of the open embeddings of bordisms into W . Hence, we have further reduced the claim to showing the π 0 (ρ i (m, ℓ )) is a weak equivalence in the Joyal model structure on simplicial sets. We redefine ρ i to be the map π 0 (ρ i (m, ℓ )), i.e., we will omit m, ℓ and π 0 in the notation below.
For Bord d , after applying the weak equivalence of Remark 6.2.2, the set B i (m, ℓ ) l can be described as ∆ l -indexed smooth families of embedded bordisms in W . In this case, we redefine ρ i to mean this map.
By Proposition 5.5.2, it suffices to show that for each pair of vertices x and y, the induced map is a weak equivalence. The map C nec (ρ i ) splits as a coproduct, indexed by the germs of cores of ambient bordisms with endcuts x and y. (In other words, morphisms in the necklace category can only exist between necklaces whose underlying bordisms from x to y coincide.) By Proposition 6.6.6, each summand of C nec (B i−1 )(x, y) and C nec (B i )(x, y), indexed by a bordism with endcuts x and y, is contractible. Hence C nec (ρ i ) is an equivalence.
The claim about the contractibility of C nec (B i−1 )(x, y) in Proposition 6.5.1 will occupy the remainder of this section. We begin with an explicit description of the Dugger-Spivak necklace categories. These have the following objects and morphisms.
• A necklace in C nec (B i−1 )(x, y) is a composable collection of bordisms (respectively ∆ l -indexed families of bordisms), as defined in Definition 4.1.3. Each bead of the necklace specifies a bordism in B i−1 . Bordisms must coincide on the germ of each joint cut and can therefore be glued (see Milnor [1969]) along each joint. The resulting bordism is not equipped with an open embedding into W , but only a local diffeomorphism. Hence, we obtain a single bordism with a different geometric structure (i.e., a local diffeomorphism to W ). The collection of cut tuples together forms a cut tuple of the entire bordism, with the joints forming a subset of this cut tuple. We emphasize that this resulting bordism is not a vertex in any level of the filtration of Definition 6.3.1 since the map to W is not an open embedding in general.
• A morphism only exists between necklaces whose underlying bordisms (as constructed in the previous paragraph) become isomorphic after throwing away the inner cuts. A morphism is a composition of the following elementary operations.
1 The joint cut between C j and C j+1 is converted to a cut in the union C j ∪ C j+1 , provided that the resulting bead is in B i−1 .
2 New compatible cuts are added to an existing cut tuple C j .
first bead C 1 first bead C 1 (6.5.2) The necklace category associated to B i has the same description, but we require that each bead is in B i .

Bordism categories for representable geometric structures are 1-truncated
In this section, we treat the cases of Bord d and Bord d simultaneously. We assume the conventions of the proof of Proposition 6.5.1, in particular, we work with the appropriately simplified (as explained in Remark 6.2.1 and Remark 6.2.2) bordism category, evaluating at a fixed multisimplex m ∈ ∆ ×d−1 and a fixed object ℓ ∈ Γ, and passing to some n-dimensional germ with respect to Cart. For Bord d , we also fix a simplex [l] ∈ ∆. Thus, we work with a simplicial set whose k-simplices are k-tuples of composable embedded bordisms (in the case of Bord d ) or k-tuples of composable ∆ l -indexed smooth families of embedded bordisms (in the case of Bord d ). Below, we treat both cases simultaneously, so in the case of Bord d a bordism really means a smooth ∆ l -indexed family of bordisms. All arguments given below that use compactness continue to work in the case of Bord d because ∆ l is compact. Consider the monomorphism obtained by evaluating ρ i on an arbitrary object of Γ × ∆ ×(d−1) , via restriction along the functor Observe that the contractibility claim of Proposition 6.5.1 implies that the fibrant replacements (in the Rezk model structure) of B i−1 and B i are weakly equivalent to nerves of ordinary categories. This is intuitively clear, since the operation of fibrant replacements adds formal compositions and homotopies between them. However, there is a natural candidate for a composition of bordisms, given by gluing, which produces a bordism with a local diffeomorphism to W .
To prove contractibility, we will invoke the simplicial Whitehead theorem in the form of Lemma 5.4.3. However, to construct the homotopy there, we need a cutting and gluing operation between cut manifolds. Moreover, we will need our surgered manifolds to have specific properties that allow us to define a homotopy. We begin with our cutting and gluing lemmas.
Lemma 6.6.1. Let C and C ′ be cuts on an open subset N whose underlying cut manifolds C =0 and C ′ =0 intersect transversally. Then for any open neighborhood U of the intersection C =0 ∩ C ′ =0 , there is a cut C ′′ having the following properties: . 2. C ′′ is compatible with C and C ′′ is compatible with C ′ , in the sense of Definition 4.1.2.
Proof. Fix maps h, k: N → R defining the cuts C and C ′ . Consider the map Then (h, k) −1 (0) = C =0 ∩ C ′ =0 , and the preimages of the two axes are C =0 and C ′ =0 . Choose an open ball of radius δ around 0 ∈ R 2 . Since transversality is a generic property, there exists a smooth embedding ι: R → R 2 which is transversal to (h, k) and which satisfies the following: Then C ′′ =0 := (h, k) −1 (ι(R)) gives rise to a cut with the desired properties. Since the tubular neighborhood of h −1 (0) ∩ k −1 (0) is given by (h, k) −1 (B δ ), we can get an arbitrarily small radius by letting δ → 0.
From the definition of the filtration in Definition 6.3.1, every bordism in B i ([1]) can be cut in the first i-directions so that each cell is contained in some W a . However, it will be useful for us to find an explicit such cut tuple.
Definition 6.6.2. Two cuts C and C ′ on a bordism M are called transversally compatible if locally, with respect to some open cover of M , C intersects C ′ transversally or C is compatible with C ′ , i.e., for the given element U in the cover, • For each j indexing a cut in the cut tuple Ψ, the closure of Ψ (j,j+1) is contained in some W ′ a . Moreover, for any finite collection of cut tuples {C k } k∈K on M , we can find such a Ψ so that each Ψ j is transversally compatible with each C k,j ′ ∈ C k .
Proof. To construct the Ψ, we will need to use a lemma from Morse theory for manifolds with corners (see Budney [2002.b, Section 3.1] for a one page survey, Vakhrameev [1998[ .a], Handron [2002[ .a, 2004.c] for proofs). We construct Ψ by induction on the number of critical points, for some Morse function f on the core of M (which is a manifold with corners) that vanishes on the source cut of M .
For the base case, we will show that any bordism M that admits a Morse function with no critical points has such a decomposition. Fix a Morse function f : M → R with f giving the initial cut (see Definition 4.1.1). Let y denote the terminal cut of M and let s = sup x∈core(M) f (x). The gradient of f may flow outside the core, but a modification of the gradient flow (see Vakhrameev [1998.a, Theorem 2.1]) produces a flow that respects the boundary. We denote this flow by ϕ t . Partition We have Ψ k,0 = f −1 (t k−1 ) and Ψ k,B = f −1 (t k ). The cuts Ψ k,b are the desired cuts. We now prove the inductive step. Consider an isolated critical point x. If x lies in the top-dimensional stratum, then by Milnor [1969, Theorem 3.2], we can modify f to a Morse function F that coincides with f everywhere, except for an arbitrarily small neighborhood of x. By construction, this F satisfies the following property. Let y denote the critical value corresponding to x. There is a sufficiently small ǫ such that the cut tuple with endcuts Ψ 0 = f −1 (y − ǫ) and Ψ 1 = F −1 (y − ǫ) satisfies Ψ (0,1) ⊂ W ′ a , for some a ∈ A. The bordism with endcuts x and Ψ 0 and the bordism with endcuts Ψ 1 and y have fewer critical points. By the induction hypothesis, these admit cut tuples C 0 and C 1 (respectively) satisfying the hypothesis of the lemma. Taking the cut tuple C 0 ∪ C 1 gives the desired cut tuple for M .
For higher-codimensional strata, we take Milnor's decomposition on the stratum and take a product with a cone (see Goresky-MacPherson [1988.a, Theorem SMT Part B]). For a sufficiently small ǫ, we adapt the construction of the function F in Milnor [1969, Theorem 3.2] by extending the bump function µ in the normal direction. Set Ψ 0 = f −1 (y − ǫ) and Ψ 1 = F −1 (y − ǫ). This gives a cut tuple satisfying the desired properties. Then, proceeding as in the case of top-dimensional strata proves the claim.
Lemma 6.6.5. Fix a bordism M with end cuts x and y. Fix a Riemannian metric g on M . Let Ψ be an intermediate cut such that M (Ψ,y) ⊂ D ǫ (see Definition 4.1.1; bar denotes the closure) for some disk of radius ǫ. Fix a collection of necklaces {C q } q∈Q in C nec (B i−1 )(x, y) such that for all q and consecutive joints j and j ′ , the 2ǫ-neighborhood of the core of the corresponding bordism in B i−1 satisfies the condition of Lemma 6.6.3. Then for any δ > 0, there is a finite collection of necklaces {C ′ q } q∈Q in C nec (B i−1 )(x, y) M that satisfies the following conditions: 1. C ′ q has the same number of cuts as C q . 2. C ′ q,j ≤ C q,j and C ′ q,j ≤ Ψ, where ≤ is defined in Definition 4.1.1. Equality holds if and only if C q,j ≤ y. 3. If j ≤ j ′ , then C ′ q,j ≤ C ′ q,j ′ . 4. The cut tuple consisting of two joint cuts C ′ q,j and C q,j is in B i−1 . 5. The cut tuple consisting of two cuts C ′ q,j and C q,j ′ , where j and j ′ index two consecutive joints, belongs to B i−1 .
6. Each bead in C ′ q corresponding to a bead in C q is in B i−1 and for consecutive joint cuts j, j ′ , the subset C ′ q,(j,j ′ ) is contained in the δ-neighborhood of C q,(j,j ′ ) .
Proof. We construct C ′ by extending the natural order on cuts C q,j (see Definition 4.1.1) to a total order and processing cuts in this order. Equal cuts are processed simultaneously. At each stage, we invoke Lemma 6.6.1 to cut and glue C q,j and Ψ. We define C ′ q,j to be the result of this surgery. Lemma 6.6.1 ensures that C ′ q,j ≤ C q,j and C ′ q,j ≤ C for each (q, j). This proves properties 1 and 2. Since there are only finitely many C q,j , we can choose the neighborhood U in Lemma 6.6.1 so that U does not intersect any of C ′ q,j that have already been constructed. This is possible by compactness. Hence we have C ′ q,j ′ ≤ C ′ q,j for j ′ ≤ j, since also C ′ q,j ′ ≤ C q,j and C ′ q,j ≤ Ψ by construction. This proves property 3. Property 4, 5 and 6, follow immediately from Lemma 6.6.3, since M (Ψ,y) ⊂ D ǫ and the ǫ-neighborhood of core of each bead satisfies the condition of Lemma 6.6.3 by assumption.
Proposition 6.6.6. The summands of both C nec (B i−1 )(x, y) and C nec (B i )(x, y) (as defined in Proposition 6.5.1) are contractible.
Proof. We prove the claim for C nec (B i−1 )(x, y). The same construction works for C nec (B i )(x, y). In what follows, we denote the summand indexed by a bordism M , with endcuts x and y, by C nec (B i−1 )(x, y) M and C nec (B i )(x, y) M , respectively.
We will use the simplicial Whitehead theorem. Given a map we seek to construct a zig-zag of simplicial homotopies from f to where f ′ factors through C nec (B i−1 )(x, x) M . This proves the proposition, since C nec (B i−1 )(x, x) M is the nerve of a category with a terminal object. We denote the necklaces picked out by f by {C q } q∈Q , where Q is the set of vertices of Sd k ∂∆[n]. Lemma 6.6.3 provides an ǫ such that the 2ǫ-neighborhood of each bead in each necklace C q satisfies the (u 1 , . . . , u i )-axis (u i+1 , . . . , u d )-axis Figure 2: Here the origin is the critical point, given by the Morse lemma. The thick parabolas are the manifolds given by f −1 (y j − ǫ) and f −1 (y j + ǫ), respectively. The solid cut C 1 is contained in f −1 [0, y j − ǫ], while the dashed cut passes through the "bridge" (labeled by H) and is not contained in f −1 [0, y j ). The bridge H is obtained by constructing a modified Morse function and the boundary segments are smooth manifolds (see Milnor [1969, page 19]). In Milnor [1969], the purpose of H is to exhibit the saddle as a handle attachment, but we will use the boundary components of H to modify existing cuts by a surgery.
conditions of Lemma 6.6.3. Lemma 6.6.4 gives a cut tuple Ψ such that each cut Ψ j is transversally compatible with each cut in the necklaces picked out by f . Set δ = ǫ/|Ψ|. Introduce an equivalence relation on the set of pairs (q, h), which is generated by (q, h) ∼ (q ′ , h ′ ) if C q,h = C q ′ ,h ′ and q and q ′ are vertices in the same edge in Sd k ∂∆[n]. The equivalence classes have a natural partial order, induced by the partial order on cuts (see Definition 4.1.1). Extend this partial order to some total order.
We now prove by induction on j, in decreasing order, that there is a zig-zag of homotopies between f and f j , where The induction starts at f |Ψ| = f . Simplices in the nerve of a poset are uniquely determined by their vertices. Hence, it suffices to construct the zig-zag of homotopies on vertices and verify that the relevant inequalities are satisfied.
We apply Lemma 6.6.5 to the necklaces {C q } q∈Q , with ǫ and δ chosen above and the cut Ψ j , to obtain new necklaces {C ′ q } q∈Q . The properties 1-3 in Lemma 6.6.5 ensure that {C ′ q } q∈Q give rise to a map It remains to connect this map to f j+1 by a zig-zag of homotopies. This zig-zag will be a composition of zig-zags, one for each equivalence class [(q, h)] such that the corresponding cut C q,h satisfies C q,h ≤ Ψ j . We process these equivalence classes in increasing order by induction, extending an existing zig-zag of homotopies. Fix such an equivalence class. We extend the zig-zag by inserting C ′ q,h immediately before C q,h , for each (q, h) in the equivalence class. If C q,h has duplicates, i.e., the corresponding cut tuple is simplicially degenerate, we insert as many duplicates of C ′ q,h . This defines the zig homotopy in the zig-zag. We then remove the old cuts corresponding to (q, h) in the equivalence class, obtaining the zag homotopy in the zig-zag. Properties 4-6 in Lemma 6.6.5 ensure that this is produces a zig-zag of simplicial homotopies through maps to C nec (B i−1 )(x, y).

Shapes and concordances
In this section, we recall the notion of the shape of a simplicial presheaf on cartesian spaces (or smooth manifolds). We first recall the notion of a cohesive ∞-topos of Schreiber [2017.b]. These are certain toposes which participate in a quadruple adjunction (B ∫ ⊣ ∆ ⊣ Γ ⊣ ∇), with ∆ and ∇ fully faithful and B ∫ preserving products. In our context the adjunction takes the form of a quadruple Quillen adjunction with left adjoints depicted above their corresponding right adjoints. This adjoint quadruple is induced by an adjunction between Cart and the terminal category 1. The right adjoint functor 1 → Cart picks R 0 . The further left and right adjoints are given by Kan extensions. Thus, the functor Γ evaluates at R 0 and the functor ∆ takes the locally constant stack associated to a simplicial set. Using the fact that the inclusion i: Cart ֒→ Man is the inclusion of a full dense subsite, one shows that the restriction functor i * : PSh ∆ (Man) → PSh ∆ (Cart) is part of a Quillen equivalence (in fact, two Quillen equivalences, corresponding to the injective and projective model structures). Hence, the ∞-category presented by PSh ∆ (Man) loc is also cohesive.
Remark 7.1.2. The constant presheaf functor sSet → PSh ∆ (Man) does not extend to a quadruple Quillen adjunction when PSh ∆ (Man) is equipped with local weak equivalences. However, one can compute explicitly the shape functor via derived functors by first applying the restriction functor i * : PSh ∆ (Man) → PSh ∆ (Cart) and then the left derived functor of B ∫ in the projective model structure. However, a more preferable formula will be presented below.
These quadruple adjoint functors give rise to idempotent monads ∫ := ∆B ∫ ⊣ ♭ := ∆Γ ⊣ ♯ := ∇Γ , (7.1.3) each of which reflects a different nature of a smooth stack (see Schreiber [2017.b] for detailed discussion). We will focus on one of these idempotent functors, namely the leftmost adjoint ∫ := ∆B ∫ , which is also known as the shape functor. Abusing terminology, we also call the left adjoint B ∫ the shape functor. We call the value of B ∫ on a sheaf X ∈ PSh ∆ (Man) loc , the shape of X.
Remark 7.1.4. The previous adjunction goes through just as well for sheaves with values in spectra Bunke-Nikolaus-Völkl [2013.b]. In particular, for a sheaf of spectra X ∈ PSh(Man; Sp), we can take the shape B ∫ , which gives a spectrum. We can also reflect back into sheaves of spectra, by taking the shape ∫ (X) = ∆B ∫ (X) ∈ PSh(Man; Sp).
Remark 7.1.5. Berwick-Evans, Boavida de Brito, and the second author [2019] (see also [2022] for the structured case) established an explicit formula for the shape functor ∫ in (7. where the homotopy colimit is taken in the category of presheaves and ∆ n is the smooth n-simplex, viewed as a smooth stack via its diffeological space of plots. Although it is fairly easy to see that the colimit on the right is invariant under concordance, it is highly nontrivial to show that it satisfies descent. See [2022] for a version that treats sheaves valued in algebraic ∞-categories, like Γ-spaces.
Remark 7.1.7. The formula for ∫ also gives rise to a formula for B ∫ . We have where the last equivalence follows from the fact that ∆ is homotopically fully-faithful and the (derived) counit ǫ: Γ ∆ ≃ → id is an equivalence. In particular, if X is a smooth manifold, viewed as a representable smooth stack, then where on the right we have the singular simplicial set of the underlying topological space. Taking π 0 of both sides of the above identifies concordance classes with the topological connected components of the underlying space. Formula (7.1.6) has some striking consequences. For example, if ∼ con denotes the equivalence relation given by concordance, then for any smooth manifold M , we have the formula where on the right we take homotopy classes of maps and on the left we mod out by the relation of concordance. This immediately gives a classifying space construction for X. We illustrate with the following example.
Example 7.1.9. Let X = Vect be the sheaf of groupoids 1 given by vector bundles with isomorphisms between them. Because every vector bundle is locally trivial, we have an equivalence Vect(∆ n ) ≃ BGL(∆ n ) on every smooth n-simplex, where BGL(S) = d≥0 BGL d (S) is the disjoint union of deloopings of smooth general linear groups, with GL d (S) = C ∞ (S, GL(R d )). The classifying space of Vect can be computed (for a fixed summand in dimension d ≥ 0) as Here B sing(GL d ) denotes the classifying space of the simplicial group sing(GL d ), or, equivalently, the singular simplicial set of the topological classifying space of the topological group GL d . Thus, we recover the classifying space construction

Smooth field theories and their classifying spaces
In this section, we prove Theorem 1.0.2. This provides an affirmative answer to a long-standing conjecture of Stolz and Teichner. We begin by defining field theories. Recall the conventions and notations of Notation 2.1.3 and Definition 2.4.2.
Remark 7.2.1. The results in the section apply equally well to both bordism categories, Bord d and Bord d , since they only rely on the codescent property. We formulate them for Bord d only, leaving the other case implicit.
Definition 7.2.2. Fix d ≥ 0, S ∈ PSh ∆ (FEmb d ), and T ∈ C ∞ Cat ⊗ ∞,d . The smooth symmetric monoidal (∞, d)-category of d-dimensional field theories, with geometric structure S and target T , is the internal ∞-hom FFT d,T (S) := Fun ⊗ (Bord S d , T ). The assignment S → FFT d,T (S) defines a functor PSh ∆ (FEmb d ) → C ∞ Cat ⊗ ∞,d . We will mostly work with the invertible part (i.e., the core) of FFT d,T (S), which we denote by where the superscript × denotes the core of a smooth symmetric monoidal (∞, d)-category (defined by evaluating its fibrant replacement at 0 ∈ ∆ ×d , which amounts to discarding noninvertible k-morphisms for 1 ≤ k ≤ d). Thus, we have a functor FFT × d,T : PSh ∆ (FEmb d ) → PSh ∆ (Cart × Γ). A functorial field theory is a vertex in the 1st level of this sheaf of Γ-spaces evaluated at R 0 ∈ Cart.
By Theorem 6.0.1, the functor FFT × d,T sends local weak equivalences in PSh ∆ (FEmb d ) to (local) weak equivalences of sheaves of Γ-spaces. If we regard a manifold as a sheaf on FEmb d via its functor of smooth points, then Čech nerves of covers of manifolds are local weak equivalences in PSh ∆ (FEmb d ). Hence if we restrict the above functor along the embedding Man ֒→ PSh ∆ (FEmb d ) we get a well defined sheaf of smooth Γ-spaces on the site of smooth manifolds. For a fixed geometric structure S, we can also consider the functor on manifolds that sends X → Fun ⊗ (Bord X×S d , T ) × = FFT × d,T (X × S). (7.2.3) Since this functor (as a functor from Man op ) is homotopy continuous, Theorem 6.0.1 implies that this presheaf of smooth Γ-spaces satisfies descent. More generally, to allow for twisted situations, we consider an arbitrary ∞-cosheaf F : Man → PSh ∆ (FEmb d ).
Example 7.2.4. Fix d ≥ 0 and S ∈ PSh ∆ (FEmb d ). The assignment F : X → X × S is a cosheaf on Man, with values in PSh ∆ (FEmb d ). In the resulting bordism category Bord X×S d , a bordism M will be equipped with a smooth map to X and a geometric structure, i.e., a vertex in S(M ).
Example 7.2.5. We consider the cosheaf F : Man → PSh ∆ (FEmb d ) that sends a smooth manifold X to the presheaf of groupoids on FEmb d that sends (M → U ) ∈ FEmb d to the groupoid whose objects are pairs (f : G → X, p: G → U ) with G a supermanifold of superdimension d|1 and U(p: G → U ) = (M → U ), where U is the reduction functor U: SMan d|1 → Man d . The isomorphisms are isomorphisms G → G ′ that make the obvious triangles commute.
We also have a variant where G is equipped with a super-Euclidean structure (in the sense of Stolz-Teichner [2011.a, 2004). This recovers the 0|1, 1|1, and 2|1-dimensional bordism categories of Stolz and Teichner. Hence, Theorem 7.2.9 produces a classifying space of d|1-Euclidean field theories. We apply the construction of Berwick-Evans-Boavida de Brito-Pavlov [2019,2022] to the presheaf of smooth Γ-spaces FFT × d,T,F , which we convert to a presheaf of Γ-spaces by evaluating at R 0 ∈ Cart. where Map on the right side denotes the powering of a Γ-space by the singular complex of X.
Proof. By Theorem 6.0.1, the functor X → Bord F (X) d is an ∞-cosheaf on Man. Thus, the functor FFT ×

d,T,F
is an ∞-sheaf on Man valued in Γ-spaces. Hence, the main theorems of [2019] and [2022] immediately imply the claim. Taking F (X) = X × S yields Theorem 1.0.2.
In the case when all objects of T are invertible with respect to the monoidal product, the resulting Γspaces are group-like, so can be identified with connective spectra. In particular, B ∫ FFT × d,T,F is a connective spectrum whose associated cohomology theory allows for a concrete model for cohomology classes over X: these are precisely concordance classes of field theories over X.
Other constructions on cohomology classes, such as cup products and other operations also allow for such geometric models in terms of field theories. In the next section, we explore one such model for power operations.

Power operations in the Stolz-Teichner program
Remark 7.3.1. Remark 7.2.1 continues to apply in this section: all results below are formulated for Bord d , but apply equally well to Bord d .
As explained at the end of the previous section, we can expect geometric models for various operations on cohomology classes in terms of field theories. In this section, we explore one such model for power operations, building upon the work of Barthel-Berwick-Evans-Stapleton [2020.b].
Definition 7.3.2. Fix d ≥ 0. The nth power cooperation (n ≥ 0) is a morphism Bord X ×n //Σn d → Bord X d , where X is a simplicial presheaf on FEmb d . The notation −//Σ n denotes the homotopy quotient of simplicial presheaves presented as a bar construction, where Σ n acts on X ×n by permutations. Fixing an object in ∆ ×d × Γ × Cart, the cooperation on individual bordisms is defined using a pull-push construction along X ×n //Σ n ← (X ×n ×n)//Σ n → X, wheren is the set on which Σ n acts, the left map is induced by the projectionn → 1, and the right map is given by evaluation (X ×n ×n)//Σ n = (Xn ×n)//Σ n ev −→ X.
Observe that the left map is an n-fold covering, so pulling back a map M → X ×n //Σ n produces an n-fold coveringM → M (in our strict modelM = M ×n → M is the projection map, but nontrivial n-fold coverings emerge after fibrantly replacing the bordism category) together with a mapM → (X ×n ×n)//Σ n defining the geometric structure. Postcomposing the latter map with the evaluation map (X ×n ×n)//Σ n → X yields the desired bordism.
We now explain how to recover power operations in cohomology via Definition 7.3.3. Fix d ≥ 0, T ∈ C ∞ Cat ⊗ ∞,d and F : Man → PSh ∆ (FEmb d ). Then the functor FFT × d,T,F is an ∞-sheaf on the site of smooth manifolds with values in smooth Γ-spaces. More generally, for an arbitrary ∞-sheaf of Γ-spaces X ∈ PSh ∆ (Man × Γ), we set FFT × d,T,F (X) := Fun ⊗ (X, FFT × d,T,F ) × ∈ PSh ∆ (Cart × Γ). Now for two ∞-sheaves of Γ-spaces X and Y , we have a comparison map ι: Hom(X, Y )(R 0 ) → Hom(B ∫ (X), B ∫ (Y )) ∈ PSh ∆ (Γ), defined as the adjoint of the map In expanded form, the latter map reads Hom(X, Y )(R 0 ) × hocolim n∈∆ op X(∆ n ) → hocolim n∈∆ op Y (∆ n ), which can be rewritten as the map hocolim n∈∆ op (Hom(X, Y )(R 0 ) × X(∆ n )) → hocolim n∈∆ op Y (∆ n ). The latter is defined objectwise for each n-separately, as the composition map. The ι assemble together to form a natural transformation (in both X and Y ). Observe that the n-fold monoidal product (FFT × d,T,F ) ×n → FFT × d,T,F gives rise to a map m n : Hom((B ∫ X) ×n , (B ∫ FFT × d,T,F ) ×n ) ≃ Hom((B ∫ X) ×n , B ∫ ((FFT × d,T,F ) ×n )) → Hom((B ∫ X) ×n , B ∫ FFT × d,T,F ), where we have used the fact that B ∫ preserves finite products. We use the maps ι and m n in the following proposition, which gives refinements of power operations in cohomology. . If X is a smooth manifold, then B ∫ (X) is equivalent to the (singular simplicial set of) underlying space of X. Hence, the power operation P n takes the form P n : Hom(X, B ∫ (FFT × d,T,F )) → Hom(X ×n //Σ n , B ∫ (FFT × d,T,F )), which exhibits the n-th power operation for the connective spectrum B ∫ (FFT × d,T,F ).