Homotopy Sheaves on Generalised Spaces

We study the homotopy right Kan extension of homotopy sheaves on a category to its free cocompletion, i.e. to its category of presheaves. Any pretopology on the original category induces a canonical pretopology of generalised coverings on the free cocompletion. We show that with respect to these pretopologies the homotopy right Kan extension along the Yoneda embedding preserves homotopy sheaves valued in (sufficiently nice) simplicial model categories. Moreover, we show that this induces an equivalence between sheaves of spaces on the original category and colimit-preserving sheaves of spaces on its free cocompletion. We present three applications in geometry and topology: first, we prove that diffeological vector bundles descend along subductions of diffeological spaces. Second, we deduce that various flavours of bundle gerbes with connection satisfy (∞,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\infty ,2)$$\end{document}-categorical descent. Finally, we investigate smooth diffeomorphism actions in smooth bordism-type field theories on a manifold. We show how these smooth actions allow us to extract the values of a field theory on any object coherently from its values on generating objects of the bordism category.


Introduction and main results
Local-to-global properties are ubiquitous in topology, geometry, and quantum field theory. The prototypical example of a local-to-global, or descent, property is the gluing of local sections of a sheaf: given a manifold M with an open covering {U i } i∈I , global sections of a sheaf F on M are in bijection with families {f i ∈ F (U i )} i∈I such that f i|U ij = f j|U ij for all i, j ∈ I, with U ij := U i ∩ U j .
Geometric structures on manifolds are, in general, not described by sheaves of sets; instead, one needs to pass to sheaves of higher categories. For example, complex vector bundles form a sheaf on the site of manifolds and open coverings which is valued in categories rather than sets. In the works of Schreiber [Sch13], n-gerbes are described as sections of certain sheaves of n-groupoids, for any n ∈ N. Geometric structures and their equivalences are thus described most generally in terms of sheaves of higher categories on a category of test manifolds C which is endowed with a (pre)topology τ .
To motivate the main questions of this paper, let us temporarily focus on sheaves of ∞-groupoids. The ∞-category of these sheaves has a presentation as a left Bousfield localisation of the projective model category H ∞,0 = Fun(C op , Set ∆ ) of simplicial presheaves on C 1 . We denote its localisation at the τ -coverings by H loc ∞,0 . Our prime example in this paper is the case C = Cart, the category of cartesian spaces. It consists of the manifolds diffeomorphic to R n , for any n ∈ N 0 , and all smooth maps between them. It is an important (well-known) observation that sheaves of ∞-groupoids on Cart allow us to describe geometric structures not just on objects of Cart, but on all manifolds (and more) [Sch13]: given a fibrant object F ∈ H loc ∞,0 and a manifold M , one defines the space of (derived) sections of F on M as the mapping space H ∞,0 (QM , F ). Here, M is the presheaf on Cart consisting of smooth maps to M , the functor Q is a cofibrant replacement 2 in H ∞,0 , and H ∞,0 (−, −) denotes the Set ∆ -enriched hom functor in H ∞,0 . If F is the simplicial presheaf on cartesian spaces which describes G-bundles or n-gerbes, for instance, then the Kan complex H ∞,0 (QM , F ) is the ∞-groupoid of G-bundles or n-gerbes on M , respectively. In fact, we could use any presheaf X on Cart in place of M and thus study geometries on any such X. (Later we will even allow X to be a presheaf of ∞-groupoids itself.) We write C = Fun(C op , Set) for the category of (small) presheaves of sets on C. Motivated by the above discussion, we pose the following two questions for sheaves on a generic site (C, τ ): (1) If F satisfies descent with respect to the pretopology τ , does the assignment X → H ∞,0 (QX, F ) satisfy descent with respect to a related pretopology τ on C?
We answer both of these questions in this paper, working over general Grothendieck sites (C, τ ): any pretopology τ on C induces a pretopology τ of generalised coverings on C. We prove that with respect to such a pair of pretopologies, the answer to the first question is always 'yes'. One way of saying this is that the process of extending any geometric structure from objects of C to objects of C does not destroy the descent property; we can still think of the extended structure as having geometric flavour and perform local-to-global constructions.
In concrete terms, the induced Grothendieck pretopology τ on C consists of the τ -local epimorphisms, also called generalised coverings [DHI04], which are defined as follows. Let Y : C → C denote the Yoneda embedding of C. A morphism π : Y → X in C is a τ -local epimorphism if, for every c ∈ C and every map Y c → X, there exists a covering {c i → c} i∈I in the site (C, τ ) such that each composition Y c i → Y c → X factors through π; this is an abstract way of saying that the morphism π has local sections. and Fun( C op , M) loc at the τ -coverings and τ -coverings, respectively. That is, the fibrant objects in these localised model categories are M-valued homotopy sheaves on (C, τ ) and ( C, τ ), respectively. We show the following result (see Theorems 3.30, 3.37 and Proposition 3.38 below): Theorem 1.1 Let M be a left proper simplicial model category which is cellular or combinatorial and C a category. Let τ be a Grothendieck coverage on C, and τ the induced Grothendieck coverage of τ -local epimorphisms on C. There is a Quillen adjunction A prime application is the case of homotopy sheaves of higher categories: denoting by CSS n the model category of n-fold complete spaces [Rez01, Bar05,BSP11], let H loc ∞,n be the left Bousfield localisation of the projective model structure on Fun(C op , CSS n ) at the Čech nerves of coverings in (C, τ ). Similarly, we let H loc ∞,n denote the left Bousfield localisation of Fun( C op , CSS n ) at the Čech nerves of the τ -local epimorphisms. As a direct corollary to Theorem 1.1 we obtain: Corollary 1.2 Let n ∈ N 0 , and let C be a small category.
(1) For any fibrant F ∈ H ∞,n , the presheaf H ∞,n Q(−), F is equivalent to the homotopy right Kan extension hoRan Y F of F along the Yoneda embedding Y : C → C. In particular, H ∞,n Q(−), F presents the ∞-categorical right Kan extension of presheaves of (∞, n)-categories along Y. In Section 4.1 we deduce the following strictification and descent result (Theorem 4.18 and Theorem 4.12, respectively), thereby filling significant gaps in the literature on diffeological spaces: Theorem 1.3 Let VBun Dfg : Dfg op → Cat be the pseudo-functor which assigns to a diffeological space its category of diffeological vector bundles. The following statements hold true: (1) There is a strict functor ι * VBun str (which we construct explicitly) and an objectwise equivalence of pseudo-functors Dfg op → Cat L , (2) Diffeological vector bundles satisfy descent along subductions of diffeological spaces.
Here we use Rezk's classifying diagram functor to deduce a 1-categorical analogue of Corollary 1.2 for n = 1. This readily provides the desired descent result; the main remaining work in proving Theorem 1.3 goes into the strictification of VBun Dfg .
Further, we deduce from Corollary 1.2 that various flavours of the 2-category of bundle gerbes with connection as introduced by Waldorf [Wal07] extend to sheaves of (2, 2)-categories on Cart (Theorem 4.21); via a 2-categorical nerve construction, they further provide examples of homotopy sheaves of (∞, 2)-categories. Our motivation for working model categorically up to this point stems from future applications to field theories: these are functors out of (∞, n)-categories of cobordisms, which exists as n-fold or n-uple (complete) Segal spaces [Lur09b,CS19a,SP17]. This also feeds into another application: in smooth field theories on a background manifold, one detects the full smooth structure of diffeomorphism groups rather than only their connected components (i.e. mapping class groups).
We show how one can nevertheless coherently and smoothly reconstruct the values of a field theory on all objects from its values on only the generating objects of the bordism category (Theorems 4.34 and 4.35).
In Section 5 we answer the second question above. To do so, we pass to quasi-categorical language: let C be a quasi-category with a Grothendieck (pre)topology τ . Let C ∞ denote the quasi-category of (small) presheaves of spaces on C. We write Sh(C, τ ) for the quasi-category of τ -sheaves on C and Sh ! ( C ∞ , τ ) for the quasi-category τ -sheaves on C ∞ whose underlying functor C ∞ → S op preserves colimits.
Theorem 1.4 Let (C, τ ) be as above, and let Y * denote the ∞-categorical right Kan extension of presheaves along the Yoneda embedding Y : C op → C op ∞ . The adjunction Y * ⊣ Y * restricts to an equivalence of quasi-categories Finally, to make contact with the first part of this paper, we prove a similar result where C = NC is an ordinary category; this relates quasi-categorical τ -sheaves of spaces on NC to τ -sheaves of spaces on the 1-category of set-valued presheaves C = Fun(C op , Set L ) (Theorem 5.14).

Conventions
Sizes and universes. Throughout, we choose and fix a nested triple of Grothendieck universes S ∈ L ∈ XL 3 , and we assume that S contains the natural numbers. We write Set S and Set ∆S for the categories of S-small sets and S-small simplicial sets, respectively, and analogously for the universes L and XL. As is common in the literature, we refer to the elements in Set S simply as small sets, those of Set L as large sets and those of Set XL as extra large sets. All indexing sets will be assumed to be S-small.
Let C be a small category. Consider the category C := Fun(C op , Set S ) of Set S -valued presheaves on C.
Observe that C is no longer small, since Set S is not small. However, since Set S is a large category (its objects and morphisms each form sets in L), it follows that C is a large category.
The Yoneda embedding Y : C → Fun( C op , Set L ) is fully faithful; observing that an object of Set S is also an object of Set L , the standard proof applies. Further, the Yoneda Lemma holds true for both Y and Y (again by the usual method of proof). Finally, observe for any c ∈ C and for any X ∈ C, by the Yoneda Lemma, there are canonical isomorphisms Categories and higher categories. In this article we use two different models for higher categories. Section 2 and Appendix A do not make use of higher categories. In Sections 3 and 4 we use n-fold complete Segal spaces (introduced in [Bar05]) as a model for (∞, n)-categories. In particular, the term '∞-groupoid' shall always refer to a Kan complex, and '(∞, n)-category' shall always mean a n-fold complete Segal space. In Section 5, and only there, all higher categories are modelled as quasi-categories in the sense of [BV73,Joy02,Lur09a]. In order to make clear the distinction between ordinary categories (possibly endowed with model structures), n-fold complete Segal spaces, and quasicategories, we denote ordinary categories by script letters C, D, . . ., n-fold complete Segal spaces (or presheaves thereof) by ordinary capitals F, G, . . ., and quasi-categories by sans-serif capitals C, D, . . .. In particular, the quasi-categories of small spaces is denoted S S , and analogously for the universes L and XL. The term 'small space' (used on its own, as opposed to 'Segal space', for instance) shall always mean an object in the quasi-category S S of small spaces, and analogously for the terms 'large space' and 'extra large space' and the universes L and XL, respectively.
Enriched categories. If V is a monoidal category and C is a V-enriched category (or V-category), we denote the V-enriched hom-objects of C by C V (−, −). If V = Set ∆L is the category of simplicial sets, we also write C(−, −) := C Set ∆L (−, −). If V is a symmetric monoidal model category and C is a V-enriched model category (in the sense of [Hov99, Sec. 4]), we will equivalently say that C is a model V-category.
Tractability. We recall from [Bar10, Def. 1.21] that a large model category M is tractable if there exists a regular L-small cardinal λ such that the category underlying M is locally λ-presentable, and there exist L-small sets I, J of morphisms with the following properties: the source and target of the morphisms in I and J are λ-presentable and cofibrant, the fibrations (resp. trivial fibrations) in M are precisely those morphisms satisfying the right lifting property with respect to J (resp. I).
In particular, M is combinatorial. For further details and background on the formalism of enriched Bousfield localisation, we refer the reader to [Bar10].
Diagrams. For J a large category and C a L-tractable or L-combinatorial model category (cf. [Bar05,Bar10]), the projective and the injective model structures on Fun(J, C) exist and are again L-tractable (resp. L-combinatorial); we denote them by Fun(J, C) proj and Fun(J, C) inj , respectively. If J is a Reedy category, we denote the Reedy model structure on Fun(J, C) by Fun(J, C) Reedy .
Left Bousfield localisation. We recall that if M is a large simplicial model category with a chosen collection of morphisms A, then is a weak homotopy equivalence in Set ∆L , and is a weak equivalence in Set ∆L .

Grothendieck sites and local epimorphisms
We recall the definition of a Grothendieck pretopology and a site. Throughout this article, we assume that C is a small category.
Definition 2.1 We write C := Fun(C op , Set S ) for the large category of small presheaves on C.
Let Y : C → C, c → Y c , denote the Yoneda embedding of C.
(1) A coverage, or Grothendieck pretopology, on C is given by assigning to every object c ∈ C a small set τ (c) of families of morphisms {f i : c i → c} i∈I (with I ∈ Set S ) satisfying the following properties: {1 c } ∈ τ (c) for each c ∈ C, and for every family {f i : c i → c} i∈I ∈ τ (c) and any morphism g : such that for every j ∈ J we find some i ∈ I and a commutative diagram The families in {f i : c i → c} i∈I ∈ τ (c) are called covering families for c.
(3) A (Grothendieck) site is a category C equipped with a coverage τ .
Later we will use the following technical condition: We call a site (C, τ ) closed if it satisfies the following condition: let {c i → c} i∈I be any covering family in (C, τ ). Further, for each i ∈ I, let {c i,j → c i } j∈J i be a covering family in (C, τ ).
Then, there exists a covering family {d k → c} k∈K such that every morphism d k → c factors through one of the composites c i,j → c i → c.
Example 2.4 Let Cart be the category of cartesian spaces, i.e. of sub-manifolds of R ∞ that are diffeomorphic to some R n , with smooth maps between these manifolds as morphisms. This category is small and has finite products. A coverage on Cart is defined by calling a family {ι i : c i → c} i∈I a covering family if it satisfies (1) all ι i are open embeddings (in particular dim(c i ) = dim(c) for all i ∈ I), (2) the ι i cover c, i.e. c = i∈I ι i (c i ), and (3) every finite intersection ι i 0 (c i 0 ) ∩ · · · ∩ ι im (c im ), with i 0 , . . . , i m ∈ I, is a cartesian space. Definition 2.6 Let (C, τ ) be a site. A morphism π : Y → X in C is a τ -local epimorphism if for every c ∈ C and each morphism ϕ : Y c → X there exists a covering {f j : c j → c} j∈J ∈ τ (c) and morphisms One directly checks the following: Lemma 2.7 For any site (C, τ ), the class of τ -local epimorphisms is stable under pullback. In particular, the collection of τ -local epimorphisms defines a coverage τ on C.
Note that in the Grothendieck site ( C, τ ) every covering family consists of a single morphism. We list some general properties of τ -local epimorphisms: Lemma 2.8 Let (C, τ ) be a site.
(2) For any covering family {c i → c} i∈I , the induced morphism i∈I Y c i → Y c is a τ -local epimorphism.
(3) τ -local epimorphisms are stable under colimits: let J be a small category, let D ′ , D : J → C be diagrams in C, and let π : D ′ → D be a morphism of diagrams such that each component π j : D ′ j → D j is a τ -local epimorphism, for any j ∈ J. Then, the induced morphism colim π : colim D ′ → colim D is a τ -local epimorphism.
(4) If X(c) = ∅ for each c ∈ C, then the projection X × Z → Z is a τ -local epimorphism, for any Z ∈ C.  Claim (4) is immediate since by assumption the projection is surjective for each c ∈ C; we can hence use the identity covering of c to obtain the desired lift.
For (5), it readily follows from the closedness of (C, τ ) that τ -local epimorphisms are stable under composition, and thus also that ( C, τ ) is closed. On the other hand, assume that τ -local epimorphisms are stable under composition. Consider an object c ∈ C, a covering family {f i : c i → c} i∈I , and for each i ∈ I a covering family {c i,k → c i } k∈K i . By claims (2) and (3), we thus obtain τ -local epimorphisms By assumption, their composition is a τ -local epimorphism again, and hence it follows (using that C(Y c , −) preserves colimits) that (C, τ ) is closed. Finally, assume that ( C, τ ) is closed. Let f : Z → Y and g : Y → X be τ -local epimorphisms, and let p := g • f . By assumption, there exists a τ -local epimorphism Z ′ → X which factors through g; that is, there exists a morphism q : Z ′ → Z such that p • q is a τ -local epimorphism. But then p is a τ -local epimorphism by claim (1).
Let (C, τ ) be a site, and consider a covering U = {c i → c} i∈I . We can form its Čech nerve, which is the simplicial object in C whose level-n object reads aš Note that, in general, C i 0 ...in ∈ C is not representable as soon as n = 0. The simplicial structure morphisms are given by projecting out or doubling the i-th factor, respectively. Depending on the context we will view the Čech nerveČU either as a simplicial objectČU • in C or as an augmented simplicial objectČU • → Y c in C. For convenience, we recall the following definitions: Definition 2.10 Let (C, τ ) be a site, and let X ∈ C be a presheaf on C. Then, X is called a sheaf on (C, τ ) if for every object c ∈ C and for every covering family {f i : c i → c} i∈I ∈ τ (c) the diagram is an equaliser diagram in Set L . The category Sh(C, τ ) of sheaves on (C, τ ) is the full subcategory of C on the sheaves.

Homotopy sheaves and descent along local epimorphisms
Let (C, τ ) be a small site, and let Y : C → C denote its Yoneda embedding. In this section we show that the homotopy right Kan extension along Y * maps large higher τ -sheaves on C to large higher τ -sheaves on C. Here, τ is the pretopology of τ -local epimorphisms on C (see Definition 2.6 and Lemma 2.7).

Sheaves of ∞-groupoids
Let C be a small category. Let H ∞,0 := Fun(C op , Set ∆L ) denote the extra-large category of large simplicial presheaves on C . It is enriched, tensored and cotensored over Set ∆L . Composing the Yoneda embedding with the functor c • : Set L → Set ∆L , we obtain a fully faithful functor Y : C → H ∞,0 . From now on, we view the category H ∞,0 as endowed with the projective model structure. (1) The two-sided simplicial bar construction of (G, H) is the simplicial object (2) The two-sided bar construction of (F, G) is the realisation of B • (G, I, H), i.e.
For later reference, we record the following direct consequences of this definition:  (3) In the above setting, let additionally I = ∆ op . Then there is a natural morphism where diag denotes the functor taking the diagonal of a bisimplicial set. This morphism is a weak equivalence of simplicial sets.  Definition 3.6 We let Q : H ∞,0 → H ∞,0 denote the functor whose action on F ∈ H ∞,0 is given by Observe that for X a simplicially constant simplicial presheaf on C, we have a canonical isomorphism Definition 3.8 Let (C, τ ) be a small Grothendieck site, and letτ denote the class of morphisms in H ∞,0 consisting of Čech nerves of coverings in (C, τ ) (see (2.9)). Since C is small, this is a small set. We let H loc ∞,0 denote the left Bousfield localisation H loc ∞,0 := Lτ H ∞,0 of the large category H ∞,0 . A sheaf of ∞-groupoids on (C, τ ) is a fibrant object in H loc ∞,0 . The model structure on H loc ∞,0 is also called the local projective model structure on Fun(C op , Set ∆L ).
is a Kan complex for every c ∈ C, and (2) for every covering family U = {c i → c} i∈I in (C, τ ), the morphism is a weak equivalence in Set ∆L (compare (2.9)). Here we have used that Q is a left adjoint.
where the argument of the end is given by the internal hom in Set ∆L . In particular, there is a natural isomorphism H ∞,0 (Y c , F ) ∼ = F (c) for any c ∈ C and any F ∈ H ∞,0 .
Since C is a small category (hence also large, since S ∈ L) and Set S is a large category, We define the extra-large category Since Set ∆L is L-tractable, the projective model structure on H ∞,0 exists and is itself L-tractable [Bar10, Thm. 2.14]. We will always view H ∞,0 as endowed with this projective model structure.
By Proposition 2.7, if (C, τ ) is a small site, then ( C, τ ) is a large site. Let ( τ )ˇdenote the large set of Čech nerves of coverings in ( C, τ ) (this is a large set since C is large). Since H ∞,0 is left proper and L-tractable, and ( τ )ˇis L-small, we can form the left Bousfield localisation We can compute the right adjoint as (Note that here we view X ∈ C as an object in H ∞,0 via the canonical inclusions Set S ֒→ Set L ֒→ Set ∆L .) We now ask whether one of the functors Y ! or Y * maps sheaves of ∞-groupoids on (C, τ ) to sheaves of ∞-groupoids on ( C, τ ), i.e. preserves fibrant objects as a functor H loc ∞,0 → H loc ∞,0 . Since this indicates that we are looking for a right Quillen functor, we focus on the right adjoint Y * . In general, Y * will not even preserve fibrant objects as a functor H ∞,0 → H ∞,0 . However, ifQ : H ∞,0 → H ∞,0 is a functorial cofibrant replacement, then the functor Remark 3.11 The above holds true for any cofibrant replacement functorQ in H ∞,0 . However, for technical reasons we always use Dugger's functor Q from (3.7) for cofibrant replacement in the projective model structure of simplicial presheaves in the remainder of this article. We abbreviate S ∞,0 := S Q ∞,0 . ⊳ Let F ∈ H ∞,0 and X ∈ C. Using that QX is a bar construction we compute where C Set ∆L and B H ∞,0 are the cobar and bar constructions in Set ∆L and in H ∞,0 , respectively. If F is projectively fibrant, then the last expression is a model for the homotopy limit (see also recalled as Proposition 3.31 below), Now, using the Yoneda Lemma and [Rie14, Ex. 9.2.11] we conclude: Proposition 3.12 If F ∈ H ∞,0 is fibrant, S ∞,0 F is the homotopy right Kan extension of F along the (opposite of the) Yoneda embedding of C, i.e.
Remark 3.13 In view of Section 5, we emphasize that S ∞,0 is a presentation of the ∞-categorical right Kan extension of presheaves of spaces on C to presheaves of spaces on C. ⊳ In order to show that S ∞,0 is a right Quillen functor, it remains to show that it is a right adjoint. Since C is a large category, C-indexed (co)ends exist in Set ∆L , the extra-large category of large simplicial sets. Therefore, a left adjoint to S ∞,0 is given by We also set Lemma 3.14 There is a canonical natural isomorphism Thus, Re ∞,0 is also a left adjoint to S ∞,0 .
Proof. Using (3.7), we have the following natural isomorphisms where the isomorphism is an application of the Yoneda Lemma for C.
Proposition 3.15 The functors Re ∞,0 and S ∞,0 give rise to a simplicial Quillen adjunction which sits inside the following non-commutative diagram Further, there exists a natural isomorphism η : Proof. We have already shown above that Re ∞,0 ⊣ S ∞,0 and that S ∞,0 preserves fibration and trivial fibrations. The rest is then a direct application of [Dug01,Prop. 2.3]. In the present case, η as written down there turns out to be an isomorphism as a consequence of the Yoneda Lemma: recall that Re ∞,0 = Q • Y * and that there is a canonical isomorphism Proposition 3.16 The functors Re ∞,0 and S ∞,0 have the following properties: (1) The functor Re ∞,0 : H ∞,0 → H ∞ is homotopical.
(2) Let Q : H ∞,0 → H ∞,0 denote the cofibrant replacement functor on H ∞,0 defined in analogy with (3.7). For each fibrant object F ∈ H ∞,0 , there is a zig-zag of weak equivalences which is natural in F .
Proof. By Lemma 3.14, Re ∞,0 ∼ = Q • Y * . The functor Y * is homotopical since weak equivalences in both H ∞,0 and in H ∞,0 are defined objectwise. Thus, claim (1) follows since Q is homotopical. As a consequence, the morphism is a weak equivalence. For F ∈ H ∞,0 , we find that and since F is fibrant and Y c is cofibrant, the functor H ∞,0 (−, F ) preserves the weak equivalence To complete the proof, we apply the homotopical functor Q to this morphism.
We now investigate whether the Quillen adjunction Re ∞,0 ⊣ S ∞,0 between the projective model structures descends to a Quillen adjunction between the local projective model structures. We recall the following standard result:  Let C be a small category endowed with a coverage τ , and let π : Y → X be a τ -local epimorphism in C with Čech nerveČπ. The augmentation map π [•] :Čπ → X is a weak equivalence in H loc ∞,0 . The proof of Theorem 3.18 is rather technical; it requires a "wrestling match with the small object argument" ([DHI04] Subsection A.12). We refer the reader to that reference for details.

Remark 3.19
In [DHI04], Theorem 3.18 is proven for the Čech localisation of the injective model structure H i on H. However, it also holds true in H ∞,0 because the injective and projective model structures on H (and H) have the same weak equivalences, and so do the associated local model structures (see also [Bun22,Prop. 2 The top morphism is an objectwise weak equivalence, and the diagonal morphisms are weak equivalences in H loc ∞,0 . Proof. The top morphism is induced by the Bousfield-Kan, or last-vertex map. It is an objectwise weak equivalence by Lemma 3.4(3). The right-hand side is a local weak equivalence by Theorem 3.18. Thus, it remains to show that the triangle commutes. We check this explicitly: for each object c ∈ C, there are canonical natural isomorphisms where N denotes the nerve functor. For X ∈ Set ∆S , recall the last-vertex map N ∆ /X −→ X of small simplicial sets (see, for instance, [Cis19, Lemma 7.3.11, Par. 7.3.14]). We describe this explicitly in our case, i.e. for X =Čπ(c). An n-simplex in this simplicial set is explicitly given by a sequence of morphisms α 01 α 01 α n−1,n κ of small simplicial sets. We denote these data by (α, κ). Consider the map ϕ α : The top morphism in diagram (3.21) acts by sending the n-simplex (α, κ) to the n-simplex ofČπ(c) which is given by the composition where the first map is the one constructed above from the data (α, κ). Explicitly, for a given n-simplex α ∈ N∆, the resulting map is the projection onto the Y -factors in positions ϕ α (0), . . . , ϕ α (n). This map commutes with the maps to X.
Proof. Let π : Y → X be a τ -local epimorphism in C, and let F ∈ H loc ∞,0 be fibrant. We need to show that the canonical morphism is a weak equivalence in Set ∆L . Substituting the definition of S ∞,0 , this is the same as the canonical map Observe that there is a canonical isomorphism in the homotopy category HoSet ∆L (in fact, this can even be modelled as an isomorphism in Set ∆L in light of Lemma 3.4(1) and Proposition 3.31 below). Since the functor Q is a bar construction (see (3.7)), it commutes with the homotopy colimit (which is also a bar construction) by Lemma 3.4(1). Then, the morphism is the image of the left-hand diagonal morphism in the commutative triangle (3.21) under the functor Combining this with Lemma 3.20, we obtain that the above morphism is a weak equivalence in Set ∆L .
Combining Proposition 3.17 and Proposition 3.22, we obtain ⊥ In Section 5 we provide a version of Theorem 3.23 for fully coherent diagrams of spaces in a quasicategorical language; on the underlying quasi-categories, we will show that S ∞,0 is fully faithful and we identify its essential image.

Homotopy sheaves with values in simplicial model categories
We now extend the results from Section 3.1 to homotopy sheaves on a small site (C, τ ), valued not just in Set ∆L , but in any large, combinatorial or cellular simplicial model category. We begin by recalling some technology for enriched model categories.
Recall the notion of a V be a symmetric monoidal model category. A V-enriched model category, or model V-category M (see, for instance, [Hov99, Def. 4.2.18]). We denote the three functors which are part of the enrichment by We import the following definition (using [Rie14, Thms. 7.6.3, 11.5.1]): Definition 3.24 Let I be a small category.
(1) The weighted I-colimit functor of M is the functor (2) The weighted I-limit functor of M is the functor

Consider the functors
They form an adjoint pair Proposition 3.30 The adjunction (3.29) is a Quillen adjunction with respect to the projective model structures.
Proof. Let X ∈ C, and consider the functor Here we have used the notation from (3.26). By Theorem 3.27 and [Hov99, Rmk. 4.2.3] this functor is right Quillen, for each X ∈ C. Since fibrations and weak equivalences in the projective model structure on Fun( C op , M) are defined objectwise, it follows that the functor S M preserves projective fibrations and trivial fibrations.
We recall some useful technology for computing homotopy limits from [Rie14] (see, in particular, Chapters 4-6). Let M be a large simplicial model category (enriched, tensored and cotensored over Set ∆L ). Let I be a large category and consider functors G : I → Set ∆L and F : I → M. The (two-sided) cosimplicial cobar construction associated to these data is the cosimplicial object in M whose l-th level reads as where NI ∈ Set ∆L denotes the nerve of I. The (two-sided) cobar construction C(G, I, F ) in M is the totalisation formed in M. If the ambient model category M is not clear from context, we will write C M for the cobar construction in M.
Proposition 3.31 [Rie14, Cor. 5.1.3] The cobar construction is a model for the homotopy limit: let M be a large simplicial model category with fibrant replacement functor R M , and let F : I → M a functor as above. Then, holim i∈I Definition 3.33 Suppose M is left proper, as well as cellular or combinatorial. Let τ be a Grothendieck coverage on C, and τ the induced Grothendieck coverage of τ -local epimorphisms on C.
(1) Let S(M, τ ) be the collection of morphisms in Fun(C op , M) of the form (2) Let S(M, τ ) be the collection of morphisms in Fun( C op , M) of the form in M is a weak equivalence (whereČ k U is viewed as a simplicially constant presheaf on C).
(2) An object G ∈ Fun( C op , M) is S(M, τ )-local if and only if it is projectively fibrant and, for each τ -covering π : Y → X in C, the induced morphism Remark 3.35 Suppose that (C, τ ) is such that, for each τ -covering U = {c i → c} i∈I , each presheaf C i 0 ···i k from (2.9) is again representable. In that case, the Čech nerveČU is projectively cofibrant (it is then objectwise a coproduct of representables), and so we can omit the cofibrant replacement functor Q from Definition 3.33(1) and Proposition 3.34. Furthermore, the enrichment and the Yoneda Lemma provide a canonical isomorphism is a weak homotopy equivalence in Set ∆ (note that the left-hand arguments in both functors above are cofibrant in Fun(C op , M), and so the above simplicially enriched hom spaces indeed compute mapping spaces). By adjointness, this morphism is isomorphic to the morphism Next, we use the isomorphisms in the homotopy category HoSet ∆L , Here we have used that we can commute Q and the homotopy colimit, since both are bar constructions. The claim now follows from the fact that a morphism f : a → b in M between fibrant objects is a weak equivalence if and only if, for each cofibrant m ∈ M, the induced morphism  ⊥ Proof. By Proposition 3.17, it suffices to show that the right adjoint S M preserves local objects. To that end, let F ∈ Fun(C op , M) be S(C, τ )-local, and let π : Y → X be a τ -local epimorphism in C. By Proposition 3.34 we thus need to show that the canonical morphism The problem is thus equivalent to showing that the canonical morphism is a weak homotopy equivalence. Here we have used that F is projectively fibrant, so that the functor {Q(−), F } C op takes values in M f ib , and that the homotopy limit of an objectwise fibrant diagram is again a fibrant object. Using the simplicial enrichment and the two-variable Quillen adjunction from Proposition 3.28, the above problem is further equivalent to showing that, for each cofibrant m ∈ M, the morphism ) is a weak homotopy equivalence. Note that this morphism is the same as the canonical morphism . By Theorem 3.23 it now suffices to show that the simplicial presheaf To that end, let U = {c i → c} i∈I be a τ -covering in C. We have to check that the canonical morphism is a weak homotopy equivalence. Again using the two-variable Quillen adjunctions, this is equivalent to checking that is a weak equivalence in M. However, that is true by the assumption that F is S(M, τ )-local.
Finally, we record a generalisation of Proposition 3.12: Proposition 3.38 In the setting of Theorem 3.30, for each projectively fibrant F ∈ Fun(C op , M) and any X ∈ C, there is a canonical natural weak equivalence Proof. We have canonical weak equivalences In the second-to-last line we have used the enriched cobar construction for the simplicially enriched category M. The final weak equivalence follows from Proposition 3.31.

Sheaves of (∞, n)-categories
The specific instances of Theorem 3.37 in which we are interested most arise from choosing the target model category M to be a model for (∞, n)-categories. Various such model categories exist in the literature, including the model structures for n-fold complete Segal spaces [Rez01, Bar05,BSP11] and Θ n -spaces [Rez10a,Rez10b] (see also [BSP11] for a general characterisation of such models, as well as [BR13a,BR20] for a comparison between the aforementioned two models). Theorem 3.37 applies to both homotopy sheaves of n-fold complete Segal spaces and Θ n -spaces. Since this paper is motivated by applications to functorial field theories, where n-fold complete Segal spaces are the prevalent model for (∞, n)-categories, we will focus on the case of n-fold complete Segal spaces.
For each n ∈ N 0 , we consider the category Further, the category s n Set ∆L is cartesian closed, with internal hom given by where Y ∆ n : ∆ n → Set ∆L is the Yoneda embedding of ∆ n . For each n ∈ N, there is a monoidal adjunction ..,k n−1 . Therefore, the category s n Set ∆L is enriched, tensored, and cotensored over s k Set ∆ for any 0 ≤ k ≤ n.
Moreover, for each k, l ∈ N there is a functor Iterating this, we have a functor For k ∈ N, k ≥ 2, define the k-th spine inclusion in Set ∆L . Localising at maps of this type will implement the Segal conditions.
Let K ∈ Set ∆L denote the pushout where the top morphism consists of the inclusions of ∆ 1 into ∆ 3 as the edge 0 → 2 and 1 → 3, respectively. Let κ denote the collapse map Localising at maps of the type κ will implement the completeness conditions below.
Definition 3.39 Let n ∈ N. The model category for (large) n-fold complete Segal spaces is obtained by left Bousfield localisation of s n Set ∆L at the following morphisms: (1) (Segal conditions) For each k ∈ N, i ∈ {0, . . . , n − 1} and each k 0 , . . . , k i−2 , k i , . . . , k n−1 ∈ N 0 , we localise at the morphism (2) (Completeness) For each i ∈ {0, . . . , n − 1} and each k 0 , . . . , k i−2 , k i , . . . , k n−1 ∈ N 0 , we localise at the morphism (3) (Essential constancy) For each i ∈ {0, . . . , n − 1} and each k 0 , . . . , k i−2 , k i , . . . , k n−1 ∈ N 0 , we localise at the morphism We denote the resulting model category by CSS n . Fibrant objects in CSS n are called n-fold complete Segal spaces, or (∞, n)-categories. (1) The homotopy limit of D can be computed levelwise: for each k ∈ N 0 , there exist canonical natural isomorphisms holim CSSn (2) The homotopy limit of D can be computed objectwise when viewing the diagram as a functor D : J × ∆ n → Set ∆L . That is, for any k ∈ ∆ n there is an isomorphism (natural in k ∈ ∆ n ) holim CSSn We use the explicit model for homotopy limits in simplicially enriched categories from Proposition 3.31. For the sake of brevity, for l ∈ N 0 , we set Since D is objectwise fibrant, a model for the homotopy limit of D is given by the two-sided cobar construction C( * , I, D) ∈ CSS n [Rie14, Cor. 5.1.3].
For any n ∈ N 0 , there are canonical natural isomorphisms of simplicial sets The last identity relies on [Rie14, Cor. 5.1.3] (see also Proposition 3.31), together with the following facts: the functor D : I → CSS n is objectwise fibrant. In particular, for each i ∈ J, the object D(i) ∈ Fun(∆ op , s n−1 Set ∆L ) is fibrant in the Reedy model structure (equivalently, the injective model structure), where s n−1 Set ∆L carries the Reedy (equivalently, injective) model structure. Forgetting the localisations in Definition 3.39 at morphisms which are non-trivial in the first ∆-factor, we further obtain that (Di) m is fibrant in CSS n−1 , for each i ∈ I and m ∈ ∆. Consequently, the functor D m : ∆ → CSS n−1 , i → D m (i) = (Di) m is also fibrant in the projective model structure on Fun(∆ op , CSS n−1 ). Thus, the cobar construction in the second-to-last line indeed models its homotopy limit. Part (2) follows in analogy with part (1), replacing Y n with Y (i.e. replacing ∆ k ⊠ ∆ 0 ⊠ · · · ⊠ ∆ 0 by ∆ k 0 ⊠ · · · ⊠ ∆ k n−1 ) and using that injectively fibrant diagrams are, in particular, projectively fibrant.
Definition 3.42 Let C be a small category.
(1) We write H ∞,n := Fun(C op , CSS n ) and view this as endowed with the projective model structure. A fibrant object in H ∞,n is called a presheaf of (∞, n)-categories on C.
A fibrant object in H loc ∞,n is called a sheaf of (∞, n)-categories on (C, τ Analogously to Sections 3.1, H ∞,n is symmetric monoidal whenever C has finite products. Definition 3.44 Let C be a small category. As before, we write C = Fun(C op , Set S ).
(1) We write H ∞,n := Fun( C op , CSS n ) for the category Fun( C op , CSS n ), endowed with the projective model structure. A fibrant object in H ∞,n is called a presheaf of (∞, n)-categories on C.
(2) In the notation of Definition 3.33, we define the left Bousfield localisation H loc ∞,n := L S(CSSn, τ ) H ∞,n . A fibrant object in H loc ∞,n is called a sheaf of (∞, n)-categories on ( C, τ ). Remark 3.45 Lemma 3.41 implies that an object F ∈ Fun(C op , CSS n ) is S(CSS n , τ )-local if and only if, for each [ k] ∈ ∆ n the object F k 0 ,...,k n−1 ∈ Fun(C op , Set ∆L ) is τ -local, and analogously for ( C, τ ) in place of (C, τ ). ⊳ In the notation of (3.29), we further write We can compute this functor more explicitly as follows: for [ k] ∈ ∆ n , we have canonical isomorphisms We have the following direct corollary of Proposition 3.38: Proposition 3.46 On any fibrant object F ∈ H ∞,n , the functor S ∞,n agrees with the homotopy right Kan extension of F : C op → CSS n along the Yoneda embedding, As a direct application of Proposition 3.30, we have: Now let (C, τ ) be a site. One checks that, given a presheaf F : C op → Cat L and a τ -covering family U = {c i → c} i∈Λ such that all C i 0 ...in are representable (see (2.9) for the notation), there is an equivalence in CSS 1 , where the homotopy limit is computed in the canonical model structure on Cat L (see, for instance, [Rez96]). We define a functor Given a τ -local epimorphism π : Y → X, we thus have a commutative diagram Since N rel reflects weak equivalences, it now follows from Corollary 3.48 that S 1,1 F satisfies τ -descent whenever F satisfies τ -descent. ⊳

Applications in geometry and field theory
Here we apply the results of Section 3 to diffeological vector bundles, gerbes with connection, 2-vector bundles and smooth field theories on background manifolds.

Strictification of diffeological vector bundles and descent along subductions
Vector bundles form a sheaf of categories on the site of manifolds and surjective submersions. One can show this either by providing an explicit descent construction, or by using a strictification of vector bundles which describes them in terms of their transition functions.
Diffeological spaces form a popular generalisation of manifolds; their category Dfg sits in a sequence of canonical fully faithful inclusion functors Apart from the first inclusion, these functors each have a left adjoint. Diffeological spaces have been introduced in [IZ13] and are increasingly used to carry out smooth constructions in geometry and topology (see, for instance, [Kih20,Min22]). Despite this, there is currently no descent theorem for diffeological vector bundles. That is a large gap in the literature, which we close here: as the main step, we extend the strictification of vector bundles on manifolds to diffeological vector bundles. The desired descent property of diffeological vector bundles then follows readily from Corollary 3.48 and Remark 3.49.

Background on diffeological spaces
We start by recalling the category of diffeological spaces, mostly following [BH11]. As a slight variation to the standard literature, we define diffeological spaces as concrete sheaves on a site which differs from the usual choice, but this is just for technical convenience (cf. Remark 4.7). Throughout this section, we let C be a small category.  (1) The functor Ev * = C( * , −) : C → Set S is faithful.
(2) Each covering {f i : c i → c} i∈I ∈ τ (c) is jointly surjective: we have i∈I (f i ) * C( * , c i ) = C( * , c) . In other words, a concrete presheaf X has an underlying set X( * ), and, for each c ∈ C, the set X(c) can canonically be described as a subset of the maps of sets Y c ( * ) → X( * ).
We commit an abuse of notation and denote the underlying set of a diffeological space X also by X; it will be clear from context whether we are referring only to the underlying set, or to the underlying set endowed with the structure of a diffeological space. from Example 2.5 is concrete.

⊳
The relation between Sections 3 and Section 4 relies on the following observation.
Remark 4.6 The category Dfg(C, τ ), together with the collection of those τ -local epimorphisms whose source and target are (C, τ )-spaces, form a site which is contained in the site ( C, τ ) as a full subcategory. The τ -local epimorphisms between (C, τ )-spaces are also called subductions [IZ13]. We denote the site of (C, τ )-spaces with this coverage by (Dfg(C, τ ), τ subd ). ⊳ Remark 4.7 In [IZ13], diffeological spaces are defined as concrete presheaves on (Op, τ op ), whereas here we define them on (Cart, τ dgop ). However, the two categories are equivalent because the canonical inclusion of (Cart, τ dgop ) into (Op, τ op ) is an inclusion of a dense subsite. For any manifold M , the presheaf M from Example 2.4 is a diffeological space. ⊳

Strictification and descent of diffeological vector bundles
We now specialise to the case (C, τ ) = (Cart, τ dgop ) and to Dfg = Dfg(Cart, τ dgop ). We will denote the underlying set Y c ( * ) of a cartesian space c ∈ Cart again by c.
Definition 4.9 Let X ∈ Dfg be a diffeological space.
(1) A (complex rank-k) vector bundle on X is a pair (E, π) of a diffeological space E and a morphism π : E → X in Dfg with the structure of a C-vector space on each fibre E |x := π −1 ({x}), for x ∈ X, satisfying the following condition: for each plot ϕ : c → X there exists an isomorphism of diffeological spaces such that pr c • Φ = pr c and such that for every y ∈ c the restriction Φ |y : C k → E |ϕ(y) is linear.
Example 4.10 If M is a smooth manifold, which we view as a diffeological space, then the category VBun Dfg (X) is canonically equivalent to the ordinary category of vector bundles on M .
⊳ Any morphism f ∈ Dfg(X ′ , X) gives rise to a pullback functor where π ′ is the pullback of π along f .

Definition 4.11
We let VBun Dfg : Dfg op → Cat L denote the pseudo-functor resulting from the above assignments (where Cat L is the 2-category of large categories).
The goal of this subsection is to prove the following theorem: Remark 4.13 We use the following strategy for the proof of Theorem 4.12: (1) Replace the pseudo-functor VBun Dfg by a strictification, i.e. find a (strict) functor Theorem 4.12 then readily follows from the equivalences in points (2) and (3).
We will achieve the above goals by means of Remark 3.49: we give a strict functor VBun triv : Cart op → Cat L satisfying the following two properties: • The functor VBun triv : Cart op → Cat L satisfies τ dgop -descent.

Remark 3.49 then ensures that points (2) and (3) above are satisfied. ⊳
We now carry out the above strategy: Definition 4.14 We define a strict presheaf VBun triv of categories on Cart as follows: for c ∈ Cart, the category VBun triv (c) has objects (c, n), where n ∈ N 0 , and a morphism (c, n) → (c, m) is a smooth function ψ : c → Mat(m×n, C) from c to the vector space of complex m-by-n matrices; to make clear that ψ is defined over c we also write (c, ψ) instead of just ψ. A smooth map f : c ′ → c acts via Using the functor S 1,1 : Fun(C op , Cat L ) −→ Fun( C op , Cat L ) from Remark 3.49 we also define the strict functor Proposition 4.15 Both N rel • VBun triv and N rel • VBun str are fibrant objects in H loc ∞,1 and in H loc ∞,1 , respectively. Equivalently (by Remark 3.49), VBun triv satisfies τ dgop -descent, and VBun str satisfies τ dgop -descent.
Proof. We only need to show that VBun triv ∈ H loc ∞,1 is fibrant. That VBun str is fibrant in H loc ∞,1 will then follow from Corollary 3.48. First, VBun triv is fibrant in H ∞,1 : for each c ∈ Cart, the bisimplicial set VBun triv (c) = N rel (VBun triv (c)) is the classification diagram of an ordinary category and hence a complete Segal space by [Rez01, Prop. 6.1]. In order to show that VBun triv is also fibrant in H loc ∞,1 , it suffices to show that the presheaf of categories VBun triv satisfies descent with respect to τ dgop , which is standard: for instance, one can see this by using that vector bundles glue along isomorphisms over open coverings of manifolds. Thus, given descent data for VBun triv with respect to a differentiably good open covering of a cartesian space c, we obtain a vector bundle on c. This vector bundle is, in general, non-trivial, but since c ∼ = R n for some n ∈ N 0 , it is isomorphic to a trivial vector bundle. This provides the essential surjectivity of the descent functor for vector bundles. Its full faithfulness follows since morphisms of trivial vector bundles are the same as smooth matrix-valued functions, which form sheaves of sets with respect to open coverings of manifolds.
For the proof of Theorem 4.12 we are thus left to show that VBun str ∈ Fun( Cart op , Cat L ) restricts to a strictification of the pseudo-functor VBun Dfg on the full subcategory Dfg ⊂ Cart. To that end, we first identify a more concrete description of the functor ι * VBun str = ι * S 1,1 (VBun triv ), where ι : Dfg ֒→ Cart denotes the canonical inclusion: it is the homotopy right Kan extension of VBun triv along the inclusion Cart ֒→ Dfg: Lemma 4. 16 We can compute the values of ι * VBun str : Dfg op → Cat L as Proof. This follows directly from Remark 3.49.

Remark 4.17
In this proof we first show that each diffeological vector bundle is determined by its strictified transition data (i.e. matrix-valued functions), and then use that the latter satisfy descent, using Remark 3.49. Alternatively, we could have shown first an analogue of Lemma 4.16 for VBun Dfg and then compared ι * VBun Dfg and VBun str on Cart ⊂ Dfg only. However, the proof of the first step is not significantly less complicated than the proof of Theorem 4.18 below. Additionally, from the route chosen here it becomes evident how one can assign a total space to any vector bundle on any diffeological space from its transition data, as well as that N rel VBun str is a classifying object in H ∞,1 for vector bundles. ⊳ We obtain the following strictification theorem for diffeological vector bundles: Theorem 4.18 There is an objectwise equivalence of (non-strict) presheaves of categories on Dfg Proof. The proof Theorem 4.18 consists of several individual steps, which we present in detail in Appendix A: first, we fix an arbitrary diffeological space X and restrict our attention to the evaluation of (4.19) at X, i.e. to the functor Using Lemma 4.16 we define the functor A |X on objects and (Definition A.4) and show in Proposition A.5 that it indeed produces diffeological vector bundles on X. Next, we define A |X on morphisms (Definition A.7) and show that, for each X ∈ Dfg, it is fully faithful (Proposition A.8) and essentially surjective A.9. Only then do we let X ∈ Dfg vary and complete the proof by showing that the functors A |X we constructed for each X ∈ Dfg assemble into a morphism of pseudo-functors Cart op → Cat L . Since we have already shown that this morphism of pseudo-functors is objectwise an equivalence of categories, we obtain the desired equivalence result.
This completes the proof of Theorem 4.12.

Descent for the (∞, 2)-sheaf of gerbes with connection
An important example of higher geometric structures consists of (higher) gerbes with connection. The simplicial presheaf of n-gerbes with connection on Cart is obtained by applying the Dold-Kan correspondence to the chain complex with the sheaf U(1) of smooth U(1)-valued functions in degree zero. The resulting simplicial presheaf is denoted B n+1 ∇ U(1). It satisfies descent with respect to τ dgop , and thus S ∞,0 (B n+1 ∇ U(1)) ∈ H ∞,0 is a fibrant object. It classifies n-gerbes on objects of C.
However, this only captures invertible morphisms of n-gerbes with connection. At least for n = 1 there also exists a theory of interesting non-invertible morphisms, due to Waldorf [Wal07]. There, Waldorf showed that the morphisms between two fixed gerbes on a manifold satisfy descent along surjective submersions. This was later improved upon by Nikolaus-Schweigert in [NS11]; they proved that the presheaf of 2-categories which assigns to a manifold its gerbes and their 1-and 2-morphisms satisfies descent along surjective submersions of manifolds. Descent properties for gerbes and their non-invertible morphisms on more general spaces are so far unknown.
In particular, restricting to objects X ∈ C of the form X = M for some manifold M , we directly obtain the result of [NS11] that gerbes with connection form a sheaf of 2-categories on the site of manifolds and surjective submersions.

Descent and coherence for smooth functorial field theories
We now apply our findings to smooth functorial field theories (FFTs), a family-version of topological quantum field theories (TQFTs).

Background on presentations of topological quantum field theories
We briefly recall some well-known background on TQFTs, highlighting only those technical details which are relevant to this paper. TQFTs were introduced in [Ati88, Seg87]; for some modern introductions, we refer the reader to [Koc04,Saf17]. The formulation of d-dimensional TQFTs rests on the d-dimensional cobordism category Bord d . Its objects are (d−1)-dimensional closed, oriented manifolds. Its morphisms are diffeomorphism classes of d-dimensional oriented cobordisms between these. Com-position is given by gluing bordisms along the mutual boundary. The category Bord d is symmetric monoidal under disjoint union of manifolds. If T is a symmetric monoidal category, a d-dimensional T-valued TQFT is a symmetric monoidal functor Z : If Y is an object in Bord d , then any orientation-preserving diffeomorphism f : Y → Y induces a morphism C f : Y → Y in Bord d . Isotopic diffeomorphisms induce the same morphism. Via the TQFT functor Z, the object Z(Y ) ∈ T thus carries an action of the mapping class group of Y . Let cMfd or d−1 denote the groupoid of closed oriented (d−1)-manifolds and their orientation-preserving diffeomorphisms. We let cMfd or d−1 /∼ denote the groupoid with the same objects, but with morphisms given by isotopy classes of diffeomorphisms. Observe that there are functors cMfd or d−1 −→ cMfd or d−1 /∼ −→ Bord d . If Y 0 ∈ cMfd or d−1 and MCG(Y 0 ) is its mapping class group, let Y 0 //MCG(Y 0 ) denote the associated action groupoid. The inclusion Y 0 //MCG(Y 0 ) ֒→ cMfd or d−1 /∼ is an equivalence onto a connected component, and hence the inclusion is an equivalence. Consequently, we can recover the values of any TQFT Z on all of cMfd or d−1 from its restrictions to Y 0 //MCG(Y 0 ), where Y 0 ranges over all diffeomorphism classes of closed, oriented (d−1)-manifolds. Further, considering a generating set of bordisms between the chosen representatives Y 0 , we can then recover the full TQFT up to natural isomorphism from its values on the Y 0 , the respective mapping class group actions and the generating bordisms.

Diffeomorphism actions in smooth functorial field theories
The picture changes when we pass to smooth FFTs. These differ from TQFTs in several ways: most importantly, we would like to decorate the manifolds underlying objects and morphisms in Bord d with additional geometric structure, and we would like to keep track of how the value of a field theory varies when we vary this additional geometric structure. In the most common and relevant case the additional structure consists of a smooth map to a fixed background manifold M . We restrict our attention to this case here. Smooth FFTs were introduced by Stolz and Teichner in [ST11]. For background, we refer the reader there; the specific formalism employed in this paper was developed in [BW21].
Remark 4.23 One can set up smooth FFTs with geometric structures in a fully general way: the additional data is modelled by a section of a (higher) sheaf on a site of families of d-manifolds and embeddings; we refer the reader to [LS21,GP20] for the full framework. ⊳ In this setup, the bordism category Bord d is replaced by a presheaf of symmetric monoidal categories Bord M d : Cart op → Cat ⊗ ; this assigns to c ∈ Cart the symmetric monoidal category Bord M d (c), whose objects are essentially pairs of a closed, oriented d-manifold Y and a smooth map f : c × Y → M . Its morphisms (Y 0 , f 0 ) → (Y 1 , f 1 ) can be described as bordisms Σ : Y 0 → Y 1 , together with a smooth map σ : c × Σ → M which restricts to f 0 and f 1 on the respective incoming and outgoing boundaries of the bordism (for the full details of this model for Bord M d , see [BW21]). Finally, for any smooth map ϕ : c × Σ → c × Σ ′ which is a fibrewise diffeomorphism of bordisms Y 0 → Y 1 over c, one identifies (Σ, σ) and (Σ ′ , σ • ϕ) as morphisms in Bord M d (c). This defines a strict functor Cart op → Cat L . We further enhance this to a strict functor taking values in the 2-category Cat ⊗ L of symmetric monoidal categories: the symmetric monoidal structure on Bord d (c) takes disjoint unions of underlying manifolds on objects as well as morphisms; that is, on objects we set , and on morphisms For X, X ′ ∈ Mfd or , we let D(X, X ′ ) denote the presheaf on Cart consisting of diffeomorphisms from X to X ′ (this is even a diffeological space). Concretely, an element of D(X, X ′ )(c) is a smooth map ϕ ⊣ : c × X → X ′ which is a diffeomorphism at each x ∈ c and such that the map of pointwise inverses is also smooth. We equivalently write this as a morphism ϕ : Y c → D(X, X ′ ). This establishes both Mfd or and M Y as categories enriched in Cart; the enriched mapping objects are M Cart Y (Y 0 , Y 1 ) = D(Y 0 , Y 1 ). We also use the shorthand notation D(Y ) := D(Y, Y ).
where ∆ + is the simplex category ∆ with an initial object [−1] adjoined to it.
Example 4.26 In the context of smooth field theories on a manifold M , the relevant choice of P is M (−) , which assigns to Y ′ ∈ M Y the mapping presheaf M Y ′ . Given two diffeomorphic manifolds

This makes M (−) into an enriched functor. ⊳
This setup allows us to form, for each n ∈ N 0 , the enriched two-sided simplicial bar construction [Rie14, Section 9.1], which produces a simplicial object B • G n , M op Y , P ∈ Cart (∆ op ) . Explicitly, We now consider the case of G = D(Y, −) : M Y → Dfg. Define morphisms Φ n as the composition where ∆ denotes the diagonal morphism, and where Ev : is defined via the tensor adjunction. The morphism δ acts as the action groupoid of the D(Y )-action via Ev on P (Y ) ∈ Cart. We further set . Lemma 4.27 For every n ∈ N 0 , the morphism Proof. This follows directly from the compatibility of Ev with compositions.
We further define morphisms Ψ k : (1) The morphism (2) The morphism Ψ k is a τ dgop -local epimorphism, and the simplicial object Proof. Part (1) is immediate from the construction of Ψ k . For part (2), we only need to observe that Ψ k is a coproduct of projections onto a factor in a product and apply Lemma 2.8.
For k, l ∈ N 0 , we introduce the short-hand notation Corollary 4.29 We obtain a bisimplicial object in Cart which is augmented in each direction, and where the vertical simplicial objects are the Čech nerves of the subductions Ψ k .
The morphisms Φ • and Ψ • have the following properties: (1) Each Φ n induces a weak equivalence in H ∞,0 : (2) Each Ψ k induces a weak equivalence in H loc ∞,0 : Proof. Ad (1): We claim that the augmented simplicial object Φn in H ∞,0 admits extra degeneracies. To see this, we define morphisms in Cart where the image lies in the summand labelled by Y ∈ M Y . For k ∈ N 0 , we set where the image lies in the component of the coproduct labelled by (Y 0 , . . . , Y k , Y ). These morphisms yield the desired extra degeneracies.
For claim (2), we observe that we have a commutative diagram The top morphism is a local weak equivalence by Proposition 4.31, and the vertical morphisms are projective weak equivalences by Proposition 3.4. It follows that the bottom morphism is a local weak equivalence.
Definition 4.33 Let n ∈ N 0 , and let F ∈ H ∞,n be a presheaf of (∞, n)-categories on Cart.
(1) We define (∞, n)-categories where we make use of the simplicial enrichment of CSS n . We call F(P ) D(Y ) the (∞, n)-category of equivariant sections of F over P (Y ) ∈ Cart.
(2) We define (∞, n)-categories Note that by Proposition 3.31 F(P ) D(Y ) and F(P ) coh are models for the homotopy limits By Lemma 3.32 these are indeed both (∞, n)-categories, i.e. fibrant objects in CSS n . The first homotopy limit describes the (∞, n)-category of homotopy fixed points of the smooth action of D(Y ) on sections of F over P (Y ). In particular, the homotopy fixed point data depends smoothly on the diffeomorphisms in D(Y ). The second homotopy limit can be described as follows: for each manifold Y 0 ∈ M Y , we can form the (∞, n)-category of sections of F over P (Y 0 ); it is given by S ∞,n (F)(P (Y 0 )). Given another manifold Y 1 ∈ M Y , there is a small presheaf (even a diffeological space) D(Y 0 , Y 1 ) of diffeomorphisms from Y 0 to Y 1 . We can pull back sections of F over P (Y 1 ) along any such diffeomorphism, and the resulting section over P (Y 0 ) should depend smoothly on the diffeomorphism. The homotopy limit F coh can be understood as the (∞, n)-category of families of sections of F(Y 0 ) over P (Y 0 ), for all Y 0 ∈ M Y ; these sections further are homotopy coherent with respect to all diffeomorphisms Y 0 → Y 1 , and their homotopy coherence data depends smoothly on these diffeomorphisms.
After this preparation, we give two different proofs (in Theorems 4.34 and 4.35) that the (∞, n)categories of equivariant sections and coherent sections are equivalent. The first proof relies on the fact that the inclusion BD(Y ) ֒→ M Y of groupoid objects in Cart is an equivalence, whereas the second proof relies on the descent results from Section 3. Both of these proofs have different advantages: the first is more slick and less involved. However, the choice of an inverse equivalence to the inclusion BD(Y ) ֒→ M Y is by no means canonical, thus making the equivalences obtained in this proof less helpful in practice. In the second proof, the inverse equivalence is obtained as the inverse of a pullback along a τ dgop -local epimorphism. That is, it is given precisely by descent. In many important cases, descent functors are available, so that the second perspective is more helpful for practical purposes.

Any choice of a family of diffeomorphisms {f
for which there exist 2-isomorphisms (i.e. natural isomorphisms, coherent over Cart) p • ι = 1 and η : ι • p → 1 of presheaves of groupoids over Cart. In particular, N η determines a simplicial homotopy between the identity and N (p • ι) in H ∞,0 . This, in turn, shows that both ι and p are homotopy equivalences, weakly inverse to each other. Consequently, we obtain a pair of projective weak equivalences The weak equivalences in the claim are (Qι) * and (Qp) * .
Theorem 4.35 For any n ∈ N 0 and for any fibrant F ∈ H loc ∞,n there is a canonical zig-zag of weak equivalences between fibrant objects in CSS n , Proof. The second and third weak equivalences are direct consequences of Proposition 4.32. The first and last weak equivalences follow by a direct computation: we have canonical isomorphisms Remark 4.36 That (Q diag(Ψ)) * is a weak equivalence relies on the descent results in Section 3; these entered in the proof of Proposition 4.31 and hence also Proposition 4.32.
Remark 4.37 In the case where F is the sheaf of diffeological vector bundles from Definition 4.9 the insights from Theorem 4.35 were used in [BW21] in order to obtain a coherent vector bundle on P , where Y = S 1 and P = M (−) for some manifold M , from the equivariant structure of the transgression line bundle of a bundle gerbe with connection. Moreover, this procedure was also applied to bundles over certain spaces of paths in M , and the coherent vector bundles thus obtained were subsequently assembled into a smooth open-closed FFT on M . In this sense, that smooth FFT was built from its values on generating objects by means of the equivalence of equivariant and coherent sections of a sheaf of higher categories as explored here. There is also great interest in extended field theories, which are formulated in the language of n-fold complete Segal spaces [CS19a,Lur09b]. It is with this in mind that we prove Theorems 4.34 and 4.35 in the setting of (∞, n)-categories. ⊳

Remarks on descent for sheaves of 2-vector bundles
In this section we point out potential applications of Corollary 3.48 to the theory of categorified vector bundles. The gerbes of Section 4.2 can be considered as a model for rank-one 2-vector bundles. The theory for 2-vector bundles of higher rank, and more generally for higher categorifications of vector bundles, is currently only little understood (see below for references). Thus, this section does not contain formal results, but outlines how the results of Section 3.3 will be applicable to the theory of higher vector bundles. This will be similar to the ideas in Sections 4.1 and 4.2. We start by briefly mentioning several models for categorified vector bundles: In [NS11,Sec. 4.3] the authors consider 2-categorical presheaves of Kapranov-Voevodsky 2-vector bundles on the category of manifolds. They apply the Grothendieck plus-construction to sheafify these higher presheaves and obtain a 2-categorical sheaf which assigns to each manifold a 2-category of 2vector bundles, modelled on Kapranov-Voevodsky 2-vector spaces (as defined in [KV94]). A similar 2-category was considered in [BDR04].
In [KLW21] the authors defined a presheaf of 2-categories on the site of manifolds with surjective submersions as coverings which assigns to each manifold a 2-category 4 built from bimodule bundles over bundles of super-algebras. The authors then sheafify this presheaf of 2-categories and propose the resulting sheaf of 2-categories as a model for 2-vector bundles on the site of manifolds and surjective submersions.
Extending this, one can consider presheaves of (∞, n)-categories on a small category C which assign to each object higher Morita categories of E n -bimodules; for C = Cart or C = Mfd, these could, for instance, be A-B-bimodules for the E ∞ -algebras A = C and B the algebra of smooth functions on the underlying manifold (or derived versions thereof). Sheafifying these higher presheaves on Cart with respect to differentiably good open coverings would give viable candidates for generalisations of higher vector bundles.
In each of these cases, one is presented with a sheaf of (∞, n)-categories in the sense of Section 3.3. Then, Corollary 3.48 applies, and we obtain that each such sheaf of (∞, n)-categories of higher vector bundles on (Cart, τ dgop ) extends to a sheaf of (∞, n)-categories on ( Cart, τ ). The resulting extended higher sheaves are good candidates for a description of categorified vector bundles on generalised smooth spaces and τ dgop -coverings, such as manifolds and surjective submersions, or diffeological spaces and subductions.

The image of right Kan extension on sheaves of spaces
We now classify those presheaves of ∞-groupoids on C which are extensions of sheaves of ∞-groupoids on (C, τ ). We achieve this by passing to ∞-categorical language using quasi-categories (the model for (∞, 1)-categories used in [Lur09a,Cis19]), rather than working with model-categorical presentations, and prove an enhancement of Theorem 3.23 to an equivalence of quasi-categories of homotopy-coherent sheaves of ∞-groupoids.

On a quasi-categorical site
Let C be a quasi-category, and let (C, τ ) be a small quasi-categorical Grothendieck site (as de- Since effective epimorphisms are stable under pullback and L is left exact, τ -local epimorphisms induce a coverage-and hence a Grothendieck topology-on the homotopy category h C ∞ . Since a Grothendieck topology on h C ∞ is the same as a Grothendieck topology on C ∞ [Lur09a, Rmk. 6.2.2.3], we obtain a new quasicategorical Grothendieck site ( C ∞ , τ ), whose topology is generated by the τ -local epimorphisms in C ∞ . The corresponding quasi-category Sh( C ∞ , τ ) of sheaves on C ∞ is the localisation of PSh( C ∞ ) at the morphisms We thus obtain another reflective localisation ⊥ L whose left adjoint L is left exact. Note that since the Yoneda embedding preserves limits and C ∞ is complete, there is a canonical equivalence YČ f ≃Č Y f . This directly leads to: Let PSh ! ( C ∞ ) = Hom ! ( C op ∞ , S L ) denote the full sub-quasi-category of PSh( C ∞ ) on those functors which preserve small limits (equivalently the small-colimit -preserving functors C ∞ → S op L ). We also define a quasi-category Sh ! ( C ∞ , τ ) as the pullback of quasi-categories is the full sub-quasi-category on those τ -sheaves whose underlying functor C op ∞ → S L turns small colimits in C ∞ into limits in S L . The functor Y : C → C ∞ induces a functor C op → C op ∞ , and by a slight abuse of notation we again denote this functor by Y. As before, we leave the inclusion S S ⊂ S L of the quasi-category of small spaces into that of large spaces implicit. Observe that the left and right Kan extensions Y ! , Y * yield functors PSh( C ∞ ) → PSh( C ∞ ).
Lemma 5.3 Let C be a small quasi-category.
(1) For any A ∈ C ∞ ⊂ PSh(C) and G ∈ PSh( C ∞ ) there is a natural equivalence Equivalently, the restriction Y * | C∞ agrees with Y as functors C ∞ → PSh( C ∞ ).
(2) For any A ∈ C ∞ ⊂ PSh(C) and G ∈ PSh ! ( C ∞ ) there is a natural equivalence Proof. Using the equivalence X ≃ colim c∈C /X Y c , for X ∈ C ∞ , we obtain proving the first equivalence. To see the second equivalence, we use that In the second-to-last step we have used that G turns colimits in C ∞ into limits in S L .
Lemma 5.4 Let D : J → C ∞ be any small diagram. There is a canonical equivalence Proof. This follows by a repeated application of the Yoneda Lemma and the fact that colimits in presheaf categories are computed objectwise. In particular, we have and which yields the desired equivalence.
We then have the following quasi-categorical enhancement of Theorem 3.23. This fully unveils the importance of the topology induced on C ∞ by the τ -local epimorphisms.
Theorem 5.5 Let C be a small quasi-category, and let Y * : PSh(C) → PSh( C ∞ ) denote the right Kan extension along Y : (2) More generally, let τ be a Grothendieck topology on C. The adjunction Y * ⊣ Y * restricts to an equivalence Proof. Claim (1) is an explicit incidence of a well-known theorem: there is a canonical equivalence Hom(C, D) ≃ Hom ! ( C ∞ , D) between the quasi-categories of functors from C to any S-cocomplete quasi-category D and (small-)colimit-preserving functors C ∞ → D. For D = S op L , this equivalence is established by Y * . In particular, Y * is fully faithful even as a functor PSh(C) → PSh( C ∞ ). The adjunction Y * : PSh( C ∞ ) ⇄ PSh(C) : Y * restricts to an adjunction Y * : PSh ! ( C ∞ ) ⇄ PSh(C) : Y * , whose right adjoint Y * is an equivalence. Thus, the left adjoint Y * is an equivalence as well [Cis19, Prop. 6.1.6].
We now prove (2): first, we show that Y * maps Sh(C, τ ) to Sh( C ∞ , τ ), i.e. that it maps τ -sheaves on C to τ -sheaves on C ∞ . Let F ∈ Sh(C, τ ), and let f : Y → X be a τ -local epimorphism in C ∞ . We need to prove that the morphism is an equivalence. By Lemma 5.3(1), Lemma 5.4, and because Y * is fully faithful, we have equivalences Using that F is a τ -sheaf and that L is a left exact left adjoint, we then compute Thus, Y * F is a τ -sheaf whenever F is a τ -sheaf. In particular, Y * restricts to a fully faithful functor Finally, we claim that the functor Y * : PSh ! ( C ∞ ) → PSh(C) restricts to a functor Y * : Sh ! ( C ∞ , τ ) → Sh(C, τ ); by part (1), this restriction is automatically fully faithful. To that end, let G ∈ Sh ! ( C ∞ , τ ). We have to show that Y * G ∈ PSh(C) is τ -local, i.e. that the presheaf Y * G is a τ -sheaf. That is the case precisely if is an equivalence for every τ -covering g : Y → Y c . (Note that if g is a τ -covering, then L(|Čg| → Y c ) is an equivalence, i.e. g is a τ -local epimorphism.) On the right-hand side, we have canonical equivalences The first equivalence is simply the adjunction Y ! ⊣ Y * , the second equivalence is Lemma 5.3(2), which applies since G turns colimits into limits. This property also provides the third equivalence. The final equivalence arises from Lemma 5.2, as G is a τ -sheaf.
Corollary 5.6 If τ 0 is the trivial topology on C (whose coverings are generated solely by the identity morphisms), then Proof. This follows from Theorem 5.5(1) since pullbacks of equivalences in a quasi-category are again equivalences.
Remark 5.7 In particular, if G ∈ PSh( C ∞ ) is colimit-preserving, then it satisfies τ 0 -descent. One can also see this directly: the τ 0 -local epimorphisms are exactly the effective epimorphisms in C ∞ , i.e. those morphisms p : Y → X whose induced morphism |Čp| → X is an equivalence in C ∞ . Thus, for G ∈ PSh ! (C), we have that G(X) → G(|Čp|) ≃ lim ∆ G(Čp) is an equivalence, and hence that G satisfies τ 0 -descent. ⊳ We summarise various characterisations of the sheaves in the image of Y * : Let denote the full sub-quasi-category on those limit-preserving functors C op ∞ → S L (equivalently colimit -preserving functors C ∞ → S op L ) which send all morphisms in W τ to equivalences.
Theorem 5.8 There are equivalences of quasi-categories

On a 1-site
For many geometric applications it is more convenient to work not with ∞-presheaves on a quasicategory, but only with Set L -valued presheaves on an ordinary category. For instance, many geometric problems are set on manifolds or diffeological spaces (such as in Sections 4.1 and 4.3), and it is often more efficient (though less general) to treat these as particular presheaves of sets. Thus, in this section we prove a version of Theorem 5.2 where C is an ordinary (1-)category and we replace C ∞ by the ordinary category C = Fun(C op , Set S ) of Set S -valued presheaves on C. The quasi-category of S S -valued presheaves on C is NC ∞ = Set ∆L (NC, S S ).
We let C = Fun(C op , Set S ) denote the category of ordinary, small presheaves on C. Let NC denote the quasi-category given by the nerve of C. The Yoneda embedding of NC factorises as where Y 0 is the 1-categorical Yoneda embedding of C and NC ∞ = Set ∆L (NC, S S ) is the quasi-category of small presheaves of spaces on NC. Note that C ֒→ NC ∞ is a reflective localisation with localisation functor π 0 . As before, we write PSh(NC) = Set ∆XL (C op , S L ).
Lemma 5.10 Let F ∈ PSh(NC). There is a natural equivalence Y 0 * F ≃ ι * Y * F of objects in PSh(N C). Proof. This is a direct application of [CS19b, Lemma 3.13] (noting that N C is large and the target S L is L-cocomplete).
Applying the functor ι * , we obtain a natural equivalence ι * Y 0 * F ≃ ι * ι * Y * F . Lemma 5.11 For G ∈ PSh ! ( NC ∞ ), there is an equivalence ι * ι * G ≃ G. 5 We thank the anonymous referee for pointing out this argument, which streamlined the proof in an earlier version.
Proof. By Theorem 5.5, there exists F ∈ PSh(NC) and an equivalence Y * F ≃ G. We have equivalences The second equivalence is Lemma 5.10 and the second-to-last equivalence is the factorisation (5.9).
In particular, we have that ι * ι * Y * F ≃ Y * F , for each F ∈ Fun(NC op , S L ), by Theorem 5.5. Combining this with Lemma 5.10, we thus obtain that the canonical morphism Theorem 5.12 Let X denote the full sub-quasi-category of PSh(N C) on those presheaves F where ι * F ∈ PSh ! ( NC ∞ ). The functor Y 0 * : PSh(NC) → X is an equivalence with inverse (Y 0 ) * . Proof. Combining the observation preceding Theorem 5.12 with Theorem 5.5(1) shows that Y 0 * takes values in X. Since Y 0 is fully faithful, so is Y 0 * ; it remains to check that Y 0 * is essentially surjective. To that end, let G ∈ X. By Theorem 5.5(1) and the definition of X there exists an F ∈ PSh(NC) and an equivalence ι * G = Y * F . Lemma 5.10 implies that The first equivalence is (the inverse of) the counit of the adjunction ι * ⊣ ι * ; this is an equivalence since ι-and therefore ι * -is fully faithful.
Corollary 5.13 The functor ι * : X → PSh ! ( NC ∞ ) is an equivalence with inverse ι * . That is, presheaves in PSh ! ( NC ∞ ) are fully determined by their restriction to N C.
This merely reflects the fact that NC ∞ is generated under small colimits by the image of the inclusion Theorem 5.14 Let X τ ⊂ PSh(N C) be the full sub-quasi-category on those presheaves X where ι * X ∈ Sh ! ( NC ∞ , τ ). There is a canonical equivalence Sh(C, τ ) ≃ X τ .
Proof. Consider the diagram By definition of X and X τ and the pasting law for pullbacks, both squares in this diagram are pullback squares. We also consider the diagram It suffices to show that the outer square in this diagram is a pullback. Theorem 5.12 states that the right-hand square in this diagram is a pullback square of quasi-categories. Additionally, Theorem 5.5 implies that the left and centre vertical morphisms are equivalences of quasi-categories, so that the left-hand square is a pullback as well. The claim now follows by the pasting law for pullbacks.
This provides a very general perspective on the statement that open coverings and surjective submersions define the same quasi-categories of sheaves on the category Mfd of manifolds: Its Čech nerve, taken in Mfd, reads asČ is not an isomorphism in Mfd, since manifolds are allowed to be non-connected. However, for F : Mfd op → S L sending coproducts in Mfd to products, the canonical morphism is an equivalence, for each k ∈ N 0 . The left-hand side are the terms in the descent condition for F with respect to the open covering U , whereas the right-hand side are the terms in the descent condition for F with respect to the surjective submersion i∈I U i → M . Since we assumed F to satisfy descent with respect to surjective submersions, the latter descent condition is satisfied. By the above observation, it thus follows that F ∈ Sh(NMfd, τ op ).
We are left to check that the inclusions in the claim are in fact essentially surjective. For the first inclusion, we claim that τ ssub,⊔ ⊂ τ op . Indeed, let π : Y → X be a τ ssub,⊔ -local epimorphism in Mfd. Therefore, given any morphism f : Y 0 M → X from a representable presheaf, there exists either an open cover by disjoint subsets of M and lifts of f to Y over the patches, or a surjective submersion p : N → M together with a commutative square Thus, we have that π ∈ τ op . An analogous argument shows that τ op ⊂ τ dgop . We thus have canonical inclusions the diagram in Set S commutes for every i, j ∈ I. Since (C, τ ) is a concrete site, the family {Y c i ( * ) → Y c ( * )} i∈I is jointly surjective, so that these data determine a unique map ϕ : Y c ( * ) → Z( * ) such that all diagrams commute. We claim that ϕ ∈ Z(c). Finally, we need to show that Z is a colimit of the diagram D : J → Dfg(C, τ ). To see this, let {ψ j : D(j) → A} j∈J be a cocone under D in the category Dfg(C, τ ). Evaluating at * ∈ C, we obtain a cocone ψ j| * : D(j)( * ) → A( * ) under the diagram Ev * •D in Set S . By construction, the set Z( * ) presents a colimit of that diagram; hence these data induce a unique map of sets ψ : Z( * ) → A( * ). Recall that for any pair of objects X, Y ∈ Dfg(C, τ ) the map Ev * : Dfg(C, τ )(X, Y ) → Set S (X( * ), Y ( * )), φ → φ | * is injective. It follows that if the map ψ gives rise to a morphism of (C, τ )-spaces, then that morphism is the unique morphism in Dfg(C, τ ) inducing a morphism of cocones under D.
Therefore, we are left to show that ψ : Z( * ) → A( * ) gives rise to a morphism in Dfg(C, τ ). That is equivalent to showing that composition by ψ sends plots of Z to plots of A. Thus, let ϕ : Y c → Z be an arbitrary morphism. As before, by definition of Z, we find a covering family {f i : c i → c} i∈I and lifts ϕ i : Y c i → D(j i ) of ϕ along the morphisms f i as in (A.2). We claim that {ψ j i • ϕ i } i∈I is a compatible family of morphisms Y c i → A in C. For i, k ∈ I, consider the diagram in C. The left-hand triangle commutes by definition of the pullback. The two central squares commute by definition of ϕ i . We do not yet know whether ψ is actually a morphism in C. However, evaluating the whole diagram at * ∈ C, we obtain a diagram in Set S in which the two right-hand triangles also commute, since ψ is a morphism of cocones under the diagram Ev * • D in Set S . Thus, we infer that the outer hexagon in (A.3) commutes as maps on the underlying sets. Recalling that the map which sends morphisms in Dfg(C, τ ) to maps of underlying sets is injective, it thus follows that the outer hexagon is commutative already as a diagram in Dfg(C, τ ) (i.e. before evaluating at the terminal object).
Consequently, {ψ j i • ϕ i } i∈I is indeed a compatible family of morphisms Y c i → A. Thus, as A is a sheaf, it defines a unique morphism ̺ : Y c → A in Dfg(C, τ ). It follows from the construction of ̺ that Ev * ̺ = ̺ | * = ψ • ϕ | * ; hence, composition with ψ sends plots of Z to plots of A, so that ψ gives rise to a morphism in Dfg(C, τ ).
We start by describing the functor A on objects: recall the computation of the values of ι * VBun str in Lemma 4.16. Consider the category Cart /X , whose objects are plots ϕ ∈ X(c) for any c ∈ Cart and whose morphisms (ϕ ∈ X(c ′ )) → (ϕ ′ ∈ X(c)) are smooth maps f : c → c ′ such that f * ϕ ′ = ϕ. Given an object (n, h) ∈ ι * VBun str (X) we define a functor D (n,h) : Cart /X → Dfg, which acts as We then set and denote the canonical morphism D (n,h) (ϕ) → E by ι ϕ . The object X ∈ Dfg is a cocone under D (n,h) , which is established by the morphism of diagrams π : D (n,h) → X , π |ϕ : D (n,h) (ϕ) = c × C n pr −→ c ϕ −→ X .
We let π : E → X denote the unique morphism induced on the colimit.
Proposition A.5 For each object (n, h) ∈ ι * VBun str (X), the colimit A(n, h) = (E, π) is a diffeological vector bundle on X, Proof. To see this, let ψ ∈ X(d) be a plot of X over d ∈ Cart. Consider the morphism of diffeological spaces Note that Φ ψ is linear on the fibres. We need to show that it is an isomorphism. For x ∈ X( * ) and c ∈ Cart, let c x : pt → X( * ) denote the constant plot of X with value x. Every element (y, [ϕ, z, v]) ∈ (d × X E)( * ) has a unique representative of the form y, [ϕ, z, v] = y, [ψ, y, w] . This is established by the pairs (z, h z ) and (y, h y ) (from the data of (n, h)) associated to the morphism z : c ψ(y) → ϕ and y : c ψ(y) → ψ in Cart /X , i.e. to the commutative diagram Proposition A.9 For each X ∈ Cart, the functor A |X : : ι * VBun str (X) −→ VBun Dfg (X) is essentially surjective.
Proof. Let π F : F → X be a diffeological vector bundle on X. For each plot ϕ : c → X choose a trivialisation Ξ ϕ : c × C n ∼ = −→ c × X F . (A.10) Given a morphism f : ϕ 0 → ϕ 1 in Cart /X , the universal property and the pasting law for pullbacks determine a canonical isomorphism c 0 × X F ∼ = c 0 × c 1 (c 1 × X F ). This allows us to define an isomorphism The map h f is linear on fibres and hence determines a unique smooth map h f : c 0 → GL(n(ϕ 0 ), C).
Since the morphisms labelled ' ∼ =' in the above diagram are chosen as canonical isomorphisms between different representatives for the same limits, the collection of morphisms h f assembles into an object (n, h) ∈ ι * VBun str (X).
We claim that there is an isomorphism A(n, h) ∼ = −→ F in VBun Dfg (X). By a convenient abuse of notation, we denote this isomorphism by Ξ. We set where Ξ ϕ was chosen in (A.10), and where pr F : c × X F → F is the projection to F . This defines a map Ξ : A(n, h) → F , which is linear on fibres. We need to show that it is smooth. Consider a plot ̺ : c → colim Dfg D (n,h) = A(n, h). By Proposition A.1, there exist a covering {f i : c i ֒→ c} i∈I of c and lifts ̺ i : c i → D (n,h) (ψ i ) for some plots ψ i : d i → X. Consider the diagram The left-hand square commutes by definition of ̺ i , and the right-hand triangle commutes by construction of Ξ and of (n, h). Thus, the map Ξ • ̺ is locally given by plots pr F • Ξ ψ i • ̺ i . By the sheaf property of diffeological spaces, Ξ • ̺ is a plot of F itself.
Lemma A.11 A is compatible with pullbacks of vector bundles along morphisms F : X → Y in Dfg. That is, A : ι * VBun str −→ VBun Dfg is a morphism of presheaves of categories on Dfg.
Proof. Let (n, h) ∈ ι * VBun str (Y ). As a diffeological space, we have We have to compare this to A F * (n, h) = colim Cart /X Dfg D F * (n,h) .
To that end, we consider the map where ϕ : c → X is a plot, z ∈ c is some point, and where v ∈ C n is a vector. We also define a map where x ∈ X( * ) is any point in the underlying set of X, w ∈ C n(F (x)) is a vector, and where we denote a constant plot pt → X by its value in X( * ). We readily see that Ξ F and Ξ ′ F are mutually inverse maps, fibrewise linear, compatible with morphisms in ι * VBun str , and that the diagram commutes for every morphism G ∈ Dfg(W, X). It thus remains to show that both Ξ F and Ξ ′ F are morphisms of diffeological spaces.
We start with Ξ F : let ̺ : c → A(F * (n, h)) be a plot. Let {f i : c i ֒→ c} i∈I , ψ i : d i → X, and ̺ i : c i → D F * (n,h) (ψ i ) be lifting data for ̺ as before. Then, we have a commutative diagram which shows that Ξ F • ̺ is smooth (the A(n, h)-valued component factors through D (n,h) locally).
For Ξ ′ F , consider a plot ̺ : c → X × Y A(n, h). It decomposes into a plot ̺ X : c → X and a plot ̺ A : c → A (n,h) such that π • ̺ A = F • ̺ X , where π : A (n,h) → Y is the vector bundle projection. As before, for the plot ̺ A there exists a covering {f i : c i ֒→ c} i∈I , plots ψ : d i → Y , and lifts ̺ A,i : c i → D (n,h) (ψ i ) such that ι ψ i • ̺ A,i = ̺ A • f i for each i ∈ I. Using the morphisms h ̺ A,i , it is in fact always possible to choose d i = c i and ̺ A,i : c i → c i × C n(ψ i ) to be a section, i.e. to satisfy pr c i • ̺ i,A = 1 c i . Observe that then ψ i = F • ̺ X • f i factors through F . We obtain a commutative diagram This shows that Ξ ′ F • ̺ is a plot of A(F * (n, h)) by Proposition A.1, which completes the proof. This also completes the proof of Theorem 4.18.