The $\mathbb{R}$-Local Homotopy Theory of Smooth Spaces

Simplicial presheaves on cartesian spaces provide a general notion of smooth spaces. There is a corresponding smooth version of the singular complex functor, which maps smooth spaces to simplicial sets. We consider the localisation of the (projective or injective) model category of smooth spaces at the morphisms which become weak equivalences under the singular complex functor. We prove that this localisation agrees with a motivic-style $\mathbb{R}$-localisation of the model category of smooth spaces. Further, we exhibit the singular complex functor for smooth spaces as one of several Quillen equivalences between model categories for spaces and the above $\mathbb{R}$-local model category of smooth spaces. In the process, we show that the singular complex functor agrees with the homotopy colimit functor up to a natural zig-zag of weak equivalences. We provide a functorial fibrant replacement in the $\mathbb{R}$-local model category of smooth spaces and use this to compute mapping spaces in terms of singular complexes. Finally, we explain the relation of our fibrant replacement to the concordance sheaf construction introduced recently by Berwick-Evans, Boavida de Brito and Pavlov.


Introduction and overview
Topological spaces and simplicial sets can be used to construct the same homotopy theory. This is made rigorous by the fact that the singular complex and geometric realisation functors form a Quillen equivalence between the standard model structure on the category Top of topological spaces and the Kan-Quillen model structure on the category Set ∆ of simplicial sets. Both of these model categories formalise what is often called the homotopy theory of spaces, or ∞-groupoids (which are the same according to Grothendieck's homotopy hypothesis). The two models differ significantly in their features, though, in that topological spaces encode homotopies via the notion of continuity, while simplicial sets are inherently combinatorial. Consequently, each of these two models for the homotopy theory of spaces has its own merits in different contexts and applications.
Apart from continuity or combinatorics, another important feature spaces can possess, and which is relevant in many problems in mathematics, is smoothness. The prime example of a category of smooth spaces is the category Mfd of manifolds and smooth maps, which underlies much of geometry. There exists a notion of smooth homotopy within the category Mfd, and one can find smooth versions of many topological concepts, such as cohomology, which are invariant under these smooth homotopies. However, the category Mfd is poorly behaved in many ways. For instance, it is far from being complete or cocomplete, thus making it unable to admit a model structure in the sense of Quillen [Qui67].
The way to cure this is to weaken-and therefore to generalise-the concept of a manifold. Here, we take the following approach to smooth spaces, with the main goal of simultaneously capturing the notions of manifold and (higher) stack. We start from the category Cart of cartesian spaces: its objects are all smooth manifolds that are diffeomorphic to R n for any n ∈ N 0 . Its morphisms are all smooth maps between these manifolds. We define a smooth space to be a simplicial presheaf on Cartinformally, we understand the sections of a simplicial presheaf over c ∈ Cart as c-parameterised families of simplices in a space. We denote the category of simplicial presheaves on Cart by H. It contains many geometrically interesting objects that are not manifolds or even diffeological spaces [IZ13] (for instance the presheaf of k-forms, or the simplicial presheaf of G-bundles with connection, for any Lie group G). The category of manifolds, the category of diffeological spaces and the category of simplicial sets each include fully faithfully into H.
The category H carries two natural model structures, namely the projective and the injective model structure on functors Cart op → Set ∆ (where the category Set ∆ of simplicial sets carries the Kan-Quillen model structure). We denote the projective and injective model categories by H p and H i , respectively, and we write H p/i to refer to either of these model structures simultaneously. The projective and injective model structures are canonically Quillen equivalent via the identity functors H p ⇄ H i , but they are not Quillen equivalent to Set ∆ . In that sense, the model structures H p/i do not yet define smooth versions of the homotopy theory of spaces. To achieve that, one needs a weaker notion of equivalence in H.
There exist (at least) two candidates for such weakened versions of equivalences in H. First, in [MW07,GTMW09] a notion of weak equivalence has been introduced on sheaves on Mfd as follows: let ∆ k e ∼ = R k denote the smooth extended (affine) k-simplex. Extending the usual face and degeneracy maps of the topological standard simplices, this gives rise to a cosimplicial cartesian space ∆ e : ∆ → Cart ⊂ Mfd. Via precomposition, this induces a functor from (pre)sheaves on manifolds to simplicial sets. In [MW07,GTMW09], a morphism of sheaves is considered a weak equivalence of (pre)sheaves whenever it becomes a weak equivalence of simplicial sets under this functor. We adapt this to our setup as follows: for technical reasons, we work with presheaves on cartesian spaces rather than manifolds, and we work with simplicial (pre)sheaves instead of ordinary (pre)sheaves. Let δ : ∆ → ∆ × ∆ denote the diagonal functor. We define the smooth singular complex functor as the composite where the first functor evaluates F ∈ H on the extended simplices to obtain a bisimplicial set, of which the second functor then takes the diagonal. Let S −1 e (W Set ∆ ) denote the class of morphisms in H that are mapped to a weak equivalence by S e . The generalisation to simplicial presheaves of the homotopy theory from [MW07,GTMW09] is then the left Bousfield localisation L S −1 e (W Set ∆ ) H p/i . The second notion of weak equivalence in H is motivated by motivic homotopy theory (see [Voe98,MV99, DLØ + 07], for instance). Let I denote the class of all morphisms in H of the form c × R → c, where c ranges over all objects in Cart, and where the morphism is the identity on c and collapses R to the point. The left Bousfield localisation H p/i I := L I H p/i is then a version in smooth geometry of motivic localisation. We call H p/i I the R-local model category of simplicial presheaves on Cart, or equivalently of smooth spaces. This localisation has appeared before in [Sch,Dug01b] and other works of these authors. For presheaves with values in stable ∞-categories, this type of localisation has been investigated in [BNV16]. Our first main result is: Theorem 1.1 There is an equality of model categories: H p/i I = L S −1 e (W Set ∆ ) H p/i . A large part of this paper is concerned establishing several explicit Quillen equivalences between these model categories and various model categories describing the homotopy theory of spaces. Concretely, these are: the Kan-Quillen model structure on simplicial sets Set ∆ , the model category of ∆generated topological spaces ∆Top and the diagonal model structure on bisimplicial sets (sSet ∆ ) diag . Further, we show in Proposition 3.19 that the model category (sSet ∆ ) diag coincides with the following two model categories: (1) the localisation L ∆ • ⊠∆ 0 sSet ∆ of the injective model structure on bisimplicial sets sSet ∆ at the collapse morphisms ∆ n → ∆ 0 , for n ∈ N 0 (where both are seen as simplicial diagrams of discrete simplicial sets), and (2) the localisation L ∆ 1 ⊠∆ 0 CSS of the model category of complete Segal spaces at the collapse morphism ∆ 1 → ∆ 0 .
To state our main theorem, we need the following notation: we let c ∆ : Set ∆ → sSet ∆ denote the functor with sends a simplicial set to a constant simplicial diagram in Set ∆ . Similarly, letc : Set ∆ → H denote the functor sending a simplicial set to a constant simplicial presheaf on Cart. Finally, we have a functor where Dc denotes the underlying topological space of c ∈ Cart, and where |−| : Set ∆ → ∆Top is the geometric realisation functor. If L : C → D : R is a pair of adjoint functors, we also express this by the notation L ⊣ R. We prove: Theorem 1.2 Each arrow in the following diagram is a Quillen equivalence (left or right as indicated) Equalities in this diagram indicate identities of model categories. Moreover, omitting the functor R e in the bottom-right triangle, the diagram is a commutative (up to canonical natural isomorphisms) diagram of Quillen equivalences (i.e. the left adjoints commute and the right adjoints commute).
The model structures in the top row are obtained by first localising H p/i at good open coverings of cartesian spaces and subsequently localising at the morphisms c × R → c. The fact that these model structures are equal to the localisation of H p/i only at the morphisms c × R → c holds true because each c ∈ Cart is contractible (see Corollary 2.12); it would not hold true if we were working with simplicial presheaves on the category Mfd of manifolds instead of Cart. This insight has many pleasant technical consequences. We provide a comparison with the theory of simplicial sheaves on manifolds in Appendix B.
We remark that there also exists a further model category for a theory of smooth spaces: the R-local model category of enriched simplicial presheaves on a simplicial category explored in [HS11] 1 . This model category is Quillen equivalent to H p/i I by [HS11,Thm. 2.4] (see also [BEBdBP,Sec. 2] for applications to R-local simplicial presheaves on the category of manifolds in particular).
Next, we prove various comparison results between the different functors in diagram (1.3). The most important one of these, stated below, underlines the significance of the functor S e : H → Set ∆ : it is a model for the homotopy colimit of diagrams Cart op → Set ∆ .
Theorem 1.4 Let Q p : H p → H p be a cofibrant replacement functor for the projective model structure. There is a zig-zag of natural weak equivalences In particular, there is a natural isomorphism in h(∆Top), and a natural isomorphism in h(Set ∆ ), On a very formal level, it allows us to identify S e as a presentation of the left adjoint in the cohesive structure on the ∞-topos of presheaves of spaces on Cart. This has also been observed recently in [BEBdBP].
Finally, we construct a fibrant replacement functor for H p/i I , motivated by the concordance sheaves introduced in [BEBdBP], generalising concepts from [MW07]. We thereby obtain explicit access to the mapping spaces in H p/i I : Theorem 1.5 Let F, G ∈ H be any simplicial presheaves on Cart. Let M ∈ Mfd be any manifold, and define M ∈ H by setting M (c) = Mfd(c, M ) for any cartesian space c ∈ Cart. There are canonical isomorphisms in hSet ∆ , the homotopy category of spaces.
Finally, we remark that during the revision of this paper, building on Theorem 1.5 and [BEBdBP], Schreiber and Sati have enhanced this result to compute the homotopy type of the mapping space is its associated sheaf, and F ∈ H is a homotopy sheaf [SS,Thm. 3.3.53] (this problem goes back to a question by C. Rezk and originally was answered by D. Pavlov 2 ). Moreover, the I-localisation of ∞-sheaves on manifolds and their fibrant replacement have also been treated extensively in [ADH] since the first version of this paper appeared.
Outline. In Section 2 we define the R-localisations H p/i I of H p/i . We show that H p/i I can also be obtained as further localisations of the Čech local model structures with respect to differentiably good open coverings. We define the functor Re : H pI → ∆Top and show that it is a left Quillen equivalence.
In Section 3, we study the smooth singular complex functor S e : H → Set ∆ . We first show that S e : H p/i I → Set ∆ is a left Quillen equivalence. Subsequently, we establish S e : H iI → Set ∆ as a right Quillen equivalence. As an intermediate step, we relate the model categories H iI and Set ∆ to localisations of the model category of complete Segal spaces.
Section 4 is concerned with the comparison of different ways of extracting spaces from simplicial presheaves on Cart. The key concept is to extend the homotopy equivalence that embeds the topological standard simplices into the smooth extended simplices to obtain natural weak equivalences between functors from H to Set ∆ and ∆Top. We show that, for each manifold M , the space S e M assigned to its associated sheaf recovers the homotopy type of M and prove Theorem 1.4.
In Section 5 we construct a fibrant replacement functor for H p/i I and use it to prove Theorem 1.5. We spell out the relation of this theorem to [BEBdBP] and works of Dugger. Then we apply Theorem 5.6 to prove the coincidence of model structures from Theorem 1.1.
We include two appendices; Appendix A contains the explicit construction of a fibrant replacement functor for the injective model structure on H, which features in the proof of Theorem 1.5. Building on this, we provide a Quillen equivalence between model categories for homotopy sheaves on Cart and homotopy sheaves on Mfd in Appendix B.

∆
The simplex category Homotopy category of a model category M • Given two categories C, I, we write Cat(I, C) or C I for the category of functors I → C.
• We will also be working with the category sSet ∆ = Cat(∆ op , Set ∆ ) of bisimplicial sets. Our convention is always to write a bisimplicial set as a functor

Acknowledgements
The author is grateful to Birgit Richter for her encouragement to carry out this project. Further, the author would like to thank Walker Stern and Lukas Müller for numerous discussions, as well as Dan Christensen, Enxin Wu, Daniel Berwick-Evans, Pedro Boavida de Brito and Dmitri Pavlov for insightful comments on an earlier version of this paper. Finally, the author is grateful to the anonymous referee, whose very careful reading and detailed comments improved this paper notably. The author was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy-EXC 2121 "Quantum Universe"-390833306.

R-local model structures and smooth spaces
We start by setting up the model-categorical background used in this paper. Partially following [Sch] and [Dugb], we consider the category of simplicial presheaves on cartesian spaces with its canonical projective and injective model structures. In analogy with A 1 -local homotopy theory, we localise this category at all the morphisms c × R → c, where c is any cartesian space and where the morphism is the projection onto the factor c. Extending ideas from [Dugb], we establish several Quillen equivalences of this localised model category with the categories of simplicial sets and topological spaces.

R-local model structures on simplicial presheaves
In this section we start by setting up the various model structures on simplicial presheaves that will play a role in this article. We start by introducing the central mathematical objects.
Definition 2.1 We let Cart denote the (small) category of submanifolds of R ∞ that are diffeomorphic to R n , for any n ∈ N 0 . These manifolds are called cartesian spaces. The morphisms c → d in Cart are the smooth maps c → d between these manifolds.
In other words, Cart is the full subcategory of the category Mfd of smooth manifolds and smooth maps on the cartesian spaces. We can view each c ∈ Cart as a manifold via the inclusion Cart ⊂ Mfd, and we have c = Y c . We view Set ∆ as endowed with the Kan-Quillen model structure. The category Cart carries a Grothendieck coverage τ of differentiably good open coverings-see [FSS12, App. A] for details. A covering of c ∈ Cart in this coverage is a collection of morphisms {ι i : c i → c} i∈Λ in Cart such that each ι i is an embedding of an open subset, the images of the maps ι i cover c (i.e. each x ∈ c lies in the image of some ι i ), and every finite intersection This is the projective (resp. injective) model structure for sheaves of ∞-groupoids on Cart. In particular, a cartesian model category is a symmetric monoidal model category whose monoidal product is the category-theoretic product.
Proposition 2.7 All model structures in Definition 2.5 are simplicial, left proper, tractable and symmetric monoidal.
Proof. Except for the claim that the model structures are symmetric monoidal, all assertions follow from [ Bar10,Thm. 4.7], building on results in [Bek00]. The model structures for sheaves of ∞-groupoids are symmetric monoidal by [Bar10,Thm. 4.58]. To see that H p/i I is symmetric monoidal, observe that the objects Y c , c ∈ Cart, form a set of homotopy generators for H p/i . Let F ∈ H p/i I be a local object, and consider the internal hom object F Y d for any d ∈ Cart. For any of the morphisms Y c × Y R → Y c in I, the internal hom adjunction yields a commutative diagram of simplicially enriched hom spaces Here we have used that Cart has finite products. The bottom horizontal morphism is induced by the morphism c × d × R → c × d, which is an element of I. Hence, the bottom morphism is a weak equivalence in Set ∆ . Therefore, it follows from [ Bar10,Prop. 4.47] that H p/i I is symmetric monoidal. The exact same proof shows that H p/i ℓI is symmetric monoidal as well. The fact that H p/i Iℓ is symmetric monoidal will follow from Corollary 2.12.
The injective case can also be found in [Rez10]. We record the following two direct observations: Proposition 2.8 There are commutative diagrams of simplicial Quillen adjunctions: where the rightwards and downwards arrows are the left adjoints. All arrows are identity functors, and all horizontal arrows are Quillen equivalences.
Proposition 2.9 Each pair of model categories defined in Definition 2.5(1)-(4) (based on either the projective or the injective model structure), respectively, has the same weak equivalences. That is, their respective underlying relative categories agree.
The reason why we also refer to fibrant objects in H p/i I as essentially constant simplicial presheaves is the following fact (the second statement is a generalisation of [Dugb, Lemma 3.4.2]): Proposition 2.10 Let F ∈ H. The following statements hold true: (1) The canonical morphism F ⊗ Y c → F is a weak equivalence in H p/i I , for every c ∈ Cart.
(2) Let F ∈ H p/i be fibrant. The following are equivalent: Proof. Ad (1): By Proposition 2.9, it suffices to show this for H iI . There, every object is cofibrant (since every object in Set ∆ is cofibrant), so that the functor F ⊗ (−) : H iI → H iI is left Quillen. Thus, it suffices to show that the morphism Y c → * is a weak equivalence. Since c ∼ = R n for some n ∈ N 0 , we can reduce to the case where c = R n .
We can write the collapse morphism R n → * as a composition where each arrow is an element of I. Thus, the claim follows.
Ad (2): Condition (ii) implies that F is fibrant in H p/i I , i.e. condition (i): since Cart has finite products, we have a commutative triangle for any c ∈ Cart. The fact that F is I-local thus follows from the two-out-of-three property of weak equivalences in Set ∆ .
We now show that (i) implies (ii): By part (1) we know that for each c ∈ Cart, the morphism Y c → * is a weak equivalence in H p/i I . The claim then follows from the enriched Yoneda Lemma: the top arrow in the commutative diagram is a weak equivalence in Set ∆ , for every c ∈ Cart. That is, an object in H p/i ℓI is fibrant precisely if it is fibrant in both H p/i ℓ and in H p/i I . In particular, this implies that F is fibrant also in H p/i I .
To prove the converse, we first introduce some notation. Given a set S, let S [·] ∈ Set ∆ denote the simplicial set whose n-simplices are n+1-tuples (i 0 , . . . , i n ) ∈ S n+1 of elements in S; its i-th face maps forget the respective i-th entry, and its j-th degeneracy maps duplicate the respective j-th entry of a tuple. Consider an object c ∈ Cart, and let U = {c i → c} i∈Λ be a differentiably good open covering of c. Let Λ n ne ⊂ Λ n+1 be the subset on those n+1-tuples (i 0 , . . . , i n ) ∈ Λ n+1 such that C i 0 ···in = ∅ (see (2.3) for the notation). One checks that this defines a simplicial subset Λ ne ⊂ Λ [·] .
Given a projectively fibrant simplicial presheaf F ∈ H, consider the maps of simplicial sets The second map is a weak equivalence because representables are already cofibrant. Now, let F ∈ H p/i I be fibrant. We need to check that F satisfies descent with respect to the Grothendieck coverage τ of differentiably good open coverings on Cart. Since F is essentially constant (i.e. fibrant in F ∈ H p/i I ), by Proposition 2.10 the collapse maps c → * induce weak equivalences F ( * ) ∼ −→ F (c). We thus have a commutative diagram in Set ∆ . We claim that the top morphism in this diagram is an equivalence: to see this, we first note that, by assumption on the open covering U, the collapse morphism Sing(C i 0 ...in ) → * is a weak equivalence in Set ∆ for any i 0 , . . . , i n ∈ Λ such that C i 0 ...in is non-empty. For any fibrant K ∈ Set ∆ , we thus obtain a weak equivalence Since F ( * ) ∈ Set ∆ is fibrant, the product of these morphisms indexed by i 0 , . . . , i n ∈ Λ is still a weak equivalence, so we obtain a commutative diagram Sing(C i 0 ...in ) · · · , F ( * ) .
is a weak equivalence in Set ∆ . This is preserved by the right Quillen functor Set ∆ (−, F ( * )), and thus the claim follows.
Remark 2.14 Corollary 2.12 fails if one considers simplicial (pre)sheaves on manifolds rather than on cartesian spaces: the proof we give above relies on the fact that the left-hand vertical morphism in Diagram (2.13) is a weak equivalence. This is true because every c ∈ Cart has an underlying topological space which is contractible. In contrast, consider a simplicial presheaf on manifolds, G : Proof. This follows from Proposition 2.10 and Theorem 2.11.

Evaluation on the point
Here we present the first of several ways of extracting a space from an object F ∈ H and show that it provides a Quillen equivalence between H p/i I and the Kan-Quillen model category Set ∆ .

Consider the Quillen adjunctionc
: Set ∆ H p/i : ev * , ⊥ whose left adjointc sends a simplicial set K to the constant simplicial presheaf with value K, and whose right adjoint evaluates a simplicial presheaf at the final object * ∈ Cart. (Indeed, the adjunction is Quillen for both targets H p and H i ; in the projective case, we readily see that ev * is right Quillen, and in the injective case we see thatc is left Quillen.) Composing this with the localisation adjunction Proof. For any F ∈ H, the morphism e |F of simplicial presheaves is the morphisms F (c) → F ( * ) in Set ∆ induced by the collapse maps c → * . It readily follows from Proposition 2.10 that e |F is an objectwise weak equivalence whenever F is fibrant.
Proof. The claim follows from Proposition 2.15 since the morphismcK → R ic K is an objectwise weak equivalence.
We can now prove a version of [Dugb,Thm. 3.4.3] in the context of simplicial presheaves on cartesian spaces rather than on manifolds. (There, the proof is outlined for simplicial presheaves on manifolds, where several additional steps are necessary. Since we work over cartesian spaces, we can employ a slightly different strategy in our proof that allows us to avoid these additional steps.) Let cc p/i : 1 H ∼ −→ Cc p/i be a functorial fibrant replacement in H p/i I (we provide an explicit construction in Section 5). Note that once a fibrant replacement Cc p is given, cc i : 1 H ∼ −→ Cc i can be defined as the composition where r i : 1 H → R i is a fibrant replacement in the injective model structure. We always take Cc i to be of this form.
Lemma 2.19 For each K ∈ Set ∆ , the morphism cc p/ĩ cK :cK −→ Cc p/ic K is an objectwise weak equivalence.
Proof. Let R denote a fibrant replacement functor in Set ∆ (such as the Ex ∞ -functor). Consider the commutative diagramc The left vertical morphism is an objectwise weak equivalence. The right vertical morphism is a weak equivalence in H pI between fibrant objects (in H pI ). Thus, it is also a weak equivalence in H p . The bottom left object is fibrant in H pI by construction. Hence, the bottom horizontal morphism, which is only a weak equivalence in H pI a priori, is even an objectwise weak equivalence. Therefore, by the two-out-of-three property of weak equivalences in H p , the top horizontal morphism is an objectwise weak equivalence as well. The injective case follows since cc i is the composition of this morphism by the objectwise weak equivalence r ĩ cK . Theorem 2.20 The Quillen adjunctionc ⊣ ev * from (2.16) is a Quillen equivalence.
Proof. We will show that both the derived unit and derived counit of the Quillen adjunctionc ⊣ ev * are weak equivalences. This implies the claim by [Hov99, Prop. 1.3.13]. For the derived counit, let F ∈ H p/i I be fibrant. Since all objects in Set ∆ are cofibrant, it suffices to check that is a weak equivalence in Set ∆ , where e |F is the component at F of the counit ofc ⊣ ev * . This holds true by Lemma 2.17. For the derived unit, let K ∈ Set ∆ and consider the composition The morphism η K is an isomorphism, and the morphism cc p/ĩ cK is an objectwise weak equivalence by Lemma 2.19.
Corollary 2.21 Let W Set ∆ denote the weak equivalences in Set ∆ and W I those in H p/i I . The functor c preserves and reflects weak equivalences as a functor of relative categories (Set ∆ , W Set ∆ ) −→ (H, W I ). Example 2.22 Let G be a Lie group with Lie algebra g. Consider the object Bun ∇ G,0 ∈ H p , whose value on c ∈ Cart is the nerve of the following groupoid (in particular, Bun ∇ G,0 is fibrant in H p by construction): its objects are g-valued 1-forms A ∈ Ω 1 (c, g) such that dA + 1 2 [A, A] = 0, and its morphisms A → A ′ are smooth maps g : In other words, A is a flat G-connection on a trivial principal Gbundle on c, and g is equivalently a morphism of flat principal G-bundles on c. In particular, any such morphism g is actually a constant map g : c → G. Observe that Bun ∇ G,0 ( * ) is the nerve of the groupoid with one object and the group underlying G as its morphisms. It hence follows that the functor is fully faithful (on the underlying groupoids), for any c ∈ Cart. Since any flat G-bundle on c is isomorphic to the trivial flat G-bundle (because c ∼ = R n for some n ∈ N 0 ), the functor Bun ∇ G,0 ( * ) −→ Bun ∇ G,0 (c) is also essentially surjective. Since the nerve of an equivalence of groupoids is an equivalence of Kan complexes, it follows that Bun ∇ G,0 is a fibrant object in H pI . ⊳

Topological realisation
In this subsection we further build on and extend ideas from [Dugb] to investigate a second way of obtaining a space from a simplicial presheaf on Cart. This time, we send a simplicial presheaf to a certain coend valued in topological spaces.
More precisely, we let ∆Top denote the category of ∆-generated topological spaces (see [Duga,Vog71] for background). We will be working with ∆Top as our choice of category of topological spaces throughout; however, most of the theory in this paper also works with the category of Kelley spaces (also known as k-spaces; see, for instance, [Hov99]), except for where we work explicitly with diffeological spaces (Lemma 2.32, Remark 4.13).
We provide some very compact background on ∆-generated topological spaces. A topological space X is ∆-generated precisely if its topology coincides with the final topology induced by all continuous maps |∆ n | → X, for all n ∈ N 0 . (Here, |∆ n | is the standard topological n-simplex.) In particular, the category of ∆-generated topological spaces and continuous maps, denoted ∆Top, is cartesian closed [Vog71]. The product in ∆Top, however, is not the usual product of topological spaces-one has to pass to the ∆-generated topology after taking the usual product of topological spaces.
The category ∆Top carries a cofibrantly generated model structure, having the same generating cofibrations and generating trivial cofibrations as the standard model structure on topological spaces [Duga,Har15,FR08,SYH]. The geometric realisation functor |−| : Set ∆ → ∆Top takes values in ∆-generated spaces, since ∆Top is closed under colimits of topological spaces and contains |∆ n | for each n ∈ N 0 . By construction of the model structure on ∆Top, the induced adjunction |−| : Set ∆ ⇄ ∆Top : Sing is a Quillen adjunction (|−| sends generating (trivial) cofibrations to (trivial) cofibrations). Further, by the same proof as in [Hov99, Lemma 3.18], it follows that |−| preserves finite products. Then, the proof of [Hov99, Prop. 4.2.11] applies as well, showing that ∆Top is a symmetric monoidal (even cartesian) model category. It also follows that ∆Top is a simplicial model category and that |−| is a monoidal left Quillen functor. Finally, the Quillen adjunction |−| ⊣ Sing is even a Quillen equivalence, since the inclusion of ∆Top into Kelley spaces (or all topological spaces) is a Quillen equivalence [Duga]; the claim then follows from the two-out-of-three property of Quillen equivalences.
In this section, we provide a left Quillen equivalence H pI → ∆Top. The main ideas for this section stem from [Dugb]; there the full proof is technically rather involved. Again, we circumvent these problems here by working over cartesian spaces rather than over the category of manifolds.
Let Dfg denote the category of diffeological spaces (see [Sou80,IZ13]); we use the conventions of [Bunb], so that Dfg is the full subcategory of Cat(Cart op , Set) on the concrete sheaves with respect to the Grothendieck coverage τ (see the beginning of Section 2.1). Concretely, a diffeological space can be defined as a pair (X, Plot X ), where X ∈ Set, and where Plot X assigns to every cartesian space c ∈ Cart a subset Plot X (c) ⊂ Set(c, X) of the maps from the underlying set of c to X. These maps are called plots of X and have to satisfy that (1) Plot X ( * ) = X (every constant map is a plot), (2) for every f ∈ Cart(c, d) and every g ∈ Plot X (d), we have that g • f ∈ Plot X (c) (i.e. Plot X is a presheaf on Cart), and (3) the presheaf Plot X is a sheaf with respect to τ . We will often identify a diffeological space (X, Plot X ) with the sheaf it defines (see [Bunb] for more background), and we will denote this simply by X.
Example 2.23 For any manifold M ∈ Mfd, the presheaf M , given by c → Mfd(c, M ), is a diffeological space. In particular, this applies to every cartesian space d ∈ Cart; for these, we have d = Y d as (pre)sheaves on Cart. ⊳ Definition 2.24 Let D : Dfg → ∆Top be the functor defined as follows: for X ∈ Dfg, we let DX be the underlying set of the diffeological space X ∈ Dfg, endowed with the final topology defined by its plots c → X, where c ranges over all cartesian spaces. A morphism f ∈ Dfg(X, Y ) is sent to the map it defines on the sets underlying X and Y . We call D the diffeological topology functor and D(X) the underlying topological space of X.
The ∆-generated topological spaces are in fact precisely those topological spaces that arise as the underlying topological spaces of diffeological spaces [SYH,CSW14]. The following proposition consists of results that can already be found in [CSW14]; we only include the proofs here for completeness. Part (2) follows readily from Part (1) together with the fact that M × N is again a manifold.
Part (3) is merely [CSW14, Lemma 4.1] and the remarks following that lemma. For completeness, we fill in the details omitted there. In [CSW14] it is proven that the natural map D(X ×Y ) → DX ×DY is a homeomorphism whenever DX is locally compact Hausdorff. Since Dc is locally compact Hausdorff for any c ∈ Cart, and since D preserves colimits, we have the following canonical isomorphisms in ∆Top: let X, Y ∈ Dfg be arbitrary. Using that Dfg and ∆Top are cartesian closed, we compute In the third isomorphism we have used the above-mentioned result [CSW14, Lemma 4.1].
Remark 2. 27 We point out that we only use manifolds without boundary or corners here. For manifolds with boundary, part (1) of Proposition 2.26 fails-see, for instance, [CW14,Cor. 4.47]. ⊳ Since each cartesian space c ∈ Cart is diffeomorphic to R n for some n ∈ N 0 , and since R n is (isomorphic to) a CW complex for any n ∈ N 0 , it follows that Dc is cofibrant in ∆Top for every c ∈ Cart. We have the following version of [Dug01b, Prop. 2.3]: Theorem 2.28 There exists a Quillen adjunction Re ⊣ S, sitting inside a weakly commutative diagram Proof. The functor Re is defined as the (enriched) left Kan extension of D along Y in digram (2.29). Explicitly, we can write Since Dc is cofibrant in ∆Top and Sing : ∆Top → Set ∆ is right Quillen, it follows that S maps fibrations (resp. trivial fibrations) in ∆Top to objectwise fibrations (resp. trivial fibrations) in H. Thus, S is right Quillen.
For the second claim, we observe the canonical isomorphisms for all d ∈ Cart. The statement now follows from Proposition 2.26(1).

Lemma 2.31
The adjunction Re ⊣ S has the following properties: (1) It is a simplicial adjunction.
Proof. Part (1) holds true since geometric realisation preserves finite products of simplicial sets and since the functor K ⊗ (−) : ∆Top → ∆Top is a left adjoint, for any K ∈ Set ∆ . Part (2) holds true since S is right adjoint and ∆Top is cartesian monoidal. Proof. For X ∈ Dfg, we have canonical natural isomorphisms Combining this with Proposition 2.26 completes the proof. Proposition 2.34 The functor Re from diagram (2.29) has the following properties: (2) For each differentiably good open covering U = {c a → c} a∈A in Cart, the functor Re sends the Čech nerveČU → Y c to a weak equivalence in ∆Top.
Proof. Ad (1): The morphism is a weak equivalence in H pI between cofibrant objects by Proposition 2.10. Therefore, the claim follows from Proposition 2.33.
Ad (2): LetČU → Y c denote the Čech nerve of the open covering U. We view this as a morphism from a simplicial presheafČU to a simplicially constant presheaf Y c . Since U is a differentiably good open covering,ČU is levelwise a coproduct of representable presheaves on Cart; hence,ČU is cofibrant in H p . By construction of the Čech model structure H pℓ , the morphismČU → Y c is a weak equivalence in H pℓ . By Corollary 2.12, this is also a weak equivalence in H pI . The result now follows from Proposition 2.33 and since bothČU and Y c are cofibrant.
We now prove an important property of the model categories H p/i I which allows us to detect I-local weak equivalences. Dugger calls this property rigidity in [Dugb]. Proof. Since ψ is a morphism between local objects in a left Bousfield localisation of H p/i , it is an equivalence in H p/i I if and only if it is an objectwise weak equivalence in H p/i . For each c ∈ Cart, the morphism c → * = R 0 induces a commutative square Since F and G are R-local, the claim now follows from Proposition 2.10.
Theorem 2.36 There is a commutative diagram of simplicial Quillen equivalences wherec, Re, and |−| are the left adjoints.
Proof. It is well-established that the pair |−| ⊣ Sing is a simplicial Quillen equivalence (see e.g. [Hov99]). We have also seen in Theorem 2.20 that the adjunctionc ⊣ ev * is a simplicial Quillen equivalence. The commutativity of (2.37) follows from the definitions (2.30) of the functors Re and S, which use |−| and Sing, respectively. The fact that Re ⊣ S is a Quillen equivalence then follows from the two-out-of-three property of Quillen equivalences.
Remark 2.38 A slightly different version of Theorem 2.36 has been found previously in [Dug01b,Dugb], working over Mfd instead of Cart. We found that Cart has several technical advantages (in particular due to Corollary 2.12) and provides a sufficiently large category of parameter spaces to describe geometric and topological structures, as Theorem 2.36 shows (see also [Sch] for various applications of this formalism). ⊳

The singular complex of a simplicial presheaf
In this section we introduce the smooth singular complex, sometimes also called the concordance space, of a simplicial presheaf on Cart. We investigate its homotopical properties-for instance, it sends smooth homotopies to simplicial homotopies-and we establish it both as a left Quillen equivalence H p/i I → Set ∆ and as a right Quillen equivalence H iI → Set ∆ .

Extended simplices and the smooth singular complex
In a fashion similar to motivic homotopy theory (see e.g. [MV99, Voe98, DLØ + 07]), we consider the extended affine simplices in order to build our smooth singular complex functor. However, we purely rely on the smooth manifold structure of the affine cartesian simplices rather than on their function algebras.
Definition 3.1 The extended n-simplex is the cartesian space Face and degeneracy maps are defined as the affine linear extensions of the face and degeneracy maps of the standard simplices |∆ n |. The extended simplices thus define a functor ∆ e : ∆ → Cart.
By construction, the topological standard simplex is a subset of the extended simplex ∆ n e , for any n ∈ N 0 . This inclusion |∆ n | ֒→ ∆ n e is compatible with the face and degeneracy maps. Recalling the functor D : Dfg → ∆Top from Definition 2.24, we see that there is a morphism ι : |∆| → D∆ e of functors ∆ → ∆Top. In particular, the diagram in ∆Top commutes for every morphism σ ∈ ∆([n], [k]).
The extended simplices functor ∆ e induces a Quillen adjunction of functors sSet ∆ → Set ∆ , where δ * (X) n = X n,n is the pullback along the diagonal functor.
Proof. This is a standard application of the Yoneda Lemma in the (co)end calculus.
Corollary 3.5 The diagonal functor is a left Quillen functor In particular, it is homotopical, i.e. it preserves all weak equivalences.
Consequently, we can define a left Quillen functor as the composition Consider a complete and cocomplete category E, two categories C, D, and a functor F : C → D. Recall that, in this situation, the functor F * : Cat(D, E) −→ Cat(C, E) has a left adjoint F ! and a right adjoint F * , which are given by the left and the right Kan extension along F . By the construction of S e as a composition of pullback functors which act on categories of simplicial presheaves, we infer: Proposition 3.6 The functor S e = δ * • ∆ * e has both adjoints. We thus obtain a triple of adjunctions L e ⊣ S e ⊣ R e , where L e and R e are given by the compositions and R e = ∆ e * • δ * .
The adjunction S e ⊣ R e is a simplicial Quillen adjunction.
Definition 3.7 We call the functor S e : H p/i → Set ∆ the smooth singular complex functor. For F ∈ H, the simplicial set S e F is called the smooth singular complex of F .

S e as a left Quillen equivalence
We further investigate the homotopical properties of the smooth singular complex functor S e . So far, we know that the adjunction S e : H p/i ⇄ Set ∆ : R e is Quillen. Our goal here is to show that this Quillen adjunction descends to the localisation H p/i I and that there it even forms a Quillen equivalence.
Definition 3.8 Let F, G ∈ H be two simplicial presheaves on Cart, and let f 0 , f 1 : F → G be a pair of morphisms. A smooth homotopy from f 0 to f 1 is a commutative diagram in H, where the vertical inclusions are induced by the maps * → R, given by * → 0 and * → 1.
Lemma 3.10 The functor S e : H → Set ∆ maps smoothly homotopic morphisms to simplicially homotopic morphisms.
Proof. The projection (t 0 , t 1 ) → t 0 yields a diffeomorphism ψ : ∆ 1 e → R of cartesian spaces. Observe that there is a morphism of simplicial sets , defined by sending the generating non-degenerate 1-simplex of ∆ 1 to the 1-simplex ψ. Hence, using the fact that S e preserves products, we apply S e to diagram (3.9) and augment it using ν to obtain a commutative diagram This establishes a simplicial homotopy S e h • (1 X × ν) from S e f 0 to S e f 1 .
Lemma 3.10 can be seen as a generalisation of [CW14, Lemma 4.10] from diffeological spaces to simplicial presheaves. Indeed, the composition is precisely the smooth singular functor from [CW14].
Proposition 3.11 For any c ∈ Cart, the functor S e sends the collapse morphism c : Y c → * to a weak equivalence in Set ∆ .
Proof. Let c ∈ Cart, and let x ∈ c be any point. The inclusion x : * → c induces a smooth homotopy equivalence * ⇄ c. The functor S e maps this to a simplicial homotopy equivalence according to Lemma 3.10.
Corollary 3.12 The functor S e induces Quillen adjunctions We call a functor (C, W C ) → (D, W D ) between relative categories homotopical if it preserves weak equivalences (see also [DHKS04,Shu,Rie14] for more background on homotopical categories and functors). Proof. S e : H iI → Set ∆ is homotopical because it is left Quillen and every object in H iI is cofibrant. The corresponding statement for the projective model structure now follows from Proposition 2.9.
Note, in particular, that by Proposition 2.12 the functor S e also sends weak equivalences in the Čech local model structures H p/i ℓ to weak equivalences in Set ∆ . Proof. By Proposition 3.10, S e sends smooth homotopy equivalences to simplicial homotopy equivalences, which are, in particular, weak equivalences in Set ∆ . Thus, the claim follows from Corollary 3.15.
Remark 3.17 Let W Set ∆ denote the class of weak equivalences in Set ∆ , and let S −1 e (W Set ∆ ) denote the class of morphisms in H whose image under S e is in W Set ∆ . Corollary 3.15 lets us suspect that there is an equivalence of model categories H p/i I ≃ L S −1 e (W Set ∆ ) H p/i . Using properties of local weak equivalences in Bousfield localisations should allow us to prove that conjecture here already, but instead we give a very direct proof later in Theorem 5.7. ⊳

S e as a right Quillen equivalence
The goal of this subsection is to establish the smooth singular functor as a right Quillen functor S e : H iI → Set ∆ . Apart from having convenient technical implications on the functor S e : H iI → Set ∆ , the appearance of several intermediate model structures of bisimplicial sets sheds additional light on the functor S e . We already know from Proposition 3.6 that S e = δ * •∆ * e has a left adjoint L e = ∆ e! •δ ! . We will show that both its constituting functors ∆ e! and δ ! are left Quillen functors.

Model structures for ∞-groupoids on the category of bisimplicial sets
We start by analysing the functor δ ! in more detail. Let denote the spine-inclusion of the n-simplex ∆ n , for n ≥ 1. (Note that for n = 1 the morphism ι 1 is an isomorphism.) We write sSet ∆ = Cat(∆ op , Set ∆ ) for the category of bisimplicial sets. There exists a bifunctor We view a bisimplicial set X as a simplicial diagram m → X m,• in Set ∆ . Let denote the set of all spine inclusions, viewed as maps of vertically constant bisimplicial sets. Let J ∈ Set ∆ denote the nerve of the groupoid with two objects and a unique isomorphism between them.
Definition 3. 18 We define the following model structures on the category sSet ∆ of bisimplicial sets: (1) We view sSet ∆ = Cat(∆ op , Set ∆ ) as endowed with the injective model structure. Recall that this coincides with the Reedy model structure [Hir03].
(2) We let SSp := L Sp sSet ∆ be the left Bousfield localisation of sSet ∆ at the spine inclusions. This is the model category for Segal spaces.
(3) The model category for complete Segal spaces is the localisation CSS := L J⊠∆ 0 SSp.
Let L ∆ • ⊠∆ 0 sSet ∆ denote the left Bousfield localisation of the injective model category of bisimplicial sets at all collapse morphisms (Compare these localisations to those in Proposition 2.15.) Finally, let L ∆ 1 ⊠∆ 0 SSp and L ∆ 1 ⊠∆ 0 CSS denote the left Bousfield localisations of SSp and CSS, respectively, at the morphism ∆ 1 ⊠ ∆ 0 → ∆ 0 ⊠ ∆ 0 . We will mostly be using the model category L ∆ • ⊠∆ 0 sSet ∆ , but for conceptual clarity and for an interpretation as model categories for ∞-groupoids, we include the following proposition.

Proposition 3.19
The following left Bousfield localisations yield identical model categories: Proof. By Theorem 2.11 it suffices to check that all four model categories have the same cofibrations and fibrant objects. For cofibrations, this is trivial since each of the model categories is a left Bousfield localisation of sSet ∆ . It thus remains to check that the fibrant objects of the three model categories coincide.
Identity (1) is a direct consequence of the two-out-of-three property of weak equivalences.
For identity (2), let X ∈ L ∆ • ⊠∆ 0 sSet ∆ be fibrant. That is, X is injective fibrant in sSet ∆ and the canonical map X 0 → X n is a weak equivalence in Set ∆ for any n ∈ N 0 . We have to show that X satisfies the Segal condition, i.e. that for every n ≥ 2 the spine inclusion Sp n ֒→ ∆ n induces a weak equivalence (As pointed out in [Rez01], the strict pullback is a homotopy pullback here because X is Reedy fibrant.) Consider the commutative diagram Since X is Reedy fibrant, the pullbacks on the right-hand side are homotopy pullbacks. Therefore, both vertical maps in (3.20) are weak equivalences. It follows by the commutativity of the diagram that X satisfies the Segal condition. Then, X is fibrant in L ∆ 1 ⊠∆ 0 SSp since, by assumption, the morphism X 0 → X 1 is a weak equivalence.
Conversely, if X is fibrant in L ∆ 1 ⊠∆ 0 SSp, then the top horizontal morphism in diagram (3.20) is a weak equivalence because X satisfies the Segal condition, and the right-hand vertical morphism is a weak equivalence because X is injective fibrant and X is local with respect to ∆ 1 ⊠ ∆ 0 → ∆ 0 ⊠ ∆ 0 . It thus follows by the commutativity of the diagram that also the left vertical morphism is an equivalence, for any n ≥ 2, so that X is fibrant in L ∆ • ⊠∆ 0 sSet ∆ . For identity (3), recall that in any Segal space X there is a notion of when a morphism f ∈ X 1 is invertible (or a 'homotopy equivalence' in the language of [Rez01]). One defines the space X weq of homotopy equivalences in X to be the union of those connected components of X 1 that contain invertible morphisms (by [Rez01, Lemma 5.8], if X ∈ SSp is fibrant, then a connected component of X 1 contains a homotopy equivalence if and only if it consists purely of homotopy equivalences). For any Segal space, the degeneracy morphism s 0 : X 0 → X 1 factors as Since every fibrant object in CSS is also fibrant in SSp, this implies that every fibrant object in Conversely, let Y ∈ SSp be fibrant. Then, Y is local in L ∆ 1 ⊠∆ 0 SSp precisely if the morphism s 0 : Y 0 → Y 1 is a weak equivalence. We need to show that, in that case, the morphism s 0 : Y 0 → Y weq is a weak equivalence in Set ∆ . However, since ι Y : Y weq → Y is the inclusion of a union of connected components of Y 1 , diagram (3.21) and the fact that s 0 is a weak equivalence imply that ι Y hits every connected component of Y 1 . Therefore, ι Y is a weak equivalence; it follows from the two-out-of-three property that s 0 : Y 0 → Y weq is a weak equivalence as well.
Remark 3.22 Let X ∈ SSp be fibrant. Since ι X : X weq → X 1 is the inclusion of a union of connected components of X 1 , it follows that ι X is a weak equivalence precisely if it is an isomorphism, i.e. precisely if X weq = X 1 . In other words, a fibrant object in both L ∆ 1 ⊠∆ 0 SSp and L ∆ 1 ⊠∆ 0 CSS is a complete Segal space with all 1-morphisms invertible. In that sense, the fibrant objects in these model categories are ∞-groupoids. The model category L ∆⊠∆ 0 sSet ∆ can be seen as the model category of essentially constant simplicial diagrams of spaces, in analogy to how L Cart H p/i describes essentially constant simplicial presheaves. Proposition 3.24 The diagonal δ * : sSet ∆ → Set ∆ induces a Quillen adjunction ⊥ From now on, we will understand the adjunction δ ! ⊣ δ * as the above Quillen adjunction. There is another Quillen adjunction that relates L ∆ • ⊠∆ 0 sSet ∆ to the model category of simplicial sets, in analogy with Theorem 2.20. (2) Composing the Quillen adjunction from (1) with the localisation adjunction sSet ∆ ⇄ L ∆ • ⊠∆ 0 sSet ∆ yields a Quillen equivalence It is straightforward to see that c ∆ : Set ∆ → sSet ∆ preserves cofibrations and further preserves as well as reflects weak equivalences. This proves claim (1). Part (2) proceeds entirely in parallel to the proofs of Lemma 2.19 and Theorem 2.20.
3.3.2 The functors ∆ e! and L e Next, we show that ∆ e! : sSet ∆ → H iI is left Quillen, and that the Quillen adjunction ∆ e! ⊣ ∆ e * descends to the localisation L ∆⊠∆ 0 sSet ∆ = L ∆ • ⊠∆ 0 sSet ∆ . The functor ∆ e! acts as In particular, for bisimplicial sets in the image of (−) ⊠ (−) we find that

It follows that
for any L ∈ Set ∆ , wherec : Set ∆ → H is the constant-presheaf functor.
Proof. For n = 0, 1 this is straightforward. Consider the presentation of ∂∆ n as a coequaliser, Since ∆ e! preserves colimits, we obtain Let f, g ∈ ∂∆ n e (c) be any two elements, and assume that ι n e • f = ι n e • g, where ι n e : ∂∆ n e −→ ∆ n e is the canonical morphism. Observe that ι n e is injective as a map on the underlying sets ∂∆ n e ( * ) ֒→ ∆ n e ( * ). Since every section f : Y c → ∂∆ n e is, in particular, a map Y c ( * ) → ∂∆ n e ( * ) of the underlying sets, and analogously a section Y c → ∆ n e is, in particular, a map Y c ( * ) → ∆ n e ( * ) of underlying sets, it follows that ι n e : ∂∆ n e → ∆ n e is an objectwise monomorphism.
Remark 3.31 We point out that ∂∆ n e , as defined in (3.30), is not a diffeological space for n ≥ 2. For instance, consider a differentiably good open covering c = c 0 ∪ c 1 of a cartesian space c. We denote the intersection c 0 ∩ c 1 by c 01 ∈ Cart. Let f i : c i → ∆ n−1 e be smooth maps, for i = 0, 1, to adjacent faces of ∆ n e , such that f i|c 01 : c 01 → ∆ n−2 e factors through the n−2-simplex which joins the two faces. These data to not lift to a section f ∈ ∂∆ n e (c), since such an f must factor through only one of the faces ∂∆ n−1 e . That is, ∂∆ n e does not satisfy the sheaf condition. ⊳ Let C, D, E be categories. Recall the notion of an adjunction of two variables C × D → E (see, for example, [Hov99, Def. 4.1.12]). We will denote an adjunction of two variables only by its tensor functor ⊗ : C × D → E. If E has pushouts, then there is an induced pushout product, or box product, on morphisms: given morphisms f : A → B in C and g : X → Y in D, their pushout product (relative to ⊗) is the induced morphism in E given by (1) if both f and g are cofibrations, then so is f g, and (2) if, in addition, f or g is a weak equivalence, then so is f g.
Definition 3.33 [Hov99, Def. 4.2.6] A (symmetric) monoidal model category is a closed (symmetric) monoidal category (C, ⊗) together with a model structure on the underlying category C such that: (1) the closed monoidal structure is a Quillen adjunction of two variables ⊗ : C × C → C.
(2) Let u ∈ C be the monoidal unit, and let q u : Q C u ∼ −→ u be a cofibrant replacement. Then, tensoring with any cofibrant object from the left or the right sends q u to a weak equivalence.
Example 3.34 The model category sSet ∆ with the injective model structure is cartesian (i.e. symmetric monoidal with ⊗ = ×). Similarly, each of the model categories H p/i , H p/i ℓ , and H p/i I is cartesian by Proposition 2.7. In each of the monoidal model structures we encounter here, the monoidal unit is already cofibrant, so that the second axiom of Definition 3.33 is trivially satisfied. ⊳ The injective model structure on sSet ∆ is cofibrantly generated (see e.g. [Rez01]), with generating cofibrations and generating trivial cofibrations which is an injective cofibration, and that which is an injective trivial cofibration.
It suffices the check that ∆ e! sends the generating (trivial) cofibrations of sSet ∆ to (trivial) cofibrations in H i (see, for instance, [Hov99, Lemma 2.1.20]). For n, m ∈ N 0 , the pushout-product is the canonical morphism Note that, for A, B, C, D ∈ Set ∆ , there is a canonical natural isomorphism in sSet ∆ . Thus, the above pushout-product is canonically isomorphic to to morphism Since the functor ∆ e! preserves pushouts, and using (3.27), this is sent to the morphism which coincides with the pushout product (∂∆ n e ֒→ ∆ n e ) (c∂∆ m ֒→c∆ m ) in H i . By Lemma 3.29 and the fact that the model category H i is symmetric monoidal (even cartesian), this pushout-product is a cofibration in H i . Thus, ∆ e! sends the generating cofibrations of Set ∆ to cofibrations. An analogous argument shows that it also sends the generating trivial cofibrations to trivial cofibrations.
Proof. This is a direct consequence of the fact that ∆ e! (∆ n ⊠ ∆ 0 ) ∼ = ∆ n e . The collapse morphism ∆ n e → * is a weak equivalence in H iI by Proposition 2.10. Corollary 3.37 The adjunction L e : Set ∆ ⇄ H iI : S e is a Quillen adjunction.
Proof. This is a direct consequence of Proposition 3.24 and Proposition 3.36.
Theorem 3.38 There is a commutative triangle of Quillen equivalences where c ∆ , ∆ e! , andc are the left adjoints.
Proof. We know from Theorem 2.20 that the bottom adjunction is a Quillen equivalence, and we know from Proposition 3.25 that the left diagonal adjunction is a Quillen equivalence. Further, it is evident that the diagram of right adjoints commutes strictly. Thus, the claim follows from the two-out-of-three property of Quillen equivalences.
(2) The right derived natural transformation of γ is a natural weak equivalence.
Proof. Consider first the functors ∆ * e , c ∆ • ev * : H p/i I −→ sSet ∆ . For any F ∈ H, the collapse map ∆ n e → * induces a morphism γ |F,n : F ( * ) → F (∆ n e ) of simplicial sets. Since * ∈ Cart is final, this induces a natural transformation γ : c ∆ • ev * −→ ∆ * e . Applying the diagonal functor δ * to this natural transformation, we obtain a natural transformation . This shows part (1). Part (2) then follows from the fact that, whenever F ∈ H p/i I is fibrant, the morphism F ( * ) → F (∆ n e ) is a weak equivalence for every n ∈ N 0 . Therefore, if F is fibrant, then γ |F : c ∆ (F ( * )) −→ ∆ * e F is an objectwise weak equivalence in sSet ∆ . The claim now follows from the fact that the diagonal functor δ * is homotopical.
Theorem 3.40 The Quillen adjunction ⊥ is a Quillen equivalence.
Proof. Lemma 3.39 provides a natural weak equivalence between the right derived functors of ev * and S e . By Theorem 2.20, the functor ev * is a right Quillen equivalence, and it follows that so is S e .
Corollary 3.41 The Quillen adjunction ⊥ is a Quillen equivalence.
Proof. This follows from the fact that L e = ∆ e! • δ ! , together with Theorem 3.38 and the two-out-ofthree property for Quillen equivalences.
Corollary 3.41 becomes particularly interesting in light of Proposition 3.19: it establishes a Quillen equivalence between the Kan-Quillen model structure on simplicial sets and each of the model structures in Proposition 3.19. In other words, Corollary 3.41 shows that each of the model categories from Proposition 3.19 is a model category for ∞-groupoids.

Comparison of spaces constructed from simplicial presheaves
In Sections 2 and 3 we have seen several ways of extracting a space from a simplicial presheaf on Cart. The main goal of Section 4.1 is to establish comparisons between the resulting spaces. We use these to show that S e is a model for the homotopy colimit of Cart op -shaped diagrams of simplicial sets in Section 4.2. In Section 4.3 we show that applying S e to the presheaf associated to a manifold reproduces the homotopy type of the topological space underlying the manifold.

Comparison results
The right adjoint of S e = δ * • ∆ * e is given as R e = ∆ e * • δ * . We start by making this functor more explicit: consider a simplicial set K ∈ Set ∆ and a cartesian space c ∈ Cart. Since the adjunction S e ⊣ R e is simplicial, there are natural isomorphisms The following lemma is then immediate: Lemma 4.2 There exist canonical natural isomorphisms S e •c ∼ = 1 Set ∆ , and ev * •R e ∼ = 1 Set ∆ .
It follows that there exists a natural isomorphism ev * •R e • Sing ∼ = Sing.

Lemma 4.3 There is an isomorphism
Proof. Since the adjunction |−| ⊣ Sing is simplicial (because |−| preserves finite products), we have binatural isomorphisms Here we have used that ∆Top is cartesian closed and simplicially enriched.
Recall the diffeological topology functor from Definition 2.24. Since ι • is a morphism of cosimplicial topological spaces, and since the maps ϕ M |n are defined by precomposition by ι n , it readily follows that ϕ M |n is natural in both M ∈ Mfd and n ∈ ∆. Thus, we obtain the desired morphism of simplicial sets  Proof. This follows readily from the observation that, for any cartesian space c ∈ Cart, both |S e Y c | and D(Y c ) are weakly equivalent to * ∈ ∆Top. Hence, by the two-out-of-three property of weak equivalences any morphism |S e Y c | −→ D(Y c ) is a weak equivalence. in H, which is natural in T ∈ ∆Top.
In order to see that the morphism Sing(T ψ ) is a weak equivalence in H pI , we first observe that both S(T ) and R e • Sing(T ) are fibrant in H pI . We have that is the identity. The fact that η is a natural weak equivalence now follows from the fact that the right Quillen equivalence ev * : H pI → Set ∆ reflects weak equivalences between fibrant objects (which was also the content of Proposition 2.35).
Corollary 4.7 There is a natural weak equivalence Proof. The functor S e : H p/i I → Set ∆ is homotopical. Applying it to the natural weak equivalence from Proposition 4.6, we obtain a natural weak equivalence Let e : S e • R e −→ 1 Set ∆ denote the counit of the adjunction S e ⊣ R e . The fact that every object in Set ∆ is cofibrant, together with the fact that S e ⊣ R e is a Quillen equivalence imply that the morphism e |K : S e • R e (K) −→ K is a weak equivalence in Set ∆ for every fibrant simplicial set K.
Since Sing : ∆Top → Set ∆ takes values in fibrant simplicial sets, it follows that the composition Sing is a natural weak equivalence.
Corollary 4.8 Let Q p : H pI → H pI be a cofibrant replacement functor, with associated natural weak equivalence q p : Q p → 1 H . There is a zig-zag of natural weak equivalences of functors H pI −→ Set ∆ .
Proof. The left-facing natural transformation is a weak equivalence since S e is homotopical. The right-facing morphism is the composition where u : 1 ∆Top −→ S • Re is the unit morphism of the adjunction Re ⊣ S. Since this adjunction is a Quillen equivalence and since every object in ∆Top is fibrant, it follows that u F : F → S • Re(F ) is a weak equivalence for every cofibrant object F ∈ H pI .
Proposition 4.10 There exists a zig-zag of natural weak equivalences of functors H pI −→ ∆Top. In particular, there exists a natural isomorphism of total left derived functors between homotopy categories Observe that S e and |−| are already homotopical, so that we do not need to precompose them by a cofibrant replacement in order to obtain their total left derived functors.
Proof. We readily obtain a zig-zag as in (4.11) by applying the functor |−| to the zig-zag (4.9) and then postcomposing by the counit e : |−| • Sing Remark 4.12 There is an alternative way of obtaining a zig-zag as in (4.11) directly and explicitly, when Q p is Dugger's cofibrant replacement functor for H p [Dug01b]. It sends a simplicial presheaf F to the two-sided bar construction in the notation of [Rie14]. (The superscript indicates in which simplicial category we are forming the bar construction.) Using that Re is simplicial and commutes with colimits, and that there is a natural isomorphism Re • Y c ∼ = Dc for any cartesian space c ∈ Cart (cf. Lemma 2.32), we obtain a canonical isomorphism On the other hand, since both |−| and S e are left adjoints, we have a natural isomorphism Now, the morphism q p : Q p ∼ −→ 1 H , together with the fact that both |−| and S e preserve weak equivalences, yield the claim. ⊳ Remark 4.13 Recall the embedding ι : Dfg ֒→ H of diffeological spaces into simplicial presheaves. By Lemma 2.32, the composition Re • ι agrees with the functor D : Dfg → ∆Top that sends a diffeological space to its underlying topological space, whose topology is induced by its plots. It is interesting to ask whether the homotopy type of DX agrees with that of the smooth singular complex S e ι(X) of X, for any diffeological space X ∈ Dfg. So far, however, we only see that the homotopy type of S e ιX agrees with the cobar construction rather than with the underlying topological space DX of X. This is in accordance with-and maybe provides some further insight to-results from [CSW14,OT] that the smooth singular complex of a diffeological space X is not in general equivalent to the smooth singular complex of DX. ⊳

Homotopy colimits of smooth spaces
We can interpret the functor Re-and because of Proposition 4.10 also the functor S e -in the context of the cohesive ∞-topos H of presheaves of spaces on Cart as follows (see [Sch] for more background and [Buna] for a brief introduction). From the proof of Proposition 4.10 we see that there are canonical weak equivalences Using the fact that the topological realisation of a bisimplicial set is independent of which simplicial direction one realises first (up to canonical isomorphism), we obtain canonical weak equivalences Combining this with Proposition 4.10, we obtain It follows that S e models the homotopy colimit of diagrams Cart op −→ Set ∆ .
Therefore, each of these functors presents the ∞-categorical colimit functor for diagrams of spaces indexed by Cart op . Consequently, on the level of the underlying ∞-categories they each present leftadjoints to the functor that sends a space to the constant presheaf whose value is that space. This means that both Re and S e provide explicit presentations for the left-adjoint Π in the three-fold adjunction which implements the cohesive structure on H (see [Sch]). A different argument for the case of S e has been given in [BEBdBP] (see the bottom of p. 2); here we establish a presentation of the ∞-categorical adjunction Π ⊣c as a Quillen adjunction H p/i ⇄ Set ∆ and a Quillen equivalence

Recovering homotopy types of manifolds
Recall the fully faithful embedding Mfd ֒→ H, M → M . In this section, we show that, for any manifold M , the smooth singular complex S e M and the ordinary singular complex Sing(DM ) of M are canonically weakly equivalent. In fact, this is a well-known and classical result in geometric topology. We nevertheless present this statement and a short proof in a language suited to this paper as a consistency check for the properties of the functor S e . Classical proofs of this statement can be found, for instance, in [Lee13,Thm. 18.7] and [War83, Secs. 5.31, 5.32] (a version with cubes instead of simplices is in [Ful95]). We can equivalently write whereČ n U = a 0 ,...,an∈A U a 0 ···an is the n-th level of the Čech nerve of U, and where we view the simplicial presheafČU as a diagram of simplicially constant presheaves on Cart. We thus arrive at a commutative diagram in Set ∆ . We have already argued that the left-hand vertical morphism is a weak equivalence. The top horizontal morphism is a weak equivalence as well: for each [n] ∈ ∆, the morphism ϕ induces a morphism between simplicial sets which are a disjoint union of contractible components. On the level of connected components, ϕ induces a bijection, and hence is a weak equivalence. Finally, the classical Nerve Theorem (see e.g. [Bor48,Ler50,Seg74,DI04]) implies that the right-hand vertical morphism is an equivalence as well.

Local fibrant replacement, concordance and mapping spaces
In this section, we present a fibrant replacement functor in the model structures H p/i I . Its construction is motivated by the concordance sheaf construction from [BEBdBP]. This functor allows us to compute mapping spaces in H p/i I in Theorem 5.6. We start by presenting the fibrant replacement functor: Lemma 5.1 Suppose that F 0 , F 1 ∈ H and that h : F 0 × R → F 1 is a smooth homotopy between morphisms f, g : F 0 → F 1 . Then, for any G ∈ H, there is a smooth homotopyh : Proof. Applying the exponential functor G (−) to the morphism h, we obtain a morphism Using the internal-hom adjunction of H then yields a morphismh = (G h ) ⊣ ∈ H(G F 1 × R, G F 0 ) with the desired properties.
We consider the following construction: let ∆ k e ∈ H denote the simplicial presheaf represented by the extended affine simplex ∆ k e ∈ Cart. This provides a functor ∆ e : ∆ → H , Given an object F ∈ H, we can compose this functor by the functor F (−) : H → H, obtaining an object F ∆ e ∈ Cat(∆ op , H). Equivalently, we can view this as a bisimplicial presheaf which we can now compose by the diagonal functor δ * : sSet ∆ → Set ∆ to obtain a new simplicial presheaf on Cart. Putting everything together, this defines a functor The collapse morphisms ∆ k e → * induce a natural transformation ∆ e → * of functors ∆ → H (this even consists of I-local equivalences by Proposition 2.10). From this we obtain a natural transformation Now, let R p/i : H → H be a fibrant replacement functor for the projective (resp. injective) model structure, with natural objectwise weak equivalence r p/i : 1 H ∼ −→ R p/i . (Observe, in particular, that a fibrant replacement functor R p for the projective model structure can be obtained by postcomposition with a fibrant replacement functor in Set ∆ , i.e. we can use R p (F ) = R Set ∆ • F for F ∈ H. An explicit model for an injective fibrant replacement functor is given in Appendix A.) We define functors for the projective and the injective model structure, respectively. We define the natural transformation By Lemma 5.1, the smooth homotopy equivalence Y c → * induces a smooth homotopy equivalence F → F Yc , which is a weak equivalence in H p/i I by Corollary 3.16. Since r p/i is a natural weak equivalence of functors valued in H p/i , its component is an objectwise weak equivalence. Consequently, for every c ∈ Cart, we have a commutative square whose vertical morphisms are induced from the collapse morphism c → * . It follows that Cc p/i F is R-local and hence a fibrant object in H p/i I . Finally, we need to show that the morphism γ |F : F −→ δ * • F ∆ e , induced by the collapse ∆ e → * , is a weak equivalence in H p/i I . Since the functor S e : H p/i I → Set ∆ preserves as well as reflects weak equivalences, that is equivalent to showing that the induced morphism is a weak equivalence of simplicial sets. More explicitly, S e (γ |F ) is the morphism induced by collapsing the extended simplices ∆ e without the tilde. Note that we have only added the tilde in order to keep track of which diagonal functor refers to which copy of ∆ e . We have canonical isomorphisms and we know from the first part of this proof that the morphism is a weak equivalence, for every k ∈ N 0 . This induces a (levelwise) weak equivalence of bisimplicial sets which under δ * maps to the morphism S e (γ |F ). Since the diagonal functor δ * : sSet ∆ → Set ∆ is homotopical, we obtain that S e (γ |F ), and thus also γ |F , is indeed a weak equivalence.
Corollary 5.3 Mapping spaces in H p/i I can be computed (up to isomorphism in hSet ∆ ) as the simplicially enriched hom spaces where Q p is a cofibrant replacement functor for the projective model structure H p .
Definition 5.5 Given an object F ∈ H, we call Cc p/i F its (projective/injective) derived concordance sheaf. For G ∈ H we refer to the spaces in (5.4) as the spaces of derived concordances of morphisms from F to G.
We can apply these insights to describe the mapping spaces in the model categories H p/i I . In particular, given any G ∈ H, part (3) of the following theorem shows that the derived sections of the derived concordance sheaf of G (in the sense of Definition 5.5) on manifolds are represented by maps from the space underlying M to the smooth singular complex of G.
Theorem 5.6 There are natural isomorphisms in hSet ∆ as follows: (1) For F, G ∈ H, we have (2) For F, G ∈ H, we have Proof. It suffices to consider one of the model structures by Proposition 2.9 and the fact that the mapping spaces in a model category depend only on the underlying relative category (see before Corollary A.11). For claim (1), we consider the injective model structure. Using that every object in H iI is cofibrant, we have the following isomorphisms in hSet ∆ : The first isomorphism is merely the fact that Set ∆ is a simplicial model category in which every object is cofibrant. In the second isomorphism, we use the fact that S e ⊣ R e is a simplicial adjunction. To see the third isomorphism, we use the weak equivalence cc i |G : G ∼ −→ Cc i (G) from Proposition 5.2. Since both S e and R Set ∆ are homotopical functors, and since R e preserves fibrant objects, the morphism R e R Set ∆ S e (cc i |G ) is a weak equivalence between fibrant objects in H iI , which is thus preserved by H(F, −). The fourth isomorphism stems from the fact that S e ⊣ R e is a Quillen equivalence, so that the canonical natural transformation 1 H −→ R e R Set ∆ S e is a weak equivalence in H iI on every cofibrant object-that is, it is a weak equivalence on every object, since all objects in H iI are cofibrant. Further, its component at the object Cc i (G) is a weak equivalence between fibrant objects in H iI , which is again preserved by H(F, −). The final isomorphism directly follows from the insight that Cc i is a fibrant replacement functor for H iI (Proposition 5.2).
Claim (2) then follows from the fact that every object in ∆Top is fibrant and that every object of Set ∆ is cofibrant: if K, L ∈ Set ∆ , then there are canonical isomorphisms in hSet ∆ Claim (3) then follows from combining part (2) with Theorem 4.15.
We can now give a direct proof of the relation between model categories announced in Remark 3.17: we characterise the homotopy theory induced on H by the smooth singular complex functor S e : Theorem 5.7 Let W Set ∆ denote the class of weak equivalences in Set ∆ , and let S −1 e (W Set ∆ ) denote the class of morphisms in H whose image under S e is in W Set ∆ . There is an identity of model categories By assumption on f , the morphism S e f is a weak equivalence in Set ∆ . Hence, it induces a weak equivalence on mapping spaces; that is, the bottom morphism in the diagram is an isomorphism in hSet ∆ . From that, it follows that also the top morphism is an isomorphism in hSet ∆ , which implies that G is S −1 e (W Set ∆ )-local, and therefore fibrant in M p/i . Remark 5.8 Part (3) of Theorem 5.6 is related to recent results by Berwick-Evans, Boavida de Brito and Pavlov from [BEBdBP]: in that paper, the authors work with the category H of simplicial presheaves on Mfd. Given G ∈ H, they consider the concordance sheaf We can now compare the derived concordance sheaves from Definition 5.5 to the concordance sheaf construction (5.9) from [BEBdBP]: let F ∈ H be any simplicial presheaf on Cart. On the one hand, we have and on the other hand, we have If M = c is a cartesian space, we readily obtain a natural weak equivalence and we further obtain natural weak equivalences The first weak equivalence uses that ∆ k e is a cartesian space, for each k ∈ N 0 , so that c × ∆ k e is representable in H, and the last morphism arises from postcomposing with a fibrant replacement functor in Set ∆ . This establishes a natural zig-zag of weak equivalences between hoRan ′ ι (Cc p F ) and B hoRan ′ ι (F In the second isomorphism, we have used that Cc p G is essentially constant and that Cc p G( * ) = R Set ∆ • S e G. The third isomorphism arises from the Quillen adjunction colim : H p ⇄ Set ∆ :c, and the fourth isomorphism stems from fact that colim•Q models the homotopy colimit. Finally, the fifth isomorphism arises from our observation at the end of Section 4 that S e F is a model for the homotopy colimit of the diagram F : Cart op → Set ∆ . Parts (2) and (3) of Theorem 5.6 then follow as in our proof above. ⊳

A An injective fibrant replacement of simplicial presheaves
From the definition of the projective and the injective model structure on H, it follows directly that there is a Quillen equivalence Both of the functors in this adjunction are the identity on H. Here, we will construct a Quillen equivalence in the opposite direction, i.e.
We start by defining the functor Q ′ . Its construction is not specific to simplicial presheaves on Cart, but works for simplicial presheaves over any small category. Thus, let C be a small category, let K denote the category of simplicial presheaves on C, and let Y : C → K denote the Yoneda embedding. We denote the projective and the injective model structures on K by K p and by K i , respectively. The conventional two-sided simplicial bar construction provides a functor [Rie14, Sec. 4.2] Lemma A.1 The functor B • ((−), C, Y) sends injective cofibrations in K to injective cofibrations in That is, if f : F → G is an objectwise cofibration of simplicial presheaves on C, then B n (f, C, Y) is a projective cofibration of simplicial presheaves, for each n ∈ N 0 .
Proof. We have that Observing that Y c ∈ K p is cofibrant for every c ∈ C and recalling that K p is a simplicial model category, we see that, in each part of the coproduct, the functor is a left Quillen functor.
In order to obtain Dugger's cofibrant replacement functor Q, one needs to take the diagonal (objectwise) of the bisimplicial presheaf B • ((−), C, Y): It is not evident to us whether taking the diagonal of a bisimplicial presheaf maps injective cofibrations in (K p ) ∆ op to projective cofibrations in K. That is the motivation for constructing the functor Q ′ below.
Let I be a small category, V a symmetric monoidal category, and M a model category category enriched, tensored and cotensored over V (i.e. a model V-category in the terminology of [Bar10]). Proof. We view the functor tensor product in the definition of Q ′ as The functor tensor product is a left Quillen bifunctor when endowing the two source categories with any of the pairs of model structures (projective, injective), (Reedy, Reedy), or (injective, projective) [Rie14, Thms. 11.5.9, 14.3.1]. Further, the functor N (∆ /(−) ) op : ∆ −→ Set ∆ is projectively cofibrant [Hir03, Prop. 14.8.8]. (Note that it is then also Reedy cofibrant.) Consequently, the functor is left Quillen with respect to the injective model structure on (K p ) ∆ op (and hence also with respect to the Reedy model structure). We can write Q ′ as the composition The model category of simplicial diagrams in the middle carries the injective model structure. We have shown in Lemma A.1 that the first functor preserves cofibrations, and it follows from our arguments above that the second functor preserves cofibrations as well.
Next, we are going to employ the Bousfield-Kan map to show that Q ′ provides a cofibrant replace-ment functor on K p . The Bousfield-Kan map is a morphism bk : N (∆ /(−) ) op −→ ∆ • of cosimplicial simplicial sets. We will use the following two statements: By Proposition A.6, the functor is a left Quillen functor for every F ∈ K. It then follows from Proposition A.5 and the arguments in the proof of Proposition A.4 that the natural transformation (A.7) is a natural weak equivalence of functors K → K p . Finally, we use that the functor B((−), C, Y) agrees with Dugger's cofibrant replacement functor Q p for K p from [Dug01b]. In particular, it comes with a natural weak equivalence q p : Q p → 1 K . Composing q p with the morphism (A.7), we obtain a natural weak equivalence q ′ : Q ′ → 1 K . Putting everything together, we have proven Proposition A.8 The functor Q ′ from (A.3), together with the natural weak equivalence q ′ , provides a cofibrant replacement functor for K p . In particular, Q ′ preserves objectwise weak equivalences.
Finally, we observe that Q ′ has a right adjoint, which is explicitly given by Theorem A.10 The functors Q ′ and R ′ satisfy the following properties: (1) The adjunction Q ′ ⊣ R ′ is a Quillen equivalence (2) There is a natural transformation r ′ : 1 K → R ′ such that r ′ |G : G → R ′ G is a weak equivalence in K i for every projectively fibrant G ∈ K.
(3) Let R Set ∆ be a fibrant replacement functor for simplicial sets. Then, G → R ′ (R Set ∆ • G) is a fibrant replacement functor on K i .
Proof. Ad (1): Proposition A.8, together with the observation that Q ′ is a left adjoint, readily implies that Q ′ is a left Quillen functor. The fact that this is a Quillen equivalence follows from the existence of the natural weak equivalence q ′ : Q ′ ∼ −→ 1 K . Formally, this implies that the composition of Q ′ by the left Quillen equivalence 1 K : K p → K i is weakly equivalent to the identity functor on K. Thus, the statement follows from the two-out-of-three property of Quillen equivalences [Hov99].
Ad (2): Let τ F,G : K(Q ′ F, G) −→ K(F, R ′ G) be the natural isomorphism that establishes the adjunction Q ′ ⊣ R ′ . We define r ′ : 1 K → R ′ to be the image under τ of the natural transformation q ′ . Now consider the weak equivalence q ′ |G : Q ′ G → G, for G ∈ K p fibrant. By part (1) and since each object in K i is cofibrant, the morphism r ′ |G : G → R ′ G is a weak equivalence. Ad (3): Let R Set ∆ be a fibrant replacement functor in Set ∆ , with associated natural weak equivalence r Set ∆ : 1 Set ∆ ∼ −→ R Set ∆ . Let G ∈ K be arbitrary and consider the composition The first morphism is an objectwise weak equivalence by definition. Since R Set ∆ • G is projectively fibrant, the second morphism is a weak equivalence as well by part (2).
Finally, we recall that the mapping spaces in a model category depend only on the weak equivalences, i.e. their homotopy types are determined completely by the underlying relative category. This was famously proven in [DK80,Prop. 4.4]; the simpler case of simplicial model categories-which applies directly to the present situation-is in [DK80,Cor. 4.7]. In particular, we record the following direct consequence for reference: Proposition A.11 Let F, G be any two objects in K. There is a canonical isomorphism in the homotopy category hSet ∆ of spaces between the mapping spaces of the projective and the injective model structures where ι : Cart ֒→ Mfd is the canonical inclusion, and where Q : H p → H p is Dugger's cofibrant replacement functor for the projective model structure on simplicial presheaves (A.2). Further, Q ′ ⊣ R ′ is the Quillen equivalence from Theorem A.10. The natural transformation (A.7) induces a natural transformation η : hoRan ι −→ hoRan ′ ι whose component η |F on every fibrant object F ∈ H p is a projective weak equivalence.

B Sheaves on manifolds and sheaves on cartesian spaces
The functor hoRan ι : H p → H p is a right Quillen functor with left adjoint Q • ι * , and this Quillen adjunction descends to a Quillen adjunction on Čech localisations, Q • ι * : H pℓ H pℓ : hoRan ι , ⊥ by the universal property of left Bousfield localisations. Analogously, hoRan ′ ι : H p → H p is a right Quillen functor (since Q ′ is homotopical and valued in projectively cofibrant simplicial presheaves). Further, the natural transformation η : hoRan ι −→ hoRan ′ ι establishes that hoRan ′ ι maps local objects in H pℓ to local objects in H pℓ , and hence that we also have a Quillen adjunction Q ′ • ι * : H pℓ H pℓ : hoRan ′ ι .
⊥ Note that we can also write the right adjoint as with R ′ : H → H defined as in (A.9), and where ι * is the right Kan extension along ι.
Lemma B.1 The derived counit of the Quillen adjunction Q ′ • ι * ⊣ hoRan ′ ι is a projective weak equivalence on every fibrant F ∈ H p .
Proof. Let Q : H p → H p be a cofibrant replacement functor with associated natural weak equivalence q : Q ∼ −→ 1 H . Consider a fibrant object F ∈ H p and the composition which is the derived counit. The first morphism is given by (Q ′ • ι * ) q |ι * •R ′ (F ) . It is a weak equivalence in H p because q |ι * •R ′ (F ) is a projective weak equivalence and ι * : H p → H p is homotopical, as is Q ′ . The second morphism is the counit of the adjunction ι * ⊣ ι * , which is an isomorphism by the Yoneda Lemma. The third morphism is a weak equivalence by Theorem A.10 and the fact that every object in H i is cofibrant.
It follows that the total derived functor R hoRan ′ ι : hH p → h H p is fully faithful, and hence that also R hoRan ′ ι : hH pℓ → h H pℓ is fully faithful. Next, we show that R hoRan ′ ι is essentially surjective. We start by recalling that coming with a natural transformation γ : 1 H → Pl which is an objectwise weak equivalence on every