Communications in Mathematical Physics The Stack of Yang – Mills Fields on Lorentzian Manifolds

We provide an abstract definition and an explicit construction of the stack of non-Abelian Yang–Mills fields on globally hyperbolic Lorentzian manifolds. We also formulate a stacky version of the Yang–Mills Cauchy problem and show that its wellposedness is equivalent to a whole family of parametrized PDE problems. Our work is based on the homotopy theoretical approach to stacks proposed in Hollander (Isr. J. Math. 163:93–124, 2008), which we shall extend by further constructions that are relevant for our purposes. In particular, we will clarify the concretification of mapping stacks to classifying stacks such as BGcon.


Introduction and Summary
Understanding quantum Yang-Mills theory is one of the most important and challenging open problems in mathematical physics. While approaching this problem in a fully nonperturbative fashion seems to be out of reach within the near future, there are recent developments in classical and quantum field theory which make it plausible that quantum Yang-Mills theory could relatively soon be constructed within a perturbative approach that treats the coupling constant non-perturbatively, but Planck's constant as a formal deformation parameter. See [BS17a,KS17,Col16] and the next paragraph for further details. The advantage of such an approach compared to standard perturbative (algebraic) quantum field theory, see e.g. [Rej16] for a recent monograph, is that by treating the coupling constant non-perturbatively the resulting quantum field theory is sensitive to the global geometry of the field configuration spaces. This is particularly interesting and rich in gauge theories, where the global geometry of the space of gauge fields encodes various topological features such as characteristic classes of the underlying principal bundles, holonomy groups and other topological charges.
Loosely speaking, the construction of a quantum field theory within this non-standard perturbative approach consists of the following three steps: (1) Understand the smooth structure and global geometry of the space of solutions of the field equation of interest.
(2) Use the approach of [Zuc87] to equip the solution space with a symplectic form and construct a corresponding Poisson algebra of smooth functions on it. (The latter should be interpreted as the Poisson algebra of classical observables of our field theory.) (3) Employ suitable techniques from formal deformation quantization to quantize this Poisson algebra. Even though stating these three steps in a loose language is very simple, performing them rigorously is quite technical and challenging. The reason behind this is that the configuration and solution spaces of field theories are typically infinitedimensional, hence the standard techniques of differential geometry do not apply. In our opinion, the most elegant and powerful method to study the smooth spaces appearing in field theories is offered by sheaf topos techniques: In this approach, a smooth space X is defined by coherently specifying all smooth maps U → X from all finite-dimensional manifolds U to X . More precisely, this means that X is a (pre)sheaf on a suitable site C, which we may choose as the site of all finite-dimensional manifolds U that are diffeomorphic to some R n , n ∈ Z ≥0 . In particular, the maps from U = R 0 determine the points of X and the maps from U = R 1 the smooth curves in X . Employing such techniques, the first two steps of the program sketched above have been successfully solved in [BS17a,KS17] for the case of non-linear field theories without gauge symmetry. Concerning step (3), Collini [Col16] obtained very interesting and promising results showing that Fedosov's construction of a -product applies to the Poisson algebra of 4 -theory in 4 spacetime dimensions, leading to a formal deformation quantization of this theory that is non-perturbative in the coupling constant. Collini's approach is different from ours as it describes the infinite-dimensional spaces of fields and solutions by locally-convex manifolds, which are less flexible. It would be an interesting problem to reformulate and generalize his results in our more elegant and powerful sheaf theoretic approach [BS17a,KS17].
The goal of this paper is to address and solve step (1) of the program sketched above for the case of Yang-Mills theory with a possibly non-Abelian structure group G on globally hyperbolic Lorentzian manifolds. A crucial observation is that, due to the presence of gauge symmetries, the sheaf topos approach of [BS17a,KS17] is no longer sufficient and has to be generalized to "higher sheaves" (stacks) and the "higher toposes" they form. We refer to [Sch13] for an overview of recent developments at the interface of higher topos theory and mathematical physics, and also to [Egg14] for a gentle introduction to the role of stacks in gauge theory. See also [FSS15] for a formulation of Chern-Simons theory in this framework. The basic idea behind this is as follows: The collection of all gauge fields on a manifold M naturally forms a groupoid and not a set. The objects (A, P) of this groupoid are principal G-bundles P over M equipped with a connection A and the morphisms h : (A, P) → (A , P ) are gauge transformations, i.e. bundle isomorphisms preserving the connections. There are two important points we would like to emphasize: (i) This groupoid picture is essential to capture all the topological charge sectors of the gauge field (i.e. non-isomorphic principal bundles). In particular, it is intrinsically non-perturbative as one does not have to fix a particular topological charge sector to perturb around. (ii) Taking the quotient of the groupoid of gauge fields, i.e. forming "gauge orbits", one loses crucial information that is encoded in the automorphism groups of objects of the groupoid, i.e. the stabilizer groups of bundles with connections. This eventually would destroy the important descent properties (i.e. gluing connections up to a gauge transformation) that are enjoyed by the groupoids of gauge fields, but not by their corresponding sets of "gauge orbits". It turns out that gauge fields on a manifold M do not only form a groupoid but even a smooth groupoid. The latter may be described by groupoid-valued presheaves on our site C, i.e. objects of the category H := PSh(C, Grpd). Following the seminal work by K. Brown, J. F. Jardine and others, in a series of papers [Hol08a,Hol08b,Hol07] Hollander developed the abstract theory of presheaves of groupoids by using techniques from model category theory/homotopical algebra, see e.g. [DS95] for a concise introduction. One of her main insights was that the usual theory of stacks [DM69,Gir71] can be formalized very elegantly and efficiently in this framework by employing homotopical techniques. In short, stacks can be identified as those presheaves of groupoids satisfying a notion of descent, which can be phrased in purely model categorical terms. With these techniques and developments in mind, we now can state more precisely the two main problems we address in this paper: (I) Understand and describe the stack of Yang-Mills fields in the framework of [Hol08a,Hol08b,Hol07]. (II) Understand what it means for the stacky version of the Yang-Mills Cauchy problem to be well-posed.
The present paper is part of a longer term research program of two of us (M.B. and A.S.) on homotopical algebraic quantum field theory. The aim of this program is to develop a novel and powerful framework for quantum field theory on Lorentzian manifolds that combines ideas from locally covariant quantum field theory [BFV03] with homotopical algebra [Hov99,DS95]. This is essential to capture structural properties of quantum gauge theories that are lost at the level of gauge invariant observables. In previous works, we could confirm for toy-models that our homotopical framework is suitable to perform local-to-global constructions for gauge field observables [BSS15] and we constructed a class of toy-examples of homotopical algebraic quantum field theories describing a combination of classical gauge fields and quantized matter fields [BS17b]. Based on the results of the present paper, we will be able to address steps (2) and (3) of the program outlined above for gauge theories and in particular for Yang-Mills theory. We expect that this will allow us to obtain first examples of homotopical algebraic quantum field theories which describe quantized gauge fields.
The outline of the remainder of this paper is as follows: in Sect. 2 we provide a rather self-contained introduction to presheaves of groupoids and their model category structures. This should allow readers without much experience with this subject to understand our statements and constructions. In particular, we review the main results of [Hol08a] which show that there are two model structures on the category H = PSh(C, Grpd), called the global and the local model structure. The local model structure, which is obtained by localizing the global one, is crucial for detecting stacks in a purely model categorical fashion as those are precisely the fibrant objects for this model structure. We then provide many examples of stacks that are important in gauge theory, including the stack represented by a manifold and some relevant classifying stacks, e.g. BG con which classifies principal G-bundles with connections. This section is concluded by explaining homotopy fiber products for stacks and (derived) mapping stacks, which are homotopically meaningful constructions on stacks that will be frequently used in our work.
In Sect. 3 we construct and explicitly describe the stack of gauge fields GCon(M) on a manifold M. Our main guiding principle is the following expectation of how the stack GCon(M) is supposed to look: Via the functor of points perspective, the groupoid GCon(M)(U ) obtained by evaluating the stack GCon(M) on an object U in C should be interpreted as the groupoid of smooth maps U → GCon(M). Because GCon(M) is supposed to describe principal G-bundles with connections, any such smooth map will describe a smoothly U -parametrized family of principal G-bundles with connections on M, and the corresponding morphisms are smoothly U -parametrized gauge transformations. We shall obtain a precise and intrinsic definition of GCon(M) by concretifying the mapping stack from (a suitable cofibrant replacement of) the manifold M to the classifying stack BG con . Our concretification prescription (cf. Definition 3.3) improves the one originally proposed in [FRS16,Sch13]. In fact, as we explain in more detail in Appendix D, the original concretification does not produce the desired result (sketched above) for the stack of gauge fields, while our improved concretification fixes this issue. We then show that assigning to connections their curvatures may be understood as a natural morphism F M : GCon(M) → 2 (M, ad(G)) between concretified mapping stacks.
Section 4 is devoted to formalizing the Yang-Mills equation on globally hyperbolic Lorentzian manifolds M in our model categorical framework. After providing a brief review of some basic aspects of globally hyperbolic Lorentzian manifolds, we shall show that, similarly to the curvature, the Yang-Mills operator may be formalized as a natural morphism YM M : GCon(M) → 1 (M, ad(G)) between concretified mapping stacks. This then allows us to define abstractly the stack of solutions to the Yang-Mills equation GSol(M) by a suitable homotopy fiber product of stacks (cf. Definition 4.4). We shall explicitly work out the Yang-Mills stack GSol(M) and give a simple presentation up to weak equivalence in H. This solves our problem (I) listed above.
Problem (II) is then addressed in Sect. 5. After introducing the stack of initial data GData( ) on a Cauchy surface ⊆ M and the morphism of stacks data : GSol(M) → GData( ) that assigns to solutions of the Yang-Mills equation their initial data, we formalize a notion of well-posedness for the stacky Yang-Mills Cauchy problem in the language of model categories (cf. Definition 5.2). Explicitly, we say that the stacky Cauchy problem is well-posed if data is a weak equivalence in the local model structure on H. We will then unravel this abstract condition and obtain that wellposedness of the stacky Cauchy problem is equivalent to well-posedness of a whole family of smoothly U -parametrized Cauchy problems (cf. Proposition 5.3). A particular member of this family (given by the trivial parametrization by a point U = R 0 ) is the ordinary Yang-Mills Cauchy problem, which is known to be well-posed in dimension m = 2, 3, 4 [CS97,C-B91]. To the best of our knowledge, smoothly U -parametrized Cauchy problems of the kind we obtain in this work have not been studied in the PDE theory literature yet. Because of their crucial role in understanding Yang-Mills theory, we believe that such problems deserve the attention of the PDE community. It is also interesting to note that Yang-Mills theory, which is of primary interest to physics, provides a natural bridge connecting two seemingly distant branches of pure mathematics, namely homotopical algebra and PDE theory.
In Sect. 6 we make the interesting observation that gauge fixings may be understood in our framework as weakly equivalent descriptions of the same stack. For the sake of simplifying our arguments, we focus on the particular example of Lorenz gauge fixing, which is often used in applications to turn the Yang-Mills equation into a hyperbolic PDE. We define a stack GSol g.f. (M) of gauge-fixed Yang-Mills fields, which comes together with a morphism GSol g.f. (M) → GSol(M) to the stack of all Yang-Mills fields. Provided certain smoothly U -parametrized PDE problems admit a solution (cf. Proposition 6.2), this morphism is a weak equivalence in H, which means that the gaugefixed Yang-Mills stack GSol g.f. (M) is an equivalent description of GSol(M).
The paper contains four appendices. In the first three appendices we work out some relevant aspects of the model category H of presheaves of groupoids that were not discussed by Hollander in her series a papers [Hol08a,Hol08b,Hol07]. In Appendix A we show that H is a monoidal model category, which is essential for our construction of mapping stacks. In Appendix B we obtain functorial cofibrant replacements of manifolds in H, which are needed for computing our mapping stacks explicitly. In Appendix C we develop very explicit techniques to compute fibrant replacements in the (−1)truncation (cf. [Bar10,Rez,ToVe05] and also [Lur09]) of the canonical model structure on over-categories H/K , which are crucial for the concretification of our mapping stacks. The last Appendix D compares our concretification prescription with the original one proposed in [FRS16,Sch13]. In particular, we show that the latter does not lead to the desired stack of gauge fields, i.e. the stack describing smoothly parametrized principal G-bundles with connections, which was our motivation to develop and propose an improved concretification prescription in Definition 3.3.

Preliminaries
We fix our notations and review some aspects of the theory of presheaves of groupoids which are needed for our work. Our main reference for this section is [Hol08a] and references therein. A good introduction to model categories is [DS95], see also [Hov99,Hir03] for more details. We shall use presheaves of groupoids as a model for stacks which are, loosely speaking, generalized smooth spaces whose 'points' may have non-trivial automorphisms.

Groupoids.
Recall that a groupoid G is a (small) category in which every morphism is an isomorphism. For G a groupoid, we denote its objects by symbols like x and its morphisms by symbols like g : x → x . A morphism F : G → H between two groupoids G and H is a functor between their underlying categories. We denote the category of groupoids by Grpd.
The category Grpd is closed symmetric monoidal: The product G × H of two groupoids is the groupoid whose object (morphism) set is the product of the object (morphism) sets of G and H . The monoidal unit is the groupoid { * } with only one object and its identity morphism. The functor G × (−) : Grpd → Grpd has a right adjoint functor, which we denote by Grpd(G, −) : Grpd → Grpd. We call Grpd(G, H ) the internal hom-groupoid from G to H . Explicitly, the objects of Grpd(G, H ) are all functors F : G → H and the morphisms from F : G → H to F : G → H are all natural transformations η : F → F . It is easy to see that a morphism in Grpd(G, H ) may be equivalently described by a functor where 1 is the groupoid with two objects, say 0 and 1, and a unique isomorphism 0 → 1 between them. (The source and target of the morphism U is obtained by restricting U to the objects 0 and 1 in 1 , and their identity morphisms, and the natural transformation η is obtained by evaluating U on the morphism 0 → 1 in 1 .) This latter perspective on morphisms in Grpd(G, H ) will be useful later when we discuss presheaves of groupoids. The category Grpd can be equipped with a model structure, i.e. one can do homotopy theory with groupoids. For a proof of the theorem below we refer to [Str00, Section 6]. Recall that a model category is a complete and cocomplete category with three distinguished classes of morphisms-called fibrations, cofibrations and weak equivalences-that have to satisfy a list of axioms, see e.g. [DS95] or [Hov99,Hir03]. With these choices Grpd is a model category.
In the following, we will need a simple and tractable model for the homotopy limit holim Grpd of a cosimplicial diagram in Grpd. See e.g. [DS95] for a brief introduction to homotopy limits and colimits. Such a model was found by Hollander in [Hol08a] in terms of the descent category.
where G n are groupoids, for all n ∈ Z ≥0 , and we suppressed as usual the codegeneracy maps s i in this cosimplicial diagram. Then the homotopy limit holim Grpd G • is the groupoid whose • objects are pairs (x, h), where x is an object in G 0 and h : in G 1 commutes.

Presheaves of groupoids and stacks.
Let C be the category with objects given by all (finite-dimensional and paracompact) manifolds U that are diffeomorphic to a Cartesian space R n , n ∈ Z ≥0 , and morphisms given by all smooth maps : U → U . Notice that U and U are allowed to have different dimensions. We equip C with the structure of a site by declaring a family of morphisms in C to be a covering family whenever {U i ⊆ U } is a good open cover of U and i : U i → U are the canonical inclusions. As usual, we denote intersections of the U i by U i 1 ...i n := U i 1 ∩ · · · ∩ U i n . By definition of good open cover, these intersections are either empty or open subsets diffeomorphic to some R n , i.e. objects in C. We note that our site C is equivalent to the site Cart of Cartesian spaces used in [FRS16,Sch13]. We however prefer to work with C instead of Cart, because covering families { i : It assigns to an object U in C the functor U : C op → Grpd that acts on objects as where Hom C are the morphism sets in C which we regard as groupoids with just identity morphisms, and on morphisms : U → U as To a morphism : U → U in C the Yoneda embedding assigns the morphism : U → U in H whose stages are for all objects U in C. Given two objects X and Y in H, their product X × Y in H is given by the functor X × Y : C op → Grpd that acts on objects as where on the right-hand side the product is the one in Grpd, and on morphisms : The product × equips H with the structure of a symmetric monoidal category. The unit object is the constant presheaf of groupoids defined by the groupoid { * }, i.e. the functor C op → Grpd that assigns U → { * } and ( : U → U ) → id { * } . In the following we denote groupoids and their corresponding constant presheaves of groupoids by the same symbols. Explicitly, when G is a groupoid we also denote by G the presheaf of groupoids specified by U → G and ( : U → U ) → id G . For example, we denote the unit object in H simply by { * }. The symmetric monoidal category H has internal homs, i.e. it is closed symmetric monoidal. To describe those explicitly, we first have to introduce mapping groupoids Grpd(X, Y ) between two objects X and Y in H. Recalling (2.1), we define Grpd(X, Y ) to be the groupoid with objects given by all H-morphisms and morphisms given by all H-morphisms where 1 is the groupoid from (2.1) regarded as a constant presheaf of groupoids. The internal hom-object Y X in H is then given by the functor Y X : C op → Grpd that acts on objects as as follows: It is the functor that assigns to an object f : For the terminal object R 0 in C, we have that Y X (R 0 ) = Grpd(X, Y ) is the mapping groupoid.
Our category H can be equipped with (at least) two model structures. To define them, let us recall that a morphism f : X → Y in H is said to have the left lifting property with respect to a morphism f : X → Y in H if all commutative squares of the form admit a lift f , i.e. the two triangles commute. Vice versa, one says that a morphism f : X → Y in H has the right lifting property with respect to a morphism f : X → Y in H if all commutative squares of the form above admit a lift f . The global model structure on H = PSh(C, Grpd) is not yet the correct one as it does not encode any information about our site structure on C. It was shown in [Hol08a] that one can localize the global model structure to obtain what is called the local model structure on H. See also [Hir03] or [Dug01, Section 5] for details on localizations of model categories. The set of H-morphisms with respect to which one localizes is given by (2.14) As before, U is the object in H that is represented by the object U in C via the Yoneda embedding. By U • we denote the simplicial diagram in H given by where denotes the coproduct in H and we suppressed as usual the degeneracy maps. Moreover, hocolim H U • denotes the homotopy colimit of the simplicial diagram (2.15) in H.
To describe the weak equivalences in the local model structure on H it is useful to assign sheaves of homotopy groups to presheaves of groupoids, cf. The sheafifications of π 0 (X ) and π 1 (X, x), for all objects x in X (U ) and all objects U in C, are called the sheaves of homotopy groups associated to an object X in H.
The following theorem summarizes the relevant aspects of the local model structure on H.
is a weak equivalence in Grpd, where holim Grpd denotes the homotopy limit of cosimplicial diagrams in Grpd. Remark 2.6. To simplify notations, we denote the cosimplicial diagram of groupoids in (2.17) also by (2.18) Then the homotopy limit in (2.17) simply reads as holim Grpd X (U • ).
Unless otherwise stated, we will always work with the local model structure on H. The reason for this will be explained below. Hence, by fibration, cofibration, and weak equivalence in H we always mean the ones in the local model structure on H.
Of particular relevance for us will be the fibrant objects in H (in the local model structure). It was shown by Hollander [Hol08a] that the fibrant condition (2.17) captures the notion of descent for stacks. In particular, fibrant presheaves of groupoids provide us with an equivalent, but simpler, model for stacks than the traditional models based on lax presheaves of groupoids or categories fibered in groupoids, see e.g. [DM69,Gir71]. Hence in our work we shall use the following definition of a stack [Hol08a].
Definition 2.7. A stack is a fibrant object X in the local model structure on H. More concretely, a stack is a presheaf of groupoids X : C op → Grpd such that the canonical morphism (2.17) is a weak equivalence in Grpd, for all good open covers {U i ⊆ U }.
We will later need the following standard Proof. This is [Hir03, Proposition 3.3.16] applied to H with the global model structure (cf. Lemma 2.3) and to its localization with respect to (2.14), i.e. the local model structure of Theorem 2.5.

Examples of stacks.
We collect some well-known examples of stacks that will play a major role in our work. We also refer to [FSS12,FRS16,Sch13] for a description of some of these stacks in the language of ∞-stacks. All of our stacks are motivated by the structures arising in gauge theories. where C ∞ (U, M) is the set of smooth maps regarded as a groupoid with just identity morphisms, and on morphisms : U → U as (2.20) Note the similarity to the Yoneda embedding (2.6), which is our motivation to use the same notation by an underline. Notice further that, when M = U is diffeomorphic to some R n , then M coincides with the Yoneda embedding U of the object U in C. Let us confirm that M is a stack, i.e. that (2.17) is a weak equivalence in Grpd for all good open covers {U i ⊆ U }. In the notation of Remark 2.6, we have to compute the homotopy limit holim Grpd M(U • ) of the cosimplicial diagram M(U • ) in Grpd. Using Proposition 2.2, we find that holim Grpd M(U • ) is the groupoid whose objects are families for all i, j, and whose morphisms are just the identities. The canonical morphism M(U ) → holim Grpd M(U • ) assigns to ρ ∈ C ∞ (U, M) the family {ρ| U i }, hence it is an isomorphism (and thus also a weak equivalence) because C ∞ (−, M) is a sheaf on C.
Denoting by Man the category of (finite-dimensional and paracompact) manifolds, it is easy to see that our constructions above define a fully faithful functor (−) : Man → H that takes values in stacks. Hence, manifolds can be equivalently described in terms of stacks.
Example 2.10 (Classifying stack of principal G-bundles). Let G be a Lie group. The object BG in H is defined by the following functor BG : C op → Grpd: To an object U in C it assigns the groupoid BG(U ) that has just one object, say * , with automorphisms C ∞ (U, G) given by the set of smooth maps from U to the Lie group G. The composition of morphisms in BG(U ) is given by the opposite point-wise product g • g := g g of g, g ∈ C ∞ (U, G). (We take the opposite product because we will be interested later in group representations of C ∞ (U, G) from the right.) Moreover, the identity morphism in BG(U ) is the constant map e ∈ C ∞ (U, G) from U to the unit element in G. To any morphism : U → U in C the functor BG assigns the groupoid morphism BG( ) : BG(U ) → BG(U ) that acts on objects as * → * and on morphisms as Let us confirm that BG is a stack. Let {U i ⊆ U } be any good open cover. Using Proposition 2.2, we find that holim Grpd BG(U • ) is the groupoid whose objects are families , for all i, and the cocycle condition The canonical morphism is a weak equivalence in Grpd (cf. Theorem 2.1) because of the following reasons: (1) It is fully faithful because C ∞ (−, G) is a sheaf on C.
(2) It is essentially surjective because all cocycles are trivializable on manifolds diffeomorphic to R n (U in this case), i.e. one can find Hence, we have shown that BG is a stack. It is called the classifying stack of principal G-bundles, see also [FSS12, Section 3.2].
Our classifying stack BG is a smooth and stacky analog of the usual classifying space of a topological group G [Seg68, Section 3], which is the topological space obtained as the geometric realization of the simplicial topological space Notice that (2.21) may be obtained by equipping the nerve of the groupoid BG({ * }) with the topologies induced by G.
Example 2.11 (Classifying stack of principal G-bundles with connections). Let again G be a Lie group, which we assume to be a matrix Lie group in order to simplify some formulas below. Let g denote the Lie algebra of G. The object BG con in H is defined by the following functor BG con : C op → Grpd: To an object U in C it assigns the groupoid BG con (U ) whose set of objects is 1 (U, g), i.e. the set of Lie algebra valued 1-forms on U , and whose set of morphisms is 1 (U, g) × C ∞ (U, G). The source and target of a morphism (A, g) ∈ 1 (U, g) × C ∞ (U, G) is as follows where d denotes the de Rham differential and A g is the usual right action of gauge transformations g ∈ C ∞ (U, G) on gauge fields A ∈ 1 (U, g). The identity morphism is (A, e) : A → A and composition of two composable morphisms in BG con (U ) is given by (2.23) To any morphism : U → U in C the functor BG con assigns the groupoid morphism BG con ( ) : BG con (U ) → BG con (U ) that acts on objects as A → * A (i.e. via pullback of differential forms) and on morphisms as Let us confirm that BG con is a stack. Let {U i ⊆ U } be any good open cover. Using Proposition 2.2, we find that holim Grpd BG con (U • ) is the groupoid whose objects are pairs of families With a similar argument as in Example 2.10 we find that the canonical morphism is a weak equivalence in Grpd and hence that BG con is a stack. It is called the classifying stack of principal G-bundles with connections, see also [FSS12, Section 3.2].
Example 2.12 (Classifying stack of principal G-bundles with adjoint bundle valued pforms). Let G be a matrix Lie group and g its Lie algebra. For p ∈ Z ≥0 , we define the object BG p ad in H by the following functor BG p ad : C op → Grpd: To an object U in C it assigns the groupoid BG p ad (U ) whose set of objects is the set of g-valued p-forms p (U, g) and whose set of morphisms is p (U, g) × C ∞ (U, G). The source and target of a morphism (ω, g) ∈ p (U, g) × C ∞ (U, G) is, similarly to Example 2.11, given by : ω → ω and the composition of two composable morphisms in BG p ad (U ) is given by To any morphism : U → U in C the functor BG p ad assigns the groupoid morphism BG p ad ( ) : BG p ad (U ) → BG p ad (U ) that acts on objects and morphisms via pullback, i.e. ω → * ω and (ω, g) → ( * ω, * g). The fact that BG p ad is a stack can be proven analogously to Examples 2.10 and 2.11. We call it the classifying stack of principal G-bundles with adjoint bundle valued p-forms. We will later use BG p ad to describe, for example, the curvatures of connections.

Fiber product of stacks and mapping stacks.
Our constructions in this paper will use a variety of techniques to produce new stacks out of old ones, e.g. out of the stacks described in Sect. 2.3. Of particular relevance for us will be fiber products of stacks and mapping stacks.
The following observation is standard in homotopy theory, however it is crucial to understand the definitions below: given a pullback diagram in H, we may of course compute the fiber product X × Z Y in the usual way by taking the limit of this diagram. (Recall that H is complete, hence all limits exist in H.) The problem with this construction is that it does not preserve weak equivalences in H, i.e. replacing the pullback diagram by a weakly equivalent one in general leads to a fiber product that is not weakly equivalent to X × Z Y . These problems are avoided by replacing the limit with the homotopy limit, which is a derived functor of the limit functor [DS95]. We denote the homotopy limit of the diagram (2.26) by X × h Z Y and call it the homotopy fiber product in H. A similar problem arises when we naively use the internal hom-object Y X in H in order to describe an "object of mappings" from X to Y ; replacing X and Y by weakly equivalent objects in H in general leads to an internal hom-object that is not weakly equivalent to Y X . These problems are avoided by replacing the internal-hom functor with its derived functor.
We now give rather explicit models for the homotopy fiber product and the derived internal hom. The following model for the homotopy fiber product in H was obtained in [Hol08b].
Proposition 2.13. The homotopy fiber product X × h Z Y in H, i.e. the homotopy limit of the pullback diagram (2.26), is the presheaf of groupoids defined as follows: For all objects U in C, If X , Y and Z in (2.26) are stacks, then X × h Z Y is a stack too. Our model for the derived internal hom-functor is the standard one resulting from the theory of derived functors, see [DS95] and also [Hov99] for details. For the theory of derived functors to apply to our present situation, it is essential that H (with the local model structure) is a monoidal model category, see Appendix A for a proof.

Proposition 2.14. The derived internal hom-functor in H is
where Q : H → H is any cofibrant replacement functor and R : H → H is any fibrant replacement functor. The following holds true:

Gauge Fields on a Manifold
Let G be a matrix Lie group and M a (finite-dimensional and paracompact) manifold. The goal of this section is to construct and study the stack GCon(M) of principal G-bundles with connections on M, which we shall also call the stack of gauge fields on M. Let us recall our main guiding principle: The stack GCon(M) is supposed to describe smoothly parametrized families of principal G-bundles with connections on M, which is motivated by the functor of points perspective. More precisely, this means that evaluating the stack GCon(M) on an object U in C, we would like to discover (up to weak equivalence) the groupoid describing smoothly U -parametrized principal G-bundles with connections on M and their smoothly U -parametrized gauge transformations. We shall obtain the stack GCon(M) by an intrinsic construction in H that is given by a concretification of the derived mapping stack from M to the classifying stack BG con of principal G-bundles with connections (cf. Example 2.11). Using Proposition 2.14, we may compute (up to a weak equivalence) the derived mapping stack in terms of the ordinary internal-hom BG con M , where M is the canonical cofibrant replacement of M (cf. Appendix B). It is important to emphasize that our concretification prescription improves the one originally proposed in [FRS16,Sch13], which fails to give the desired result, i.e. a stack describing smoothly parametrized families of principal G-bundles with connections, together with smoothly parametrized gauge transformations, see Appendix D for details.
By construction, our model BG con (−) : Man op → → H for the derived mapping stacks is functorial on the category of (finite-dimensional and paracompact) manifolds with morphisms given by open embeddings. The same holds true for the stacks of gauge fields, i.e. we will obtain a functor GCon : Man op → → H. This is an advantage compared to the usual approach to construct the stack of gauge fields by using a (necessarily nonfunctorial) good open cover to obtain a cofibrant replacement for M, see e.g. [FSS12].

Mapping stack.
We now compute the object BG con M in H, which by Proposition 2.14 is a weakly equivalent model for the derived internal-hom object from M to BG con . Its underlying functor BG con M : C op → Grpd assigns to an object U in C the mapping groupoid By definition, an object in this groupoid is a morphism f : U × M → BG con in H and a morphism in this groupoid is a morphism u : U × M × 1 → BG con in H. We would like to describe the objects and morphisms of the groupoid (3.1) by more familiar data related to gauge fields and gauge transformations. To achieve this goal, we have to explicate the stages of the H-morphisms f and u.
Let us start with f and consider the associated groupoid morphism f : of smooth maps, where V runs over all open subsets of M that are diffeomorphic to R dim(M) and the last arrow points to the right.
of smooth maps. The groupoid morphism f : U (U ) × M(U ) → BG con (U ) is then given by an assignment satisfying the following compatibility conditions: (1) For all morphisms ( , (V, ν), The property of f : U (U ) × M(U ) → BG con (U ) being the stages of an H-morphism (i.e. natural transformation of functors C op → Grpd) leads to the following coherence conditions: For all morphisms : U → U in C and all objects ( , and, for all morphisms : U → U in C and all morphisms ( , These coherences constrain the amount of independent data described by (3.4): Given which implies that In words, this means that the A of the form , determine all the others. As there are no further coherences between A's of the form A (pr U ,(V,pr V )) , it follows that the action of the functor (3.4) on objects is uniquely specified by choosing for all open subsets V ⊆ M diffeomorphic to R dim(M) . A similar argument applies to the morphisms assigned by (3.4), which are determined by for all open subsets V ⊆ M and V ⊆ M diffeomorphic to R dim(M) , and the coherences The morphisms of the groupoid (3.1) can be analyzed analogously. In summary, we obtain Lemma 3.1. The objects of the groupoid (3.1) are pairs of families (3.14) The morphisms of the groupoid where V ⊆ M runs over all open subsets diffeomorphic to R dim(M) , which satisfy the following conditions: • For all open subsets V ⊆ M diffeomorphic to R dim(M) , Moreover, the functor BG con M : C op → Grpd assigns to a morphism : This completes our description of the mapping stack BG con M .
3.2. Concretification. The mapping stack BG con M we described in the previous subsection (see in particular Lemma 3.1) is not yet the correct stack of gauge fields on the manifold M. Even though the groupoid of global points BG con M (R 0 ) correctly describes the gauge fields and gauge transformations on M, the smooth structure on BG con M , which is encoded in the groupoids BG con M (U ) for all other objects U in C, is not the desired one yet. In fact, the groupoid BG con M (U ) describes by construction gauge fields and gauge transformations on the product manifold M ×U , while the correct stack of gauge fields on M, when evaluated on U , should be the groupoid of smoothly U -parametrized gauge fields and gauge transformations on M. Hence, the problem is that BG con M (U ) includes also gauge fields along the parameter space U and not only along M.
This problem has already been observed in [FRS16,Sch13], where a solution in terms of concretification was proposed. The goal of this subsection is to work out explicitly the concretification of our mapping stack BG con M . This is achieved by using the results on fibrant replacements in the (−1)-truncation of the model structure on over-categories H/K that we develop in Appendix C. As we explain in more detail in Appendix D, the original concretification prescription of [FRS16,Sch13] fails to produce the desired result, i.e. a stack which describes smoothly parametrized families of principal G-bundles with connections, together with smoothly parametrized gauge transformations. Hence, we propose an improved concretification prescription that is valid for the case of interest in this paper, namely principal bundles and connections on manifolds M. A general concretification prescription for the ∞-stacks of higher bundles and connections is beyond the scope of this paper and will be developed elsewhere.
Crucial for concretification is existence of the following Quillen adjunction (3.20) The left adjoint functor : H → H assigns to an object X in H the object given by the following presheaf of groupoids X : C op → Grpd: To any object U in C, it assigns X (U ) = X (R 0 ), i.e. the groupoid of global points of X , and to any C-morphism : U → U it assigns the identity morphism X ( ) = id X (R 0 ) . The action of on morphisms in H is the obvious one. Loosely speaking, X is something like a 'discrete space' as it forgot all the smooth structure on X . The right adjoint functor : H → H assigns to an object X in H the object X defined as follows: To any object U in C, it assigns the groupoid where the product goes over all points p ∈ U , and to any C-morphism : U → U it assigns the groupoid morphism defined by universality of products and the commutative diagrams for all points p ∈ U , where pr denote the projection Grpd-morphisms associated to the products. The action of on morphisms in H is the obvious one. It is easy to prove that is a Quillen adjunction by using the explicit characterization of fibrations in H (in the local model structure) given by [Hol07, Proposition 4.2], see also Proposition C.3. The conceptual interpretation of X is as follows: For any object U in C, there exist isomorphisms of groupoids where we make use of the Yoneda lemma and the adjunction property of . This shows that, loosely speaking, the groupoid X (U ) is given by 'evaluating' X on the discrete space U . The key idea behind concretification is, again loosely speaking, to make use of the passage to discrete spaces U to avoid gauge fields along the parameter spaces U .
Let us now focus on our explicit example. Consider the object (BG con M ) in H, which is a stack because BG con M is a stack and is a right Quillen functor. The objects (respectively morphisms) of the groupoid (BG con M )(U ) describe by construction families of gauge fields (respectively gauge transformations) on M that are labeled by the points p ∈ U . In particular, there appear no gauge fields along the parameter space U . Unfortunately, there is no smoothness requirement on such families. To solve this problem, consider the canonical H-morphism Explicitly, the stages ζ con : BG con M (U ) → (BG con M )(U ) are the following groupoid morphisms: To any object ({A V }, {g V V }) of the source groupoid (cf. Lemma 3.1), it assigns (3.24b) where we regarded points p ∈ U as C-morphisms p : R 0 → U . To any morphism {h V } of the source groupoid (cf. Lemma 3.1), it assigns Unfortunately, the stage-wise naive image of ζ con is in general not a homotopically meaningful construction in the sense that it does not preserve weak equivalences. We thus have to solve our problem in a more educated manner to ensure that its solution is homotopically meaningful. For this let us consider the following pullback diagram BG con to the homotopy fiber product associated to (3.25). We shall use the following abstract Definition 3.3. The differential concretification of the mapping stack BG con M , which we will also call the stack of gauge fields on M, is defined by where Im 1 denotes the 1-image, i.e. the fibrant replacement in the (−1)-truncation of the canonical model structure on the over-category H (BG con We shall now explicitly compute GCon(M) in order to confirm that our abstract definition leads to the desired stack of gauge fields on M, i.e. the stack describing smoothly parametrized gauge fields and gauge transformations on M. It is practically very convenient to compute instead of GCon(M) defined in Definition 3.3-a weakly equivalent object of H that has a simpler and more familiar explicit description.
As a first step towards a simplified description of GCon(M), we notice that we actually do not have to compute the homotopy fiber product in Definition 3.3 by using the explicit construction of Proposition 2.13. The reason is as follows: Using Proposition C.3 it is easy to prove that the morphism forget : BG con → BG is a fibration (in the local model structure on H).
from the ordinary fiber product to the homotopy fiber product is a weak equivalence. Hence, we may replace the homotopy fiber product in Definition 3.3 by the ordinary fiber product P in order to find a weakly equivalent description of GCon(M). The ordinary fiber product P is much easier to compute: For any object U in C, the groupoid P(U ) has as objects all pairs of families where V ⊆ M, V ⊆ M run over all open subsets diffeomorphic to R dim(M) and p ∈ U runs over all points of U , such that ( . For a morphism : U → U in C, the associated Grpd-morphism P( ) : P(U ) → P(U ) is given by As a side remark, notice that P(U ) has the desired morphisms, however the gauge fields are not smoothly U -parametrized yet.
As a further simplification in our explicit description of GCon(M), we may combine Proposition C.10 and Proposition C.11 to compute (again up to weak equivalence) the 1-image in Definition 3.3 by the full image sub-presheaf of groupoids corresponding to the canonical morphism BG con M → P to the ordinary fiber product P. This defines the following object in H, which is weakly equivalent to GCon(M) given in Definition 3.3. (3.34) The • For all open subsets V ⊆ M diffeomorphic to R dim(M) , • For all open subsets V ⊆ M and V ⊆ M diffeomorphic to R dim(M) ,   This unfortunately does not automatically imply that our weakly equivalent simplified description of GCon(M) given in Proposition 3.4 is a stack too. One can, however, verify explicitly that the stack condition (2.17) holds true for the presheaf of groupoids GCon(M) presented in Proposition 3.4. This is a straightforward, but rather tedious, calculation using Proposition 2.2 to compute the relevant homotopy limits. As this calculation is not very instructive, we shall not write it out in full detail and just mention that it uses arguments similar to those in Example 2.11. (In particular, it uses that all cocycles on manifolds diffeomorphic to some R n may be trivialized, and that G-valued functions and g-valued forms are sheaves.) We introduce the following notation for the mapping stack BG M in order to match the notations established in Proposition 3.4 and Definition 3.5.
where V ⊆ M and V ⊆ M run over all open subsets diffeomorphic to R dim(M) , which satisfy where V ⊆ M runs over all open subsets diffeomorphic to R dim(M) , which satisfy     between the concretified mapping stacks. Using our simplified, but weakly equivalent, models for both concretified mapping stacks, the stages F M : GCon(M)(U ) → 2 (M, ad(G))(U ) of this H-morphism read as

Yang-Mills Equation
We formalize the Yang-Mills equation on globally hyperbolic Lorentzian manifolds in terms of a morphism between concretified mapping stacks. The corresponding stack of Yang-Mills solutions is defined by an appropriate homotopy fiber product and it will be worked out explicitly, up to weak equivalence in H. Our constructions are functorial on the usual category Loc m of oriented and time-oriented globally hyperbolic Lorentzian manifolds (of a fixed but arbitrary dimension m ≥ 2), which we shall review below.

Globally hyperbolic Lorentzian manifolds. Spacetimes in physics are described by globally hyperbolic Lorentzian manifolds, see [BEE96, Chapter 3], [ONe83, Chapter 14]
and also [BGP07, Section 1.3] for a more concise introduction. Recall that a Lorentzian manifold is a manifold M that is equipped with a metric of Lorentzian signature (− + + · · · +). We further assume our Lorentzian manifolds to be equipped with an orientation and a time-orientation, and that they are of a fixed but arbitrary dimension m ≥ 2. For notational simplicity, we denote oriented and time-oriented Lorentzian manifolds by symbols like M, i.e. we suppress the orientation, time-orientation, and metric from our notation.
A Cauchy surface in a Lorentzian manifold M is a subset ⊆ M such that every inextensible timelike curve in M meets exactly once. A Lorentzian manifold that admits a Cauchy surface is called globally hyperbolic. Globally hyperbolic Lorentzian manifolds M provide us with a suitable geometric framework to study hyperbolic partial differential equations, whose initial data are assigned on a spacelike Cauchy surface    .4) may be assumed to be also causally convex. We shall use this weakly equivalent description from now on.

Yang-Mills morphism.
Let us first fix some notations. Let V be an object in Loc m that is diffeomorphic (as a manifold) to R m . Consider the vector space p (V ) of p-forms on V . Because V is in particular an oriented Lorentzian manifold, we may introduce the Hodge operator * : p (V ) → m− p (V ). Using also the de Rham differential d : p (V ) → p+1 (V ), we may define the codifferential δ := (−1) m ( p+1) * d * : p (V ) → p−1 (V ). These operations extend to the vector space p (V, g) of g-valued p-forms. Given A ∈ 1 (V, g), we also define (4.6) and Interpreting A as a gauge field, the Yang-Mills equation on V reads as Given also an object U in C, the Hodge operator, the de Rham differential and the codifferential on V may be extended vertically along V × U → U to the vector space p,0 (V × U ) of vertical p-forms on V × U → U . We denote the vertical Hodge operator by * vert : p,0 (V × U ) → m− p,0 (V × U ), the vertical de Rham differential by d vert : p,0 (V × U ) → p+1,0 (V × U ) and the vertical codifferential by δ vert : and . (4.9) Interpreting A as a smoothly U -parametrized gauge field, the vertical Yang-Mills equa- (4.11) To confirm that this defines a groupoid morphism observe that δ vert    is a morphism in 1 (M, ad(G))(U ). Observe that this does not enforce the Yang-Mills equation in the strict sense as above in our naive construction (cf. (4.13)), but it demands that YM M (A, P) is isomorphic to one of the zeros 0 M ( P) in 1 (M, ad(G))(U ). It is important to notice that every morphism in 1 (M, ad(G))(U ) with target given by 0 M ( P) necessarily has to originate from an object of the form 0 M (P ), because the vanishing vertical 1-forms 0 ∈ 1,0 (V × U, g) are invariant under the adjoint action ad of gauge transformations. As a consequence, the fact that the morphism (4.19) exists implies already the strict condition that δ vert Summing up, we observe that even though a priori the homotopy fiber product enforces a weaker version of the Yang-Mills equation (i.e. up to isomorphism), a specific feature of the H-morphism 0 M turns this weaker version into a strict equality similar to (4.13). Below we shall make this statement precise by proving that 0 M is a fibration in H, hence the homotopy fiber product may be computed (up to weak equivalence) by the ordinary fiber product. This will eventually show that GSol(M) is weakly equivalent to our naive solution stack discussed at the beginning of this subsection.
As already mentioned in the remark above, there exists a weakly equivalent simplified description of the Yang-Mills stack GSol(M) given in Definition 4.4. It relies on the following observation. As a consequence of this lemma, the homotopy fiber product in Definition 4.4 is weakly equivalent to the ordinary fiber product. Therefore, similarly to Proposition 3.4, we obtain a weakly equivalent simplified description of GSol(M), which agrees with our naive solution stack from the beginning of this subsection. Summing up, we obtained

Stacky Cauchy Problem
In this section we introduce and discuss a stacky version of the Yang-Mills Cauchy problem. It turns out that well-posedness of the stacky Cauchy problem is a stronger statement than well-posedness of the ordinary Cauchy problem for gauge equivalence classes of Yang-Mills fields. In particular, the solution for each given initial datum in the stacky Cauchy problem must be unique up to a unique isomorphism, which is stronger than uniqueness of their associated gauge equivalence classes. To set up the stacky Cauchy problem, we first introduce a stack GData( ) that describes initial data on a Cauchy surface for the Yang-Mills equation and an H-morphism data : GSol(M) → GData( ) that assigns to Yang-Mills fields their initial data on . This will allow us to define well-posedness of the stacky Yang-Mills Cauchy problem using the language of model categories. We conclude explaining that this condition is equivalent to a family of parametrized PDE problems, which may be addressed by ordinary PDE-theoretical techniques.

Initial data stack.
Let M be any object in Loc m and ⊆ M any spacelike Cauchy surface. Recall that dim( ) = dim(M) − 1 = m − 1. In the usual approach, see e.g.
[CS97,C-B91], an initial datum on for the Yang-Mills equation on M is a triple (A , E, P ) consisting of a principal G-bundle P on with connection A and a 1-form E on with values in the corresponding adjoint bundle, which satisfies the Yang-Mills constraint δ A E = 0. Here δ A is the covariant codifferential on that is obtained from the induced Hodge operator on . As a refinement of the set of initial data used in [CS97,C-B91], our approach allows us to introduce a stack GData( ) of initial data on . Abstractly, this stack may be obtained by the following construction: Consider the stack of gauge fields GCon( ) on and form its tangent stack T st GCon( ) using similar techniques as in [Hep09]. Then implement the Yang-Mills constraint by a homotopy fiber product similarly to Sect. 4.3. Since for our practical purposes the construction of tangent stacks in [Hep09] is too involved, we shall not employ this abstract perspective and instead define directly the stack of initial data (up to weak equivalence) in an explicit form. where our pullback notation * is analogous to the one introduced in Proposition 3.4.

Initial data morphism.
Let M be any object in Loc m and ⊆ M any spacelike Cauchy surface. We denote by ι : → M the embedding of the Cauchy surface into M. We further choose and fix any normalized future-directed timelike vector field n on M whose restriction to ⊆ M is normal to the Cauchy surface. Given any object U in C, we denote by the same symbol n also the vector field on M × U that is obtained by extending n constantly along U . The restrictions of n to open subsets V × U of M × U are also denoted by n in order to simplify our notations.
The aim of this subsection is to define the initial data H-morphism where we used an intuitive compact notation. Explicitly, for (A, where n F vert (A D(W ) ) ∈ 1,0 (D(W ) × U, g) denotes the contraction of the vertical curvature F vert (A D(W ) ) ∈ 2,0 (D(W ) × U, g) with our fixed normalized timelike vector field n. (As explained above, the vector field n on M is extended constantly to M × U and then restricted to D(W ) × U .) Using the vertical Yang-Mills equations δ vert A V F vert (A V ) = 0, it is an elementary check that data (A, P) satisfies the vertical Yang-Mills constraint of Definition 5.1. Verifying that (5.6) indeed defines a groupoid morphism and confirming naturality in U are also elementary checks. Hence, we defined the initial data H-morphism in (5.5).

Cauchy problem.
Using the language of model categories, we can now define the concept of a well-posed stacky Cauchy problem associated to the Yang-Mills equation. We will then show that our definition is equivalent to a family of parametrized PDE problems, which generalize the ordinary Yang-Mills Cauchy problem, cf. [CS97,C-B91]. The purpose of this section is thus to establish a bridge between our model categorical perspective on Yang-Mills theory and the language more familiar to PDE theorists.
h w w n n n n n n n n n n n (A , E, P ) (5.8) in GData( )(U ) commutes.
Remark 5.4. In more explicit words, item 1. of this proposition demands that there exists for every smoothly U -parametrized initial datum (A , E, P ) a smoothly Uparametrized solution (A, P) of the Yang-Mills equation whose initial datum is isomorphic to the given one by a smoothly U -parametrized gauge transformation h : data (A, P) → (A , E, P ). Item 2. demands that any two such solutions are isomorphic by a unique smoothly U -parametrized gauge transformation h : (A, P) → (A , P ). This is clearly a stronger condition than existence and uniqueness of solutions to the Cauchy problem for gauge equivalence classes, where the uniqueness aspect in item 2. does not play a role.
Remark 5.5. The fact that the conditions in Proposition 5.3 have to hold true for all objects U in C is very important from our stacky perspective. This is because the groupoids GSol(M)(U ) and GData( )(U ) encode the smooth structure of the Yang-Mills stack and initial data stack. Thus, well-posedness of the stacky Cauchy problem does not only formalize the usual notion of well-posedness (i.e. existence and uniqueness of solutions for given initial data), but it also demands smooth dependence (in the sense of stacks) of solutions on their initial data. The latter may be interpreted as a smooth analogue of the condition of continuous dependence of solutions on their initial data, which is more familiar in PDE theory.
Remark 5.6. As a final remark, we note that the conditions in Proposition 5.3 are known to hold for U = R 0 and spacetime dimension m = 2, 3, 4 [CS97,C-B91]. However, we are not aware of any results for other objects U in C, which leads to the more complicated realm of smoothly U -parametrized PDE problems. Being crucial for a better understanding of the geometry of the stack GSol(M) of Yang-Mills fields, we believe that a detailed study of the explicit conditions of Proposition 5.3 is a very interesting and compelling PDE problem. This problem is clearly beyond the scope of the present work. However, we would like to mention that analogous results for the simpler case of smoothly U -parametrized normally hyperbolic linear PDEs can be established via the theory of symmetric hyperbolic systems [BHS]. We expect such techniques to be sufficient for proving that the stacky Cauchy problem is well-posed for Abelian Yang-Mills theory with structure group G = U (1).

Yang-Mills Stack in Lorenz Gauge
We briefly discuss how gauge fixings may be interpreted in our framework as weak equivalences of stacks. For simplicity, we shall focus on the particular example given by Lorenz gauge fixing, even though our ideas apply to other gauge fixings as well.
for all causally convex open subsets V ⊆ M diffeomorphic to R m . As a consequence of (6.1) and the conditions  Remark 6.3. For the unparametrized case U = R 0 , solvability of PDEs of the form (6.4) has been studied in [C-B87] for spacetime dimension m = 2. However, we are not aware of any results for other objects U in C. We believe that, together with the smoothly parametrized PDEs explained in Remark 5.6, these are very interesting problems for PDE theorists as they are crucial for understanding the geometry of the Yang-Mills stack. For the case of Abelian Yang-Mills theory with structure group G = U (1), we expect that similar techniques as in [BHS], which are based on the theory of symmetric hyperbolic systems, can be used to prove that the conditions in Proposition 6.2 hold true and hence that j M is a weak equivalence in H. We hope to come back to this issue in future work.
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A. Monoidal Model Structure on H
The goal of this appendix is to show that the local (as well as the global) model structure on H is compatible with the closed symmetric monoidal structure discussed in Sect. 2.2. More precisely, we show that H is a symmetric monoidal model category, see e.g. [Hov99,Chapter 4]. For this purpose we first need the symmetric monoidal model structure on Grpd. It is well-known that the closed symmetric monoidal structure and model structure on Grpd that we introduced in Sect. 2.1 define a symmetric monoidal model category structure on Grpd. As we could not find a proof of this statement in the literature, we shall provide it here. Proof. Since all objects in Grpd are cofibrant, so is the unit object { * }. Therefore, to conclude that Grpd is a symmetric monoidal model category, it is sufficient to prove that the monoidal bifunctor × : Grpd × Grpd → Grpd is a Quillen bifunctor, cf. [Hov99, Chapter 4.2].
Take two cofibrations F : G → H and F : G → H in Grpd and form their pushout product We have to show that F F is a cofibration, which is acyclic whenever either F or F is. Recall that cofibrations in Grpd are functors that are injective on objects. Computing the pushout in Grpd, one finds that objects of P(F, F ) are equivalence classes of pairs of the form (y, (By the subscript 0 we denote the set of objects of a groupoid.) Since F and F are by hypothesis injective on objects, it follows by using our equivalence relation that F F is injective on objects too. Hence, it is a cofibration. The case where one of the cofibrations is acyclic may be simplified by using that Grpd is cofibrantly generated (cf. [Hol08a]) and [Hov99, Corollary 4.2.5]. Using also symmetry of the monoidal structure, it is sufficient to show that for the generating acyclic cofibration J : { * } → 1 , given by * → 0 and id * → id 0 , the pushout product is an acyclic cofibration for any cofibration F : G → H in Grpd. The pushout groupoid P(F, J ) can be computed explicitly: Its set of objects is G 0 H 0 , i.e. an object is either an object x in G or an object y in H . Its morphisms are characterized by The groupoid morphism F J in (A.2) is the functor that acts on objects as and on morphisms as where it is important to recall the definition of morphisms in P(F, J ), see (A.3). It is clear that F J is fully faithful and essentially surjective (and of course injective on objects), hence it is an acyclic cofibration. This completes our proof.

B. Cofibrant Replacement of Manifolds in H
Let M be a (finite-dimensional and paracompact) manifold and V = {V α ⊆ M} any cover by open subsets. The goal of this appendix is to prove that the presheaf ofČech groupoids associated to V is always weakly equivalent to M in H. Moreover, it provides a cofibrant replacement of M when all V α are diffeomorphic to R dim(M) . It is important to stress that the latter statement does not require that the cover is good, in particular V α ∩ V β may be neither empty nor diffeomorphic to R dim(M) . We shall always work with the local model structure on H, see Theorem 2.5.
We define the object V in H by the following functor V : C op → Grpd: To any object U in C, it assigns the groupoid V(U ) with objects given by diagrams commutes. To any C-morphism : U → U , we assign the groupoid morphism V( ) : 3) The action of V( ) on morphisms is fixed by their uniqueness. There exists a canonical H-morphism where M is the stack represented by our manifold M, cf. Example 2.9. Explicitly, recalling that M(U ) = C ∞ (U, M), for all objects U in C, the stages of q are given by the groupoid morphisms Naturality of these stages in U is obvious by definition.  The crucial point is now to prove that the stage-wise liftings f : V(U ) → X (U ) can be chosen to form a natural transformation, thus providing the stages of a morphism f : V → X in H. In order to do so, we take advantage of the fact that each V α is assumed to be an object of C. Given any morphism of the form : U → V α in C (also regarded as a smooth map of manifolds), naturality is expressed as commutativity of the diagram V(V α ) in Grpd. Taking the object (α, id V α ) in V(V α ), this commutative diagram implies that f (α, ) = X ( ) f (α, id V α ) . (B.10) Hence, the stages f : V(U ) → X (U ) of a natural lift are uniquely determined by their actions on the objects (α, id V α ), for all α. Because there are no further coherence conditions for the restriction of f to objects of the form (α, id V α ), it follows from our discussion at the end of the previous paragraph of this proof that there exists a lift for our original problem (B.7), which is determined by (B.10) and by the choice of a preimage along f : X (V α ) → Y (V α ) of the object f (α, id V α ), for each α. As a consequence, V is a cofibrant object in H. defined using the mapping groupoids of H/K : For objects f X and f Y of H/K , the mapping groupoid Grpd H/K ( f X , f Y ) has as objects all commutative diagrams (C.1) in H and as morphisms all commutative diagrams in H, where pr X : X × 1 → X is the projection H-morphism. Proof. Using the Yoneda lemma, the objects of the source groupoid in (C.4) can be described by commutative diagrams We prove a technical lemma that provides us with a useful characterization of the S −1 -local objects up to (un-truncated) weak equivalences in H/K . Given any object f X : X → K in H/K , we denote by Im( f X ) the object in H given by the full image sub-presheaf of K . Explicitly, for any object U in C, the groupoid Im( f X )(U ) has as objects all objects f X (x) in K (U ), for all objects x in X (U ), and as morphisms all morphisms k : f X (x) → f X (x ) in K (U ). There is a canonical commutative diagram X f X ? ? ? ? ? ?
in H, i.e. a canonical H/K -morphism f X from f X : X → K to f Im( f X ) : Im( f X ) → K .
Lemma C.5. Let f X : X → K be any S −1 -local object in H/K . Then f Im( f X ) : Im( f X ) → K is an S −1 -local object in H/K and the canonical H/K -morphism f X in (C.13) is a (un-truncated) weak equivalence.
Proof. By Lemma C.2, f X : X → K is stage-wise fully faithful. By construction of Im( f X ), we then find that f X : X → Im( f X ) is stage-wise fully faithful and also stage-wise surjective on objects, hence a weak equivalence in H/K . It remains to show that f Im( f X ) : Im( f X ) → K is an S −1 -local object. For this we make use of Lemma C.2 and Proposition C.3. It is immediately clear by construction that f Im( f X ) : Im( f X ) → K is stage-wise fully faithful and a stage-wise fibration. (To prove the latter statement, use that f X : X → K is by hypothesis a stage-wise fibration.) Thus, it remains to prove that is a weak equivalence in Grpd, for all good open covers {U i ⊆ U }. The target groupoid can be computed explicitly by using Propositions 2.2 and 2.13. One finds that its objects are tuples where y is an object in K (U ), x i are objects in X (U i ), g i j : f X (x i )| U i j → f X (x j )| U i j are morphisms in K (U i j ), and k i : y| U i → f X (x i ) are morphisms in K (U i ). This data has to satisfy g ii = id f X (x i ) , for all i, g jk | U i jk • g i j | U i jk = g ik | U i jk , for all i, j, k, and g i j • k i | U i j = k j | U i j , for all i, j. The morphisms of the target groupoid are tuples where h : y → y is a morphism in K (U ) and h i : f X (x i ) → f X (x i ) are morphisms in K (U i ), satisfying g i j • h i | U i j = h j | U i j • g i j , for all i, j, and k i • h| U i = h i • k i , for all i. The canonical morphism (C.14) is explicitly given by from which one immediately observes that it is fully faithful. Using that f X is S −1 -local, we can show that the canonical morphism (C.14) is also essentially surjective and thus a weak equivalence: Given any object (y, ({ f X (x i )}, {g i j }), {k i }) of the target groupoid in (C.14), we obtain from the property that f X is stage-wise fully faithful an object (y, ({x i }, { g i j }), {k i }) of the homotopy fiber product defined by means of the H -morphism f X : X → K . Explicitly, the X (U i j )-morphisms g i j : x i | U i j → x j | U i j are uniquely determined by full faithfulness and f X ( g i j ) = g i j .
Because f X is a (local) fibration, there exists a morphism Explicitly, h : f X (x) → y is a morphism in K (U ) and h i : The weak equivalences in the (−1)-truncation of H/K are by construction the socalled S −1 -local equivalences in H/K . We say that a morphism (C.1) in H/K is an is a weak equivalence in Grpd, for all S −1 -local objects f Z : Z → K in H/K . Here Q is a cofibrant replacement functor in H and Q( f ) is the corresponding H/K -morphism Because f Im( f Z ) : Im( f Z ) → K is a (local) fibration in H, there exists a morphism in the groupoid (C.27) from an object ( f Im( f Z ) (z), ({z| U i }, {id}), {id}), where z is an object in Im( f Z )(U ), to (C.26). Using that f Im( f Z ) is a stage-wise fibration, the object z in Im( f Z )(U ) may be chosen such that f Im( f Z ) (z) = f Q(Y ) (y). Because f Im( f Z ) is stage-wise injective on objects, such z is unique and we may define the dashed arrow by setting f (y) = z. Naturality of our construction of f follows immediately from uniqueness, which completes our proof.
We developed sufficient technology to obtain a functorial fibrant replacement in the (−1)-truncation of H/K . Let f X : X → K be any object in H/K . We define a new object in H/K , which we call the 1-image of f X and denote as f Im 1 ( f X ) : Im 1 ( f X ) → K , by the following construction: For an object U in C, the groupoid Im 1 ( f X )(U ) has as objects all objects z in K (U ) for which there exist a good open cover {U i ⊆ U }, objects x i in X (U i ) and K (U i )-morphisms h i : z| U i → f X (x i ). The morphisms between two objects z and z in Im 1 ( f X )(U ) are all K (U )-morphisms k : z → z . For a morphism : U → U in C, the groupoid morphism Im 1 ( f X )( ) : Im 1 ( f X )(U ) → Im 1 ( f X )(U ) is the one induced by K ( ) : K (U ) → K (U ). (To show that K ( )(z) is an object in Im 1 ( f X )(U ) for any object z in Im 1 ( f X )(U ) use refinements of open covers to good open covers.) The H-morphism f Im 1 ( f X ) : Im 1 ( f X ) → K is then given by stage-wise full subcategory embedding. There is a canonical commutative diagram X f X ? ? ? ? ? ?
in H, i.e. a canonical H/K -morphism f X from f X : X → K to f Im 1 ( f X ) : Im 1 ( f X ) → K .
in H commutes. Recall that f X and g • f X induce epimorphisms on sheaves of 0-th