Inertia and delocalized twisted cohomology

Ulrich Bunke, Thomas Schick and Markus Spitzweck ^{∗}

January 17, 2007

Contents

1 Introduction  1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 A description of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Motivation of the definition of twisted delocalized cohomology . . . . . . . . . . . . . . . . 7
2 Inertia  9
2.1 2-limits in 2-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Loops of topological stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Loops and principal bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Gerbes and local systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6 The holonomy of G̃^{δ} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 Delocalized Cohomology of orbispaces and orbifolds  33
3.1 Definition of delocalized twisted cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Twisted de Rham cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1 Introduction

1.1 Motivation

1.1.1 In the recent mathematical literature cohomological and topological properties of orbifolds became an intensively studied subject. A considerable part of the motivation comes from the mirror symmetry program where orbifolds arise naturally. Cornerstones^{1} of the recent developments were the introduction of twisted orbifold K-theory [AR03] and the orbifold quantum cohomology [CR04] on the topological side, and the investigation of gerbes [LU04] and loop groupoids [LU02] on the geometric side. 1.1.2 Classically, orbifolds are defined like manifolds as spaces which are locally homeomorphic to a quotient of an euclidean space by a finite group. Alternatively, orbifolds are represented by proper étale smooth groupoids. Working with groupoid representations of orbifolds is like working with manifolds with a fixed atlas. In the modern coordinate invariant point of view an orbifold is a smooth stack in smooth manifolds which admits an orbifold atlas. By considering orbifolds as objects in the 2-catgeory of smooth stacks one makes the notion of morphisms^{2} and other constructions like fibre products transparent. The framework of stacks is most natural if one wants to include gerbes into the picture. 1.1.3 If one replaces smooth manifolds by topological spaces then the corresponding analog of an orbifold is an orbispace. The goal of the present paper is to show that some geometric constructions on orbifolds are in fact topological concepts and extend to orbispaces. 1.1.4 The fixed point manifolds of the elements of the local automorphism groups of an orbifold X can be assembled into a new orbifold LX called the inertia or loop orbifold or the orbifold of twisted sectors. In the present paper we show that the loop orbifold can be characterized as the 2-categorial (in the 2-category of stacks) equalizer of the pair (id_{X}, id_{X}) of the identity morphisms. The same definition applies to orbispaces in the topological context. Since 2-categorial equalizers always exist in 2-categories of stacks it is clear that LX exists as a stack. But it is not a priori clear that LX is again an orbifold (or orbispace, resp.). In the present paper we show that taking loop stacks preserves orbispaces. The corresponding result for orbifolds is well-known, but requires different, manifold specific arguments. 1.1.5 A U(1)-banded gerbe G → X over an orbifold gives rise to a U(1)-principal bundle G̃ → LX over the loop orbifold of X. This bundle has a natural reduction of structure groups to the discrete U(1)^{δ}. The traditional way to construct this reduction is to choose a connection and curving on the gerbe G → X. This geometric data induces a connection on G̃ → LX which turns out to be flat by a calculation. The flat connection gives the reduction of structure groups, which turns out to be independent of the choices of geometric structures. The sheaf L of locally constant sections of the associated flat line bundle L → LX is called an inner local system and plays an important role in the definition of the orbifold cohomology. In the present paper we give a topological construction of the reduction of the structure group of G̃ → LX to U(1)^{δ} and of the sheaf L. Furthermore, we calculate its holonomy in terms of the Dixmier-Douady class of the gerbe G → X. 1.1.6 The third concept which we generalize to the topological case is that of twisted delocalized orbifold cohomology. The usual definition in the smooth case is based on the de Rham complex of forms on LX with coefficients in L → LX. The differential of this complex involves the flat connection on L and a closed three form on X which represents the image of the Dixmier-Douady class of the gerbe f : G → X in real cohomology. Let f_{L} : G_{L} → LX denote the pull-back of the gerbe via LX → X. In the present paper we use the sheaf theory for smooth (or topological, resp.) stacks [BSSc] in order to define the twisted delocalized orbifold cohomology as sheaf cohomology H^{∗}(LX, Tw_{G}(L)), where Tw_{G}(L) := R(f_{L})_{∗}f_{L}^{∗}(L). Our main result is, that in the smooth case the twisted delocalized cohomology according to this sheaf theoretic definition is isomorphic to the former construction using the de Rham complex. In addition to the fact that it works in the topological context our sheaf theoretic definition of twisted delocalized orbifold cohomology has the advantage that it is functorial in the gerbe G → X. 1.1.7 In the remaining parts of the introduction we give a detailed description of the results of the present paper and explain how they are related to the existing literature.

∗Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, GERMANY, bunke@unimath.gwdg.de, schick@uni-math.gwdg.de, spitz@uni-mah.gwdg.de

1Here we mention those works which are relevant for the present paper. Note that there is a huge literature on orbifolds in algebraic geometry and mathematical physics.

2The right notion of a morphism between orbifolds is a representable morphism of stacks. This definition corresponds to the notion of a good morphism in the literature.

1.2 A description of the results

1.2.1 In the present paper we consider stacks in smooth manifolds or stacks in topological spaces. Our basic reference for stacks in these contexts is [Hei05], but see also [Noo], [Met] and the recent [BX]. A stack X in smooth manifolds (topological spaces, resp.) is called a smooth stack (topological stack, resp.) if it admits an atlas A → X. The atlas is called an orbifold (orbispace, resp.) atlas if the smooth (topological, resp.) groupoid A ×_{X} A ⇒ A is proper étale (very proper, étale and separated (see 2.3.7 for explanations)). An orbifold (orbispace, resp.) is a smooth (topological, resp.) stack which admits an orbifold (orbispace, resp.) atlas. We refer to [BS] for an introduction to orbispaces, and e.g. to [CR04, Sec 2.] for some basic information on orbifolds. 1.2.2 In Subsection 2.1 we review the notion of 2-categorial limits. The 2-categorial equalizer of a pair of maps is a special kind of limit. We will see that equalizers exist in the 2-catgeory of stacks on a site and in the two-category of groupoids in topological spaces. The goal of Subsections 2.2 and 2.3 of the present paper is to place the construction of the loop orbifold LX (or orbispace, resp.) into the framework of stacks in manifolds (topological spaces, resp.).

We consider the orbifold (orbispace, resp.) X as a stack and define its inertia stack IX → X as the 2-categorial equalizer of the pair (id_{X}, id_{X}). The loop stack LX is defined in an ad-hoc manner, see Definition 2.16 and Remark 2.2.6. We will see that it is canonically equivalent to IX. Though Definition 2.10 of the 2-categorial equalizer by a pull-back diagram is quite constructive we prefer to work with the simpler construction LX from now on. If X is an orbifold (orbispace, resp.), then apriori LX is a stack in smooth manifolds (topological spaces, resp.). In Lemma 2.25 (Lemma 2.31) we show that the loop stack of a topological stack (orbispace, resp.) is again a topological stack (orbispace). The main idea is to show that the existence of an (orbispace) atlas of X implies the existence of an (orbispace) atlas of LX. In the smooth case, the fact that the loop stack of an orbifold is again an orbifold is well known, see [CR04, Lemma 3.1.1] or [LU02, Cor. 2.6.2]. 1.2.3 The loop orbifold is also known as the orbifold of twisted sectors (compare [CR04, Sec. 3.1]) or inertia orbifold. It plays an important role in the construction of the delocalized orbifold cohomology. The twisted sectors first appeared in connection with the orbifold index theorem [Kaw78], [Kaw81]. In the framework of topological groupoids G the corresponding object is called the inertia groupoid ΛG which has been studied in detail in [LU02]. In order to keep our notation uniform in the present paper we will denote the inertia groupoid by LG and call it loop groupoid^{3}. 1.2.4 To a topological group G we associate the classifying stack BG := [∗/G] (see [Hei05, Example 1.5]). A G-principal bundle over a stack X is by definition a map p : X → BG^{4}. Applying the loop functor and using the canonical isomorphism LBG ≅ [G/G] we get a map Lp : LX → [G/G]. If G is abelian, then this map lifts to a function h : LX → G. We are in particular interested in the case G = U(1) and give various geometric and cohomological interpretations of this function. In the present paper, ordinary cohomology of an orbispace X is understood in the sense of [BS, Sec. 2.2]. Let A → X be an atlas and form the simplicial space A^{·} such that A^{n} := A ×_{X} ⋯ ×_{X} A (n + 1-factors). Here the fibre product is taken in stacks in topological spaces, but the stack A^{n} is in fact equivalent to a space since the map A → X is representable. The cohomology of X with integral coefficients is then defined as

H^{∗}(X; ℤ) := H^{∗}(|A^{·}|; ℤ) ,

where |A^{·}| denotes the realization of the simplicial space. Independence of the choice of the atlas has been shown in [BS, Sec. 2.2] and [Beh04]^{5}. An alternative definition of the cohomology of X could be based on the sheaf theory for orbifolds which will be discussed below. The group H^{2}(X; ℤ) classifies isomorphism classes of U(1)-principal bundles p : E → X (see [BS, Sec. 4.2] for this fact).

(1) If Γ is a finite group, then we have H^{2}([∗/Γ]; ℤ) ≅ H^{2}(Γ; ℤ) ≅ H^{1}(Γ; U(1)) ≅ Γ̂, where Γ̂ := Hom(Γ, U(1)). A class χ ∈ H^{2}([∗/Γ]; ℤ) thus gives rise to function χ̄ : L[∗/Γ] ≅ [Γ/Γ] → U(1). This construction extends to general orbispaces X and associates to each class χ ∈ H^{2}(X; ℤ) a function χ̄ : LX → U(1). A class χ ∈ H^{2}(X; ℤ) also classifies a U(1)-principal bundle and therefore gives ries to a function h : LX → U(1). We will show that χ̄ = h. This equality has the following geometric interpretation. A point in the fibre of LX → X over x ∈ X is an element γ ∈ Aut(x). The group Aut(x) acts on the fibre E_{x} := p^{−1}(x). We write this as a left action. Then we show in Lemma 2.38 that the functions h, χ̄ are both characterized by

γe = eχ̄(γ) , γe = eh(γ)

\xymatrix{E \ar[r] \ar[d] & ∗ \ar[d] \\ X \ar@{=>}[ur] \ar[r]^{p} & BG} .

3Note that the loop groupoid LG in [LU02] is a much bigger object, and it is related with ΛG by the equation LG^{ℝ} ≅ ΛG in the notation of [LU02, Prop. 3.6.6].

4Sometimes we will use a more sloppy language and say that E → X is a G-principal bundle, where E → X is defined by the pull-back

5The result in this paper is more general. The only condition on the atlas A → X is that the range and source maps of the groupoid A ×_{X} A ⇒ A are topological submersions.

for all e ∈ E_{x}.

(2) G-principal bundles can be defined in terms of cocycles. We will give an interpretation of the function h in terms of the cocyle.

(3) A third cohomological interpretation uses the transgression Tr : H^{2}(X; U(1)) → H^{1}(LX; U(1)) introduced in [ARZ], [LU], [TXb].

1.2.5 Let f : G → X be a topological gerbe with band U(1) over an orbispace X. The induced map Lf : LG → LX can be factored canonically as LG →^{p} G_{L} →^{f_{L}} LX, where G_{L} := LX ×_{X} G. Here f_{L} : G_{L} → LX is a topological gerbe with band U(1), and p : LG → G_{L} is (the underlying map of) a U(1)-principal bundle. The first main observation of Subsection 2.5 is that the bundle LG → G_{L} descends canonically to a U(1)-principal bundle G̃ → LX. The second result of this Subsection assumes that X is an orbispace and asserts that G̃ → LX has a canonical reduction of the structure group G̃^{δ} → LX to U(1)^{δ}, the group U(1) with the discrete topology. The heuristic picture is as follows. Roughly speaking, a gerbe G → X over a topological stack X associates to each point x ∈ X a U(1)-central extension 1 → U(1) → { \atop Aut(x)} → Aut(x) → 1 of the group of automorphisms Aut(x). The fibre of the canonical map LX → X over x ∈ X is the automorphism group Aut(x). The U(1)-principal bundle G̃ → LX restricts to { \atop Aut(x)} → Aut(x) over x ∈ X. If X is an orbispace, then finiteness of the groups Aut(x) provide a reduction of the structure group of this bundle to U(1)^{δ}. Let L → LX denote the complex line bundle associated to G̃^{δ} → LX. Since its structure group is discrete we can form the sheaf L of locally constant sections of L. By 2.5.4 we have actually an extension

X × U(1)^{δ} → G̃^{δ} → LX

of group stacks over X. The induced algebraic structures on L → LX turn this line bundle into an inner local system in the sense of [Rua, Def. 2.1], [LU04, Def. 2.2.2]. 1.2.6 In the framework of groupoids the construction of G̃^{δ} → LX has been previously given in [LU02, Thm. 6.4.2] and [TXa, Prop. 2.9] (with the exception of the reduction of the structure group to the discrete U(1)^{δ}). In the smooth case a reduction of the structure group of a line bundle from U(1) to U(1)^{δ} is equivalent to a flat unitary connection. It has been observed in [LU, Lemma 5.0.1] and [TXa, Prop. 3.9] that a connection on the gerbe G → X induces a flat connection on L → LX. Our original contribution here is to give a construction of this reduction of the structure group in purely topological terms. In addition to simplifications this extends the previous results to the topological case. A twisted torsion in the language of [Rua] is a class α ∈ H^{2}(π_{1}^{orbifold}(X), U(1)), i.e. an isomorphism class of central U(1)-extensions

1 → U(1) → { \atop π_{1}^{orbifold}(X)} → π_{1}^{orbifold}(X) → 1 .

The orbifold fundamental group π_{1}^{orbifold}(X) is the automorphism group of the universal orbifold covering

Y → X. The map G_{α} := [Y/{ \atop π_{1}^{orbifold}(X)}] → [Y/π_{1}^{orbifold}(X)] = X is a topological gerbe with band U(1) over X. In [Rua, Sec 4] or [LU04, Example 2.2.2] an inner local system L_{α} is associated directly to a twisted torsion α. In the philosophy of the present paper we would consider L_{α} as the bundle associated to the gerbe G_{α} → X via the U(1)^{δ}-bundle G̃_{α}^{δ} → LX. The sheaf of locally constant sections L of the line bundle L (also called inner local system) plays an importand role in the definition of twisted delocalized cohomology of an orbifold [AR03], [Rua, Def. 2.2]^{6}, [TXa, Def. 3.10].

6This is the cohomology of LX with coefficients in L with shifted grading. It is different from the gerbe-twisted delocalized cohomology

1.2.7 It is an interesting problem to calculate the holonomy of the bundle G̃^{δ} → LX in terms of the Dixmier-Douady class d ∈ H^{3}(X; ℤ). We discuss this question in a typical case in Subsection 2.6. Let π : E → X be a U(1)-principal bundle in orbispaces and G → E be a topological gerbe with band U(1) and Dixmier-Douady class d ∈ H^{3}(E; ℤ). Let χ ∈ H^{2}(X; ℤ) be the first Chern class of E → X. As explained in 1.2.4 we get a function χ̄ : LX → U(1). Let LX_{1} := χ̄^{−1}(1). We will see that the canonical map LE → LX factorizes over LX_{1}, and that LE → LX_{1} is again a U(1)-principal bundle. The holonomy of the bundle G̃^{δ} → LE along the fibres of LE → LX_{1} can be considered as a function

g : LX_{1} → U(1) .

Our main result is the following description of this function. Note that π : E → X is an oriented fibre bundle. We have an integration map π_{!} : H^{3}(E; ℤ) → H^{2}(X; ℤ). In particular we can form π_{!}(d) ∈ H^{2}(X; ℤ) and the associated function

\overline{π_{!}(d)} : LX → U(1) .

In Proposition 2.49 we show the equality of functions

g = \overline{π_{!}(d)}_{|LX_{1}} .

In the smooth case (i.e. for orbifolds) holonomy questions could be addressed using Deligne cohomology. In fact, Deligne cohomology H_{Del}^{∗}(X) for orbifolds has been introduced in [LU]. The choice of a connection on the gerbe G leads to a lift of the Dixmier-Douady class d ∈ H^{3}(X; ℤ) of G → X to a Deligne cohomology class d_{Del} ∈ H_{Del}^{3}(X) under the natural forgetful map H_{Del}^{3}(X) → H^{3}(X; ℤ). The transgression of d_{Del} according to [LU, Thm. 6.0.2] is a class Tr(d_{Del}) ∈ H_{Del}^{2}(LX). Its integral (Lπ)_{!}(Tr(d_{Del})) ∈ H_{Del}^{1}(LX_{1}) should^{7} give the function g : LX_{1} → U(1). 1.2.8 Section 3 of the present paper is devoted to twisted delocalized cohomology. We are in particular interested in a version which is the target of the Chern character from twisted K-theory. We refer to Subsection 1.3 for a detailed introduction and a motivation of the particular definition of twisted delocalized cohomology. Our main original contribution in the present paper is a construction of this cohomology in the framework of sheaf theory on topological stacks. All previous definitions used the de Rham complex and are therefore tied to the orbifold case. To a topological stack (smooth stack, resp.) X we associate a site X. The smooth case was discussed at length in [BSSc]. So let us fix our conventions for the topological case here. A detailed account for the sheaf theory on topological stacks will be given in the paper [BSSa]. An object of X is a map (ϕ : U → X) in stacks in topological spaces, where U is a topological space (or more precisely stack which is equivalent to a space), and ϕ is a representable map which admits local sections^{8}. The morphisms in X are commutative diagrams

[figure: U; V; X]

consisting of a morphism U → V and a 2-morphism. A family (U_{i} → U)_{i∈I} of morphisms in X is a covering family if all maps U_{i} → U admit local sections and the induced map ⊔_{i∈I}U_{i} → U is surjective. To the site X we can associate the category of sheaves ShX of sets and the abelian category Sh_{Ab}X of sheaves of abelian groups. A map between topological (resp. smooth) stacks f : X → Y induces an adjoint pair of functors

f^{∗} : ShY ⇔ ShX : f_{∗}

7We have not checked the details here. In this picture it is also not obvious that (Lπ)_{!}(Tr(d_{Del})) only depends on d ∈ H^{3}(X; ℤ), and not on the choice of its lift d_{Del} ∈ H_{Del}^{3}(X).

8Note that X must be small. A precise definition would either involve universes or a cardinality restriction.

relating the categories of sheaves on these sites. In the smooth case the construction of this adjoint pair was given by [BSSc, Sec. 2.1]. The construction in the case of topological stacks is very similar, see [BSSa]. The restriction f_{∗} : Sh_{Ab}X → Sh_{Ab}Y of f_{∗} to abelian sheaves is left-exact and admits a right-derived functor

Rf_{∗} : D^{+}(Sh_{Ab}X) → D^{+}(Sh_{Ab}Y)

between the lower-bounded derived categories. Let G → X be a topological (smooth, resp.) gerbe with band U(1) on an orbispace (resp. orbifold) X. It gives rise to the U(1)^{δ}-principal bundle G̃^{δ} → LX and an associated locally constant sheaf L of ℂ-vector spaces on the site LX. In Subsection 3.3 we define the G-twisted delocalized cohomology of X by

H_{deloc}^{∗}(X; G) := H^{∗}(ev ∘ Rp_{∗} ∘ f_{L}^{∗}(L)) .  (1.1)

The notation is explained by means of the following diagram

\xymatrix{∗ & G_{L} \ar[l]^{p} \ar[d]^{f_{L}} \ar[r] & G \ar[d]^{f} \\ & LX \ar@{=>}[ur] \ar[r] & X} ,

where the square is 2-cartesian, i.e. f_{L} : G_{L} → LX is the pull-back of the gerbe f : G → X via the canonical map LX → X, and the map p : G_{L} → ∗ is the canonical projection to the point. Since Site(∗) is the big site of the point, i.e. the category of all non-empty topological spaces we need the evaluation ev : D^{+}(Sh_{Ab}Site(∗)) → D^{+}(Ab) at the object (∗ → ∗) ∈ Site(∗). The functoriality of this cohomology in the data G → X is studied in Lemma 3.4. Our main result is the comparison of this sheaf-theoretic definition of G-twisted delocalized cohomology with the previous de Rham model [TXa, Def. 3.10] in the case of orbifolds. 1.2.9 We now explain the de Rham model for the twisted delocalized cohomology. Let X be an orbifold and G → X be a smooth gerbe with band U(1). In this case we can define three versions of twisted delocalized de Rham cohomology. The 2-periodic twisted delocalized cohomology is the correct target of the Chern character and will be defined in 1.3.12. The sheaf theoretic cohomology (1.1) is not 2-periodic. In the following we describe its appropriate de Rham model. We choose a closed three-form λ ∈ Ω^{3}(LX) which represents the image of the Dixmir-Douady class of G_{L} → LX in real cohomology. Then we define a sheaf Ω_{LX}[[z]]_{λ} ∈ C^{+}(Sh_{Ab}LX) of complexes which associates to each object (ϕ : U → LX) ∈ LX the complex (Ω(U)[[z]], d_{λ}), where (Ω(U), d_{dR}) is the de Rham complex of the smooth manifold U, z is a formal variable of degree 2, and d_{λ} = d_{dR} + \frac{d}{dz} ϕ^{∗}λ. Let Ω(LX; L)[[z]]_{λ} := Γ_{LX}(Ω_{LX}[[z]]_{λ} ⊗L) denote the complex of global sections (see 3.2.4 for the definition of global sections) of the tensor product of sheaves Ω_{LX}[[z]]_{λ} ⊗ L. Its cohomology is the twisted delocalized de Rham cohomology

H_{dR,deloc}^{∗}(X, (G, λ)) := H^{∗}(Ω(LX; L)[[z]]_{λ})  (1.2)

(see 3.15). 1.2.10 The twisted delocalized de Rham cohomology defined in [TXa, Def. 3.10] is related to the definition of the present paper by a duality. Let us first recall the definition [TXa, Def. 3.10]. Let u be a formal variable of degree −2 and define the complex of sheaves Ω_{LX}((u)) which associates to (ϕ : U → LX) the space of formal Laurent series of forms Ω(U)((u))_{λ} with the differential d′_{λ} := d_{dR} − uiϕ^{∗}λ. The twisted cohomology in [TXa, Def. 3.10] is the cohomology of the complex of compactly supported global sections Ω(LX; L)_{comp}((u))_{λ}^{9} of Ω_{LX}((u))_{λ} ⊗ L. Note that the multiplication by u induces an isomorphism of complexes which makes the cohomology of [TXa, Def. 3.10] two-periodic. We define the pairing (using the hermitean structure of L)

< …, ⋯ >: Ω(LX; L)_{comp}((u))_{λ} ⊗ Ω(LX; L)[[z]]_{λ} → ℂ  (1.3)

9Here we use the freedom of rescaling λ by non-zero factors as explained in [TXa, Rem. 3.11(1)].

by

< u^{n}ω, z^{m}α^{l} >= δ_{m,n}m! ∫_{LX} ω ∧ α ,

where ω ∈ Ω(LX; L)_{comp} and α ∈ Ω(LX; L). One easily checks that

< d′_{λ}ω, α >= (−1)^{|ω|+1} < ω, d_{λ}α > .

The pairing (1.3) induces an embedding of Ω(LX; L)[[z]]_{λ} into the dual complex of Ω(LX; L)_{comp}((u))_{λ}. 1.2.11 Let us now explain the relation between (1.2) and the 2-periodic version 1.3.12. Note that the complex of sheaves Ω_{LX}[[z]]_{λ} admits an action of the operation T := \frac{d}{dz} of degree −2. We consider the system

S : Ω_{LX}[[z]]_{λ} ←^{T} Ω_{LX}[[z]]_{λ}[2] ←^{T} Ω_{LX}[[z]]_{λ}[4] ←^{T} …

in the category C(Sh_{Ab}LX) of unbounded complexes. The discussion of [BSSc, 1.3.23] can be subsumed in the assertion that Γ_{LX}(lim S ⊗ L) is exactly the periodic complex (1.7). 1.2.12 Our basic result, Theorem 3.16, is an extension of [BSSc, Thm. 1.1] from smooth manifolds to orbifolds. It asserts that there is an isomorphism

R(f_{L})_{∗}(ℝ_{G_{L}}) ≅ Ω_{LX}[[z]]_{λ}  (1.4)

in the dervied category D^{+}(Sh_{Ab}LX). This isomorphism is not canonical and depends on the choice of a connection on the gerbe G → X. As a consequence of (1.4) we get in Theorem 3.17 the non-canonical isomorphism

H_{dR,deloc}^{∗}(X; (G, λ)) ≅ H_{deloc}^{∗}(X; G) .

1.2.13 The main goal of the forthcoming paper [BSSb] will be a sheaf theoretic construction of 2-periodic twisted delocalized cohomology. The idea is to define an analog T of the operation \frac{d}{dz} on the left-hand side of the derived category isomorphism (1.4). In analogy with the de Rham model we then will consider the system

T : R(f_{L})_{∗}(ℝ_{G_{L}}) ←^{T} R(f_{L})_{∗}(ℝ_{G_{L}})[2] ←^{T} R(f_{L})_{∗}(ℝ_{G_{L}})[4] ←^{T} …

in D(Sh_{Ab}LX). The sheaf-theoretic version of periodic delocalized twisted cohomology will be defined as

H^{∗}(ev ∘ Rp_{∗}(holimT ⊗ L)) .

In order to make this rough idea precise we must solve various problems, in particular

(1) The homotopy limit holimT of the digragram T in the derived category is only well-defined up to non-canonical isomorphism. In order to define a functorial periodic cohomology we must work hard to construct a much more concrete version of the system T .

(2) The push-forward Rp_{∗}(colimT ⊗ L) is not a standard derived functor since it acts between unbounded derived categories. We use a model category approach in order to construct functors like Rp_{∗}.

The main application and technical tool in [BSSb] will be T -duality. The results of Subsections 2.4 and 2.6 of the present paper will be needed in [BSSb] in a crucial way.

1.3 Motivation of the definition of twisted delocalized cohomology

1.3.1 In the present subsection we motivate the definition of twisted delocalized cohomology as the correct target for the Chern character from twisted K-theory. It is a well-known fact that the Chern character ch : K(X) → H(X; ℚ) from the complex K-theory of a space X to the rational cohomology of X induces an isomorphism K(X) ⊗_{ℤ} ℚ →^{∼} H(X; ℚ) (we consider both sides as ℤ/2ℤ-graded groups)

1.3.2 Complex K-theory and rational cohomology both have equivariant generalizations. Every generalized cohomology E theory has the Borel extension. If X is a G-space, then the Borel extension of E to G-spaces associates to X the group E_{G}^{Borel}(X) := E(EG ×_{G} X). Here EG is a universal space for G, i.e. a contractible space on which G acts freely. The Chern character induces an equivariant Chern character ch_{G} : K_{G}^{Borel}(X) → H_{G}^{Borel}(X; ℚ) which gives again a rational isomorphism. 1.3.3 The interesting equivariant extension of K-theory is not the Borel extension but the extension due to Atiyah-Segal based on equivariant vector bundles [AS69]. It will be denoted by K_{G}(X). In order to see the difference between K_{G}^{Borel} and K_{G} consider the simple example of finite group G acting trivially on the point ∗. The equivariant Atiyah-Segal K-theory is isomorphic to the representation ring R(G) of G. In [Ati61] is was shown that K_{G}^{Borel}(∗) is isomorphic to the completion { \atop R(G)}_{I} of the representation ring at the dimension ideal I, which is defined as the kernel of the homomorphism dim : R(G) → ℤ. 1.3.4 It is not true that the Atiyah-Segal equivariant K-theory is rationally isomorphic to the Borel extension of rational cohomology. In the case of discrete groups and proper actions the appropriate target of the Chern character was found in [BC88]. It will be called the delocalized cohomology in this paper. Let G be a discrete group which acts properly on a space X. Then we define a new proper G-space (sometimes called the Brylinski space)

ΛX := ⨆_{g∈G} X^{g} ,

where X^{g} ⊂ X is the subspace of fixed points of g. The action of h ∈ G on ΛX maps x ∈ X^{g} to hx ∈ X^{hgh^{−1}}. The delocalized cohomology of the G-space X is the cohomology of the quotient ΛX/G. 1.3.5 A G-space X gives rise to a topological quotient stack [X/G]. If G is a discrete group which acts properly on X, then the quotient [X/G] is an example of an orbispace (the topological variant of an orbifold). But not every orbispace can be represented in this form. We refer to [BS] for the description of the category of orbispaces. The stack [ΛX/G] has a description in the language of topological stacks. If Z is a topological stack, then we define its loop stack LZ (see 2.16 and 2.2.6)^{10} such that

L[X/G] = [ΛX/G]

for a discrete group acting properly on a space X. 1.3.6 If G is a discrete group which acts properly on a space X, then the quotient X/G is a reasonable topological space. It is the coarse moduli space of the orbispace [X/G]. The definition of the coarse moduli space extends to arbitrary orbispaces. The coarse moduli space of the orbispace Z will be denoted by |Z|. If Z^{1} ⇒ Z^{0} is a presentation of the orbispace by a proper étale groupoid, then |Z| = Z^{0}/Z^{1}. The rational cohomology of an orbispace Z is the cohomology of its coarse moduli space |Z|. Therefore we can define the delocalized cohomology of an orbispace as the cohomology of |LZ|. This generalizes the definition of the delocalized cohomology from global quotient orbispaces to general orbispaces. Note that this is not quite the definition of delocalized cohomology which we are going to use in the main part of the paper but sufficient for the present discussion. Later we prefer a sheaf-theoretic definition of the delocalized cohomology. 1.3.7 Delocalized cohomology for orbifolds appeared in connection with the index theorem for orbifolds [Kaw81]. In a completely different context of quantum cohomology for orbifolds it was constructed in [CR04], [Rua02]. Note that the grading used in [CR04] is different from the grading in the present paper. 1.3.8 A different generalization of K-theory is twisted K-theory (see [AS04]). The search for the target of an appropriate Chern character lead to the definition of 2-periodic twisted de Rham cohomology^{11}. Usually it is defined on smooth manifolds X. Given a closed three-form λ ∈ Ω^{3}(X) twisted de Rham cohomology is the cohomology of the complex

⋯ →^{d_{λ}} Ω^{even}(X) →^{d_{λ}} Ω^{odd}(X) →^{d_{λ}} Ω^{even}(X) →^{d_{λ}} …,  (1.5)

10In the present paper we use the name loop stack. In the literature it is also known under the name inertia stack

11This could also be reversed. The equations for fields associated to D-branes in string theory with B-field backgroup can be expressed in terms of the twisted de Rham differential. In this history twisted K-theory was found as a cohomology theory with a (Chern character) map to twisted de Rham cohomology giving the integrality lattice of D-brane charges [MM97], [Wit98].

where d_{λ} := d_{dR} + λ. 1.3.9 A Chern character for twisted K-theory with values in λ-twisted de Rham cohomology was constructed in [BCM^{+}02], [MS03], and [AS]. The twist of K-theory is classified by a class λ_{ℤ} ∈ H^{3}(X; ℤ). The closed form λ ∈ Ω^{3}(X) should represent the image of λ_{ℤ} in real cohomology. It was shown that this ℤ/2ℤ-graded cohomology theory is again isomorphic to twisted K-theory tensored with ℝ. 1.3.10 Twisted K-theory on orbifolds has first been considered in [AR03]. In this paper the twist was given by a so-called inner local system of twisted torsion. The natural object to be used to twist complex K-teory should a gerbe G → X with band S^{1}. Gerbe twisted K-theory for orbifolds was discussed in [LU04]. For general local quotient stacks it was defined in [FHT]. Using topological groupoids in order to represent stacks a very general definition of twisted K-theory was given in [TXLG04]. 1.3.11 The result of [BC88] in the case of global quotient orbispaces obtained from proper actions of discrete groups shows that the correct target of the Chern character has to take the topology of the fixed point sets into account. Thus the target of the Chern character from twisted K-theory of an orbifold should be a delocalized version of twisted de Rham cohomology. If X is an orbifold, then LX is again an orbifold. In particular we can consider differential forms on LX. Given a three-form λ ∈ Ω^{3}(LX) we can define the twisted delocalized de Rham cohomology as the cohomology of the complex

⋯ →^{d_{λ}} Ω^{even}(LX) →^{d_{λ}} Ω^{odd}(LX) →^{d_{λ}} Ω^{even}(LX) →^{d_{λ}} ….  (1.6)

It turned out that this cohomology is not the correct target of the Chern character. This has already been observed at the end of [AR03]. 1.3.12 Let (L, ∇^{L}) be the flat complex line bundle associated to G̃^{δ} → LX. We let Ω(LX; L) denote the differential forms with values in L, and d^{L} be the differential induced by d_{dR} and the flat connection ∇^{L}. We let λ ∈ Ω^{3}(LX) be a closed three from which represents the image of the Dixmir-Douady class λ_{ℤ} ∈ H^{3}(LX; ℤ) of the gerbe G_{L} → LX in real cohomology. We set d_{λ}^{L} := d^{L} + λ. The correct target of the Chern character on G-twisted K of the orbifold X is the 2-periodic cohomology of the complex

⋯ → Ω^{ev}(LX; L) →^{d_{λ}^{L}} Ω^{odd}(LX; L) →^{d_{λ}^{L}} Ω^{ev}(LX; L) → ….  (1.7)

This Chern character was constructed in [TXa].

2 Inertia

2.1 2-limits in 2-categories

2.1.1 In the present paper we consider stacks on some site or groupoids in some ambient category like topological spaces or manifolds. A common feature of these constructs is that they are objects in a 2-category. Of particular importance for the present paper is the notion of a 2-limit. The goal of this Subsection is to explain this notion. 2.1.2 By a 2-category we always mean a strict 2-category. In our main examples of 2-categories have the property that all 2-morphisms are isomorphisms, but in the present subsection do not assume this. For objects a and b of a 2-category we denote by Hom_{C}(a, b) the Hom-category from a to b (we will often omit the subscript and write Hom(a, b)). We will write the objects as straight arrows a → b, and the morphisms between two arrows f, g : a → b as f ⇝ g. 2.1.3 By a 2-functor we always mean a pseudo-2-functor, as explained for example in [Hov99, Definition 1.4.2]. By a strict 2-functor we mean such a functor where all unit and composition 2-isomorphisms are identities. 2.1.4 Let C be a 2-category. For any X ∈ ObC we denote by C/X the over 2-category

• with objects the 1-arrows A → X, • whose 1-morphisms are triangles filled in with a 2-morphism

[figure: A; B; X]

,

• and where 2-morphisms are the ones of C making the natural diagram commutative.

There is a version of this construction for a 2-functor D → C and an object X of C, denoted D/X. Note that if D is a 1-category then so is D/X. 2.1.5 Let C be a 2-category and D a small category. Let F, G : D → C be two 2-functors. A natural 2-transformation φ from F to G is an assignment of a 1-morphism φ(a) : F (a) → G(a) for any object a of D and a 2-isomorphism φ(f) for any f : a → b in D filling in the square

\xymatrix{F (a) \ar[r]^{φ(a)} \ar[d]^{F (f)} & G(a) \ar[d]^{G(f)} \\ F (b) \ar@{=>}[ur]^{ϕ(f)} \ar[r]^{φ(b)} & G(b)} ,

satisfying the obvious compatibility for compositions of maps in D. Let φ, ψ : F → G be two natural 2-transformations. A modification t from φ to ψ consists of a 2-morphism t(a) : φ(a) ⇝ ψ(a) for any object a of D satisfying an again obvious compatibility with the φ(f) and ψ(f) for any map f in D. With these definitions the 2-functors, the natural 2-transformations and the modifications form a 2category. For F, G as above we denote by Hom_{C^{D}}(F, G) the corresponding category of natural transformations from F to G. 2.1.6 For an object c of C we denote by \underline{D}_{c} the constant diagram on c, i.e. the (strict) 2-functor from D to C sending all objects to c and all morphisms to the identity on c.

Definition 2.1 Let F : D → C be a 2-functor. A 2-limit of F is an object c of C together with a natural 2-transformation φ : \underline{D}_{c} → F such that for any object T of D the functor

Hom_{C}(T, c) → Hom_{C^{D}}(\underline{D}_{T} , F)

given by composition with φ is an equivalence of categories.

The constant diagram functor c ↦ \underline{D}_{c} is a 2-functor C → C^{D}. Note that F ∈ C^{D}. Using \underline{D} we form the over 2-category C/F. By definition a 2-limit (c, ϕ) of F is an object of C/F. For example a 2-final object of C is an object c such that for all objects T of C the projection from Hom(T, c) to the point category is an equivalence.

Lemma 2.2 Let u : C → D be a 2-functor between 2-categories, X an object of D. Let c, f : u(c) → X be an object of C/X. Then if the functor

Hom_{C}(T, c) → Hom_{D}(u(T), X)

is an equivalence for all objects T of C the object (c, f) is 2-final in C/X. If the 2-morphisms in D and C are all 2-isomorphisms the converse holds.

Proof. Let (c′, f′) ∈ C/X be another object. Then there is a canonical 2-cartesian square

\xymatrix{Hom_{C/X}((c′, f′), (c, f)) \ar[r] \ar[d] & Hom_{C}(c′, c) \ar[d] \\ pt \ar[r]^{f′} & Hom_{D}(u(c′), X)}

in Cat. Hence the first statement follows. The second statement follows from the fact that a map φ : A → B between groupoids is an equivalence if and only if all (2-categorical) fibers over objects of B are contractible. □

2.1.7 An equivalence between two objects c and d of C are 1-arrows f : c → d and g : d → c together with 2-isomorphisms φ : id_{c} ⇝ g ∘ f and ψ : id_{d} ⇝ f ∘ g satisfying the triangular identities as for units and counits of adjunctions. 2.1.8 As particular case consider two 2-final objects c, c′ in a 2-category D. Then there is an equivalence between c and c′ which is unique up to unique 2-isomorphism. 2.1.9

Lemma 2.3 If an object (c, φ) ∈ C/F is a 2-limit of F then it is 2-final in C/F. If all 2-morphisms in C are 2-isomorphisms or if C has all small 2-limits then the converse is true. Any two choices of 2-limits are equivalent in C/F, unique up to unique 2-isomorphism, in particular the underlying objects in C are (canonically) equivalent.

Proof. The first statement follows from Lemma 2.2. The second statement under the assumption on the 2-morphisms also follows from that Lemma, under the completeness assumption it follows from the first statement and the uniqueness (up to unique isomorphism) of 2-final objects 2.1.8. The third statement is also 2.1.8. □

2.1.10

Lemma 2.4 In Cat, the 2-category of small categories, small 2-limits exist.

Proof. The usual construction gives a preferred model: For a 2-functor F : D → Cat define c to be the category whose objects are collections of objects x_{a} ∈ F (a) for any object a of D together with isomorphisms φ_{f} : (F f)(x_{a}) → x_{b} for any map f : a → b in D satisfying a compatibility condition for compositions of maps in D, and whose morphisms from (x_{a}) to (y_{a}) are compatible systems of morphisms x_{a} → y_{a}. The transformation \underline{D}_{c} → F induced by projections exhibits c as a 2-limit of F. □

2.1.11 Let us consider for example the category

D := \xymatrix{ & b \ar[d] \\ a \ar[r] & c}.  (2.5)

A functor F : D → C is a diagram

\xymatrix{ & B \ar[d]^{v} \\ A \ar[r]^{u} & C} .  (2.6)

Usually a 2-categorical fiber product of F is a diagram

\xymatrix{A ×_{C} B \ar[d] \ar[r] & B \ar[d]^{v} \\ A \ar@{=>}[ur]^{ψ} \ar[r]^{u} & C}  (2.7)

fulfilling some natural properties. Such a diagram gives in two natural ways an object in C/F (by requiring the map A ×_{C} B → C be one of the two possible compositions), and it is easily checked that the usual properties are equivalent to this object being a 2-limit. If these properties are fulfilled we call a diagram as above 2-cartesian. 2.1.12 Assume that C = Cat. A model of A ×_{C} B is then the category whose objects are triples (a, b, γ), where a ∈ Ob(A), b ∈ Ob(b) and γ : u(a) → v(b). A morphism (a, b, γ) → (a′, b′, γ′) is a pair (f : a → a′, g : b → b′) such that γ′ ∘ u(f) = v(f) ∘ γ. The 2-morphism in (2.7) is given by ψ(a, b, γ) := γ. We see in particular that 2-categorical fiber products in Cat are 2-limits. We call any diagram as 2.7 a standard model of the fiber product in a 2-category C if for any object T the functor Hom(T, ) produces a diagram which is isomorphic (with respect to an obvious map) to the model in Cat from above. Note that this is not the preferred model. 2.1.13 Like ordinary limits 2-categorial limits are characterized by a universal property for Homcategories.

Lemma 2.8 Let F : D → C be a 2-functor, (c, φ) ∈ C/F a 2-limit of F and T and object of C. Consider the 2-functor H : D → Cat given by a ↦ Hom_{C}(T, F (a)). Then the natural map \underline{D}_{Hom_{C}(T,c)} → H is a 2-limit of the functor H.

Proof. In fact Hom_{C^{D}}(\underline{D}_{c}, F) is naturally isomorphic to the preferred model of the 2-limit of the diagram a ↦ Hom_{C}(T, F (a)). □ Lemma 2.8 implies an equivalence of categories

Hom_{C}(T, 2 − lim_{a∈D} F (a)) ≅ 2 − lim_{a∈D} Hom_{C}(T, F (a)) ,

where the left 2-limit is taken in C, and the right 2-limit is taken in Cat. 2.1.14 Let C be another small category and suppose given a 2-functor F : C × D → C. For simplicity suppose that C has all small 2-limits.

Proposition 2.9 Let the notation be as above. The assignment

a ↦ 2 − lim_{b∈D} F (a, b)

can be made into a 2-functor K : C → C, and two such choices are canonically equivalent. Moreover the 2-limit of K is canonically equivalent to the 2-limit of F.

Proof. The first assertion is a consequence of Lemma 2.3. We sketch the proof of the second statement. By Lemma 2.8 we are reduced to prove the statement in Cat. But taking everywhere preferred models produces isomorphic models of the two 2-limits in question. □

2.1.15 We will assume that C has a final object and admits standard models (see 2.1.12) of all 2categorical fiber products. The absolute product × is understood as a standard model of the fiber product over the final object. Consider a pair of maps

\xymatrix{X \ar[dr]^{f} & & X \ar[dl]_{g} \\ & Y} .

Definition 2.10 The equalizer E(f, g) of the pair of maps f, g : X → Y is defined as a standard model of the 2-categorical fiber product

\xymatrix{E(f, g) \ar[d] \ar[r] & Y \ar[d]^{diag} \\ X \ar@{=>}[ur] \ar[r]^{(f,g)} & Y × Y} .

Note that on Hom-categories this definition yields in fact the preferred model of the equalizer diagram.

Definition 2.11 We define the inertia object of X as the equalizer IX := E(id_{X}, id_{X}).

2.1.16 We say that a 2-category is 2-complete if it admits a small 2-limits. There is an analogous notion of a 2-colimit, and the category is called 2-cocomplete if all small 2-colimits exist. The category is called 2-bicomplete if it is 2-complete and 2-cocomplete. The 2-category of small groupoids gpd is 2-bicomplete as well as bicomplete as a category. The same holds for the 2-category PStI of prestacks on a small category I, which is by definition the 2-category of 2-functors gpd^{I^{op}}. The 2-category of stacks StS on a small site S is 2-bicomplete. 2.1.17 We consider the 2-category gpd(U) of groupoids in a category U which has finite limits. Our basic example for U is the category Top of topological spaces.

Lemma 2.12 The category gpd(U) admits standard models of all 2-categorical fiber products.

Proof. The objects and morphisms of the standard model of a fiber product in gpd(U) can be expressed in terms of fiber products in U. □

Lemma 2.13 In gpd(U) equalizers exist for any pair of maps.

Proof. We observe that gpd(U) has a final object and admits 2-categorical fiber products (Lemma 2.12). In fact, the limit of the empty diagram in U is the final object ∗ of U. The groupoid ∗ ⇒ ∗ is the final object in gpd(U). □

2.1.18 Let C be as in 2.1.15. We consider a diagram (2.6).

Lemma 2.14 We have a natural isomorphism I(A ×_{C} B) ≅ IA ×_{IC} IB, where we use standard models for the fiber products.

Proof. We only have to check this for C = Cat since everything can be stated in terms of Hom-categories. We let D̃ be the category freely generated by two objects 0, 1, and two isomorphisms from 0 to 1, see 2.2.3. Then we have an isomorphism IA ≅ Hom(D̃, A), see also Lemma 2.15 in the case of groupoids. Since standard fiber products commute with the cotensor structure the claim follows.

2.2 Loops

In a 2-category of groupoids gpd(U) or stacks St(S) the preferred model (see 2.1.12) of the inertia IX (see Definition 2.11) of X is quite complicated. The goal of the present Subsection is the construction of a simpler model of IX which we call the loop object LX. 2.2.1 We start with the case of gpd(U). Let us assume that U is tensored and cotensored over Sets. The cotensor functor will be denoted by

\underline{Hom} : Sets^{op} × U → U .

Using the existence of finite limits in U we extend this functor to a bifunctor

                      \underline{Hom}_{Cat} : (Sets^{fin} − Cat) × (U − Cat) → (U − Cat),

where for a category A with finite limits we write (A − Cat) for the 2-category of category objects in A, and Sets^{fin} is the category of finite sets. 2.2.2 Let X ∈ gpd(U) ⊂ (U − Cat) be a groupoid in U. We consider the category

D := \xymatrix{•_{0} \ar[r]^{α} \ar[r]_{β} & •_{1}} ∈ (Sets^{fin} − Cat) .

Since X is a groupoid, \underline{Hom}_{Cat}(D, X) ∈ (U − Cat) is again a groupoid in U.

Lemma 2.15 We have a natural isomorphism

IX ≅ \underline{Hom}_{Cat}(D, X) .

Proof. We insert the standard model of the 2-categorical fibre product of gpd(U) into the definition of the equalizer in the special case that f = g = id_{X}. Then the assertion becomes obvious. □

2.2.3 Later we will have the freedom to replace groupoids by equivalent groupoids. We let D̃ be the category obtained from D by adjoining inverses. Since X is a groupoid we have

                               \underline{Hom}_{Cat}(D, X) ≅ \underline{Hom}_{Cat}(D̃, X) .

We now consider the category L with one object ∗ and infinite cyclic automorphism group generated by σ

[figure: ∗; σ] .

Then we have a natural functor i : L → D̃ which maps ∗ to •_{0} and σ to β^{−1} ∘ α. This is an equivalence of categories. It induces an equivalence of groupoids

\underline{Hom}_{Cat}(D, X) ≅ \underline{Hom}_{Cat}(D̃, X) →^{i^{∗}} \underline{Hom}_{Cat}(L, X) .

Definition 2.16 The groupoid LX := \underline{Hom}_{Cat}(L, X) will be called the loop groupoid of X.

Note that we have an equivalence of groupoids

IX → LX .  (2.17)

If f : X → Y is a morphism in gpd(U), then composition with f functorially induces a morphism Lf : LX → LY. 2.2.4 It is easy to describe the objects and morphisms of the loop groupoid LX explicitly.

Lemma 2.18 The objects LX^{0} and morphisms LX^{1} of LX are given by the following fibre products in U.

\xymatrix{LX^{0} \ar[r] \ar[d]^{δ} & X^{1} \ar[d]^{(s,r)} \\ X^{0} \ar[r]^{diag} & X^{0} × X^{0}}  (2.19)

\xymatrix{LX^{1} \ar[d]^{s} \ar[r] & X^{1} \ar[d]^{s} \\ LX^{0} \ar[r]^{δ} & X^{0}}  (2.20)

The range map is given (in the language of elements) by the map

r((x, γ), μ) := (r(μ), μ ∘ γ ∘ μ^{−1}) .

We will give another description of LX^{1} which turns out to be useful later. We define P by the cartesian diagram

\xymatrix{P \ar[d]^{(p,q)} \ar[r]^{k} & X^{1} \ar[d]^{s,r} \\ LX^{0} × LX^{0} \ar[r]^{δ,δ} & X^{0} × X^{0}} ,  (2.21)

The composition of X induces a map m : P → X^{1} defined in the language of objects by

((x_{0}, γ_{0}), (x_{1}, γ_{1}), μ) ↦ γ_{1}^{−1} ∘ μ ∘ γ_{0} ∘ μ^{−1} .

Lemma 2.22 We have a cartesian diagram

\xymatrix{LX^{1} \ar[d]^{j} \ar[r]^{i} & X^{0} \ar[d]^{1} \\ P \ar[r]^{m} & X^{1}} ,

where j := (s, r) and i := δ ∘ s.

Proof. Consider an object T ∈ U. A map f : T → LX^{1} is uniquely determined by a pair (u, v), u : T → LX^{0} and v : T → X^{1} such that δ ∘ u = s ∘ v : T → X^{0}. The map u is given by pair (a, b) of maps with a : T → X^{0} and b : T → X^{1} such that s ∘ b = r ∘ b = a. We see that u is completely determined by b. Note that δ ∘ u = s ∘ b = s ∘ v. We have j ∘ f = ((s ∘ b, b), (r ∘ v, v ∘ b ∘ v^{−1}), v) and observe that m ∘ j ∘ f = 1 ∘ i ∘ f. This construction is natural in T → LX^{0} and therefore determines a map LX^{1} → P ×_{X^{1}} X^{0}. Consider now a map g : T → LX^{1} → P ×_{X^{1}} X^{0} given by a pair (x, y) of maps x : T → P and y : T → X^{0} such that m ∘ x = 1 ∘ y. The pair (p ∘ x, k ∘ x) satisfies δ ∘ p ∘ x = s ∘ k ∘ x and therefore defines a map f : T → LX^{1}. Again, the construction is functorial in g and defines a map P ×_{X^{1}} X^{0} → LX^{1}. We leave it to the reader to check that these maps are inverses to each other. □

2.2.5 Let X ∈ gpd(U) and LX be its loop groupoid. Evaluation at the unique object ∗ of L induces a functor LX → X. Therefore LX can naturally be considered as an object of gpd(U)/X (see 2.1.4). Note that a morphism in this category is a diagram

[figure: Y; Z; X] ,

and a 2-morphism between two such maps is a 2-morphism f ⇝ g between the given 1-morphisms f, g : Y → Z commuting with the 2-morphisms. We will now consider group objects in gpd(U)/X. They together with their products (i.e. fiber products over X) will lie in a subcategory which is equivalent as a 2-category to a 1-category, so it will not be a problem to formulate what we mean by a group object in this case.

Lemma 2.23 The loop groupoid LX has a natural structure of a group object in gpd(U)/X.

Proof. We consider the category E ∈ (Sets − Cat) pictured by

[figure: •_{0}; a; b; •_{1}; c] ,

where a, c generate inifnite semigroups. By Ẽ we denote the category obtained from E by adjoining inverses. Then we observe that in the 2-category gpd(U)

LX ×_{X} LX ≅ \underline{Hom}_{Cat}(E, X) ≅ \underline{Hom}_{Cat}(Ẽ, X) .

We define a functor j : L → Ẽ which maps ∗ to •_{0} and σ to b^{−1} ∘ c ∘ b ∘ a. The pull-back

LX ×_{X} LX ≅ \underline{Hom}_{Cat}(Ẽ, X) →^{j^{∗}} \underline{Hom}_{Cat}(L, X) ≅ LX

induces the composition law. We leave it to the reader to write out the inverse, the unit and the remaining necessary verifications. □

2.2.6 Let S be a Grothendieck site. Then we can consider the category of presheaves of sets PShS. It is closed under taking arbitrary small limits. The 2-category of strict prestacks PSt^{strict}S on S is by definition the category gpd(PShS). By Lemma 2.13 in PSt^{strict}S equalizers exist for all pairs of maps. The catgeory PShS is tensored and cotensored over Sets. Hence we can apply the construction of the loop groupoid in PSt^{strict}S. We now consider the full 2-subcategory of strict stacks St^{strict}S ⊂ PSt^{strict}S of stacks on S. Recall that a stack is a prestack which satisfies descend conditions for objects and morphisms. This subcategory is closed with respect to 2-limits and preserved by the cotensor structure. For all pairs of maps in the category St^{strict}S the equalizer exists by Lemma 2.13. Moreover, the loop object of a stack is again a stack. While a strict prestack is a strict 2-functor S^{op} → gpd(Sets), a prestack is a (in general non-strict) 2functor S^{op} → gpd(Sets), i.e it preserves compositions of morphisms in S up 2-morphisms which satisfy coherence conditions for triple compositions as indicated in 2.1.3. The category of stacks is again a full subcategory of the category of prestacks on S which satisfy certain descend conditions. Note that PStS is cotensored over (Sets − Cat), i.e. we have a bifunctor

                           \underline{Hom}_{Cat} : (Sets − Cat) × PStS → PStS .

This structure is induced by the corresponding cotensor structure of (Sets − Cat), i.e. for a category D ∈ Sets − Cat and a prestack X the value of \underline{Hom}_{Cat}(D, X) on U ∈ S is given by

\underline{Hom}_{Cat}(D, X)(U) := \underline{Hom}_{Cat}(D, X(U)) ,

where X(U) ∈ Sets − Cat. If X is a stack, then \underline{Hom}_{Cat}(D, X) is also a stack. The 2-categorical fibre product of (pre)stacks is given objectwise in S by the 2-categorical fibre-product in gpd(Sets). Therefore, Lemma 2.15 remains true in the categories PStS and StS. We can furthermore define the loop (pre)stack LX of a (pre)stack as in Definition 2.16 and (2.17) still induces an equivalence of (pre)stacks

IX → LX .

Finally, Lemma 2.23 holds in the sense, that for a (pre)stack X the loops LX form a group object in the category of (pre)stacks over X. 2.2.7 Like Lemma 2.14 in the case of inertia stacks we have

Lemma 2.24 The inertia functor preserves standard 2-cartesian diagrams.

2.3 Loops of topological stacks

2.3.1 We consider the small site Top of topological spaces and open coverings. Let StTop be the 2category of stacks in topological spaces. By the observations 2.2.6 we can form the loop stack LX of a stack X ∈ StTop. In the present subsection we show that taking loops preserves topological stacks. Furthermore we show that taking loops commutes with the classifying stack functor from topological groupoids to stacks in topological spaces. We use the latter result in order to verify that LX for an orbispace is what is called the orbispace of twisted sectors in the literature. 2.3.2 We refer to [Hei05], [Noo] and also to [BS] for details about stacks (in topological spaces). Topological spaces are considered as stacks via the Yoneda embedding. A map a : A → X from a topological space to a stack X is called an atlas if it is representable, surjective and admits local sections. A topological stack is a stack which admits an atlas. We shall show that taking loops preserves topological stacks.

Lemma 2.25 If X ∈ StTop is a topological stack, then LX is a topological stack.

Let a : A → X be an atlas of X. Then we define a space W by the pull-back diagram

\xymatrix{W \ar[r] \ar[d]^{w} & A ×_{X} A \ar[d]^{(pr_{1},pr_{2})} \\ A \ar[r]^{diag} & A × A} .

We will construct a canonical map c : W → LX and show that it is an atlas of W. The map c : W → LX is defined as follows. Let T be a topological space and (f : T → W) ∈ W (T). By the definition of W this map is given by a pair (g, h) of maps g : T → A and h : T → A ×_{X} A such that diag ∘ g = (pr_{1} ∘ h, pr_{2} ∘ h). The map h : T → A ×_{X} A is given by a pair h_{1}, h_{2} : T → A and a 2-isomorphism σ : a ∘ h_{1} ⇝ a ∘ h_{2}. Combining these two facts we see that f is given by a pair (g, σ) of a map g : T → A and a 2-automorphism σ : a ∘ g ⇝ a ∘ g. Recall that an object of LX(T) is a pair (u, ϕ) of an object u ∈ X(T) and an automorphism ϕ ∈ Aut(u). We define c(f) ∈ LX(T) to be the object (a ∘ g, σ) ∈ LX(T). We now construct a 2-commutative diagram

\xymatrix{W \ar[d]^{w} \ar[r]^{c} & LX \ar[d]^{i} \\ A \ar@{=>}[ur]^{ϕ} \ar[r]^{a} & X}  (2.26)

by defining ϕ is follows. As above let (f : T → W) ∈ W (T) be given by a pair (g, σ). In X(T) we have the equalities i ∘ c(f) = i(a ∘ g, σ) = a ∘ g and a ∘ w(f) = a ∘ g. Therefore we can define ϕ(f) := σ. We claim that the diagram (2.26) is 2-cartesian. In order to see this let as above T be a space and consider a triple (u, v, θ) consisting of maps u : T → A, v : T → LX and a 2-isomorphism θ : a∘u ⇝ i∘v. To this data we must associate a unique pair of maps (f, ψ) of a map f : T → W and a 2-isomorphism ψ : c ∘ f ⇝ v such that

[figure: T; u; f; v; W; ψ; w; c; LX; i; A; θ; ϕ; a; X]

commutes. The map v is given by a pair (i ∘ v, κ) of an object i ∘ v ∈ X(T) and an automorphism κ ∈ Aut(i ∘ v). Using the description of maps T → W obtained above we define f : T → W as the map which corresponds to the pair (u, θ^{−1} ∘ κ ∘ θ) of an object u : T → A and the automorphism

θ^{−1}∘κ∘θ : a∘u ⇝ a∘u. We furthermore define 2-isomorphism ψ : c∘f = (a∘u, θ^{−1}∘κ∘θ) ⇝^{κ^{−1}∘θ} (i∘v, κ) = v.

Observe that ψ is uniquely determined by the condition that i(ψ) ∘ ϕ(f) = θ : a ∘ w ∘ f = a ∘ u → i ∘ v. This equality indeed holds for our construction since ϕ(f) = θ^{−1} ∘ κ ∘ θ and i(ψ) = κ^{−1} ∘ θ. This finishes the proof of the claim. Since A → X is an atlas the map a is representable, surjective and admits local sections. These properties are preserved under pull-back. It follows that c : W → LX is representable, surjective and admits local sections, too. Therefore it is an atlas of LX. □

2.3.3 A topological groupoid G is a groupoid object in Top. It represents the stack of G-principal bundles BG. If A → X is an atlas of a topological stack, then we form the topological groupoid A : A ×_{X} A ⇒ A. The stack of A-principal bundles is equivalent to X. We can define an equivalence X → BA which maps (T → X) ∈ X(T) to (T ×_{X} A → T) ∈ BA (we omit to write the action of A on that space over T). 2.3.4 Observe that finite limits in Top exist, and that Top is tensored and cotensored over Sets. Therefore by 2.13 for any pair of maps in gpd(Top) an equalizer exists. Furthermore, we can form the loop groupoid LA of a topological groupoid A.

2.3.5 Let A → X be the atlas of a topological stack, and let A ∈ gpd(Top) denote the associated topological groupoid.

Lemma 2.27 We have a natural equivalence of stacks LX ≅ BLA.

Proof. Let W → LX be as in the proof of Lemma 2.25. Then can form W : W ×_{LX} W ⇒ W. If we show that W ≅ LA, then the assertion follows. ¿From (2.19) we get W ≅ (LA)^{0}. Next we calculate using (2.20)

W ×_{LX} ×W ≅ (A ×_{X} LX) ×_{LX} (A ×_{X} LX)
≅ LX ×_{X} (A ×_{X} A)
≅ (LX ×_{X} A) ×_{A} (A ×_{X} A)
≅ (LA^{0}) ×_{A} A^{1}
≅ (LA)^{1} .

These isomorphisms are compatible with the groupoid structures. □

2.3.6 The following result was also shown in [Noo, Cor. 7.6].

Lemma 2.28 If X is a topological stack, then LX → X is representable.

Proof. We must show that for all spaces T and maps T → X the fibre product T ×_{X} LX is equivalent to a space. It suffices to verify this in the case that T is an atlas. We choose an atlas A → X. The assertion then follows from the following two facts:

(1) The diagram (2.26) is cartesian.

(2) W is a space.

□

2.3.7 Let us recall some notions related to orbispaces. Orbispaces as particular kind of topological stacks have previously been introduced in [BS, Sec. 2.1] and [Noo, Sec. 19.3]). In the present paper we use the set-up of [BS] but add the additional condition that an orbifold atlas should be separated. This condition is needed in order to show that the loop stack of an orbifold is again an orbifold.

(1) A topological groupoid A : A^{1} ⇒ A^{0} is called separated if the identity 1_{A} : A^{0} → A^{1} of the groupoid is a closed map.

(2) A topological groupoid A^{1} ⇒ A^{0} is called proper if (s, r) : A^{1} → A^{0} × A^{0} is a proper map.

(3) A topological groupoid is called étale if the source and range maps s, r : A^{1} → A^{0} are étale.

(4) A proper étale topological groupiod A^{1} ⇒ A^{0} is called very proper if there exists a continuous function χ : A^{0} → [0, 1] such that

(a) r : supp(s^{∗}χ) → A^{0} is proper

(b) ∑_{y∈A^{x}} χ(s(y)) = 1 for all x ∈ A^{0}.

(5) A topological stack is called (very) proper (étale, separated, resp.), if it admits an atlas A → X such that the topological groupoid A ×_{X} A ⇒ A is (very) proper (étale, separated, resp).

(6) An orbispace X is a very proper étale separated topological stack.

(7) An orbispace atlas of a topological stack X is an atlas A → X such that A ×_{X} A ⇒ A is a very proper étale and separated groupoid.

(8) If X, Y are orbispaces, then a morphism of orbispaces X → Y is a representable morphism of stacks.

2.3.8 The following Lemmas illustrates the meaning of the separatedness and very properness condition.

Lemma 2.29 Let A : A^{1} ⇒ A^{0} be a proper étale groupoid. If A^{1}, A^{0} are locally compact, then A is very proper.

Proof. The existence of the cut-off function was shown in [Tu99, Prop. 6.11].

Lemma 2.30 Let A : A^{1} ⇒ A^{0} be a topological groupoid. If A^{0} and A^{1} are Hausdorff spaces, then A is separated.

Proof. We define the Hausdorff space Q as the pull-back

\xymatrix{Q \ar[r]^{j} \ar[d] & A^{1} \ar[d]^{(r,s)} \\ A^{0} \ar[r]^{diag} & A^{0} × A^{0}} .

The property of a map between topological spaces being a closed is preserved under pull-back. Since A^{0} is Hausdorff the diagonal diag : A^{0} → A^{0} × A^{0} is a closed map. It follows that j : Q → A^{1} is a closed map. The composition ∘ in A gives the squaring map

sq : Q →^{diag} Q ×_{A^{0}} Q →^{∘} Q .

Then we have a pull-back

\xymatrix{I \ar[d] \ar[r]^{k} & Q \ar[d]^{(id_{Q},sq)} \\ Q \ar[r]^{diag} & Q × Q} .

Since Q is Hausdorff, it follows that diag and hence k are a closed maps. The composition j ∘ k : I → A^{1} of closed maps is again a closed. In a group the identity is the unique solution of the equation x^{2} = x. It follows that j ∘ k(I) = 1_{A}(A^{0}). Therefore 1_{A}(A^{0}) ⊆ A^{1} is closed. This implies that 1_{A} : A^{0} → A^{1} is a closed map. If fact, if K ⊆ A^{0} is a closed subset, then we define the Hausdorff space A_{K}^{1} ⊆ A^{1} as the pull-back

\xymatrix{A_{K}^{1} \ar[d] \ar[r]^{v} & A^{1} \ar[d]^{(r,s)} \\ K × K \ar[r]^{u} & A^{0} × A^{0}} .

Since u (the obvious embedding) is a closed map, so is v. We apply the discussion above to the restricted groupoid A_{K}^{1} ⇒ K with identity 1_{A_{K}} : K → A_{K}^{1} in order to show that 1_{A_{K}} (K) ⊆ A_{K}^{1} is closed. Hence 1_{A}(K) = v(1_{A_{K}} (K))) ⊆ A^{1} is closed. □

2.3.9

Lemma 2.31 If X is an orbispace, then LX is an orbispace and LX → X is a morphism of orbispaces.

Proof. We choose an orbispace atlas A → X. The associated groupoid A : A ×_{X} A → A is étale, proper. and separated. In order to show that LX is an orbispace it suffices to show by Lemma 2.27 that LA is étale, proper and separated. The property of a map between topological spaces being étale is preserved under pull-back. By (2.20) the fact that s : A^{1} → A^{0} is étale therefore implies that s : (LA)^{1} → (LA)^{0} is étale. Using the inversion homeomorphism I : (LA)^{1} → (LA)^{1} we can express the range map in terms of the source map: r = s ∘ I. This implies that r : (LA)^{1} → (LA)^{0} is étale, too. We thus have shown that LA is étale. We consider the pull-back

\xymatrix{P \ar[r] \ar[d]^{j} & A^{1} \ar[d]^{(r,s)} \\ (LA)^{0} × (LA)^{0} \ar[r] & A^{0} × A^{0}}

(compare (2.21) ). The property of a map between topological spaces being proper is also preserved by pull-backs. Therefore j : P → (LA)^{0} × (LA)^{0} is a proper map. The image of 1_{A} : A^{0} → A^{1} is closed. By Lemma 2.22 we can write (LA)^{1} as a closed subspace (LA)^{1} := m^{−1}(1_{A}(A^{0})) ⊂ P. In general, the restriction of a proper map to a closed subspace is still proper. Since the restriction of j to the closed subspace (LA)^{1} ⊂ P is exactly (r, s) : (LA)^{1} → (LA)^{0} × (LA)^{0} we see that the groupoid A is proper.^{12} We now show that LA is very proper. Since A is very proper there exists a continuous function χ : A^{0} → [0, 1] such that r : supp(s^{∗}χ) → A^{0} is proper and ∑_{y∈A^{x}} χ(s(y)) = 1 for all x ∈ A^{0}. Let i : LA → A be the canonical map. Then i^{∗}χ : LA^{0} → [0, 1] has corresponding properties for the groupoid LA. Finally we show that 1_{LA} : (LA)^{0} → (LA)^{1} is a closed map. By definition we have the cartesian square

\xymatrix{(LA)^{1} \ar[r] \ar[d] & A^{1} \ar[d] \\ (LA)^{0} \ar[r] & A^{0}} .

Therefore we have an embedding as a subspace (LA^{1}) ⊂ (LA)^{0} × A^{1}. Let K ⊆ (LA)^{0} be a closed subset. Then we can write 1_{LA}(K) = (LA^{1}) ∩ (K × i(A^{0})). Since A is separated the subspace (K × 1_{A}(A^{0})) ⊆ (LA)^{0} × A^{1} is closed. Therefore 1_{LA}(K) ⊂ (LA)^{1} is closed, too. In order to be a map of orbispaces LX → X must be representable. This is Lemma 2.28. □

2.3.10 We can replace the site of topological spaces Top by the site of smooth manifolds Mf^{∞}. We will call the corresponding stacks stacks in smooth manifolds. A map A → X from a manifold to a stack in smooth manifolds is called an atlas if it is representable, surjective and smooth (i.e. submersion). A stack in smooth manifolds which admits an atlas is called a smooth (or differentiable) stack. An orbifold is a proper étale smooth stack in smooth manifolds. Since manifolds are Hausdorff a smooth stack is The obvious problem to extend the proof of Lemma 2.25 from topological spaces to smooth manifolds is that in smooth manifolds fibre products only exist under appropriate transversality conditions. In fact, the map (pr_{1}, pr_{2}) : A ×_{X} A → A × A is in general not transverse to the diagonal diag : A → A × A. But it is still true that the loop stack of an orbifold is an orbifold. Proofs of this fact can be found e.g. in [Kaw78], [AR03], [CR04]. Note that for smooth stacks LX → X is in general neither smooth nor representable.

2.4 Loops and principal bundles

2.4.1 Let G be a topological group. The classifying stack BG of G-principal bundles is given as a quotient stack BG := [∗/G] of the action of G on the one point space ∗ [Hei05, Example 2.5]. The map ∗ → BG is an atlas and we have a canonical cartesian diagram

\xymatrix{G \ar[d] \ar[r] & ∗ \ar[d] \\ ∗ \ar@{=>}[ur] \ar[r] & BG} .

12It is because of this argument that in addition to the conditions used in [BS] we require an orbispace atlas to be separated.

Hence this atlas gives rise to the groupoid G : G ⇒ ∗. We see that LG is the groupoid G × G ⇒ G of the action of G on itself by conjugations. Therefore by Lemma 2.27 we have LBG ≅ [G/G]. 2.4.2 A G-principal bundle over a space Y is by definition an object of p ∈ BG(Y), or equivalently, by Yoneda’s Lemma, a map p : Y → BG. The underlying map of spaces P → Y fits into the cartesian diagram

\xymatrix{P \ar[r] \ar[d] & ∗ \ar[d] \\ Y \ar@{=>}[ur] \ar[r]^{p} & BG} .

We adopt the same definition for a G-principal bundle over a stack Y. In this case the underlying map P → Y is a representable map. 2.4.3 Let a : A → Y be an atlas such that the pull-back of the principal bundle p : Y → BG admits a trivialization. A trivialization is a lift t in the diagram

\xymatrix{ & & ∗ \ar[d] \\ A \ar@{.>}[urr]^{t} \ar[r]^{a} & Y \ar@{=>}[ur] \ar[r]^{p} & BG} .

The cocycle associated to the atlas a and the trivialization t is the induced map

Φ_{a,t} : A ×_{Y} A → ∗ ×_{BG} ∗ ≅ G .

Let A : A ×_{Y} A ⇒ A be the groupoid determined by the atlas and A^{•} denote the associated simplicial space. Let

C^{•}(A; G) := C(A^{•}, G) , δ : C^{•}(A; G) → C^{•+1}(A; G)

be the associated cochain complex (the part in degree > 2 is only defined if G is abelian). Then Φ_{a,t} ∈ C^{1}(A, G) is closed, i.e. it satisfies δΦ_{a,t} = 0. We refer to [Hei05, Sec.2] for a description of G-principal principal bundles in terms of cocycles. 2.4.4 Let p : Y → BG be a G-principal bundle over a stack Y. We apply the loop functor and get the map Lp : LY → LBG ≅ [G/G]. It is a homomorphism over the map Y → BG. If G is abelian, then it induces a homomorphism

h : LY → G .  (2.32)

2.4.5 In the following we give a heuristic description of this homomorphism. Let f : P → Y be the underlying map of stacks of the principal bundle. Furthermore let i : LY → Y denote the canonical map. For a point y ∈ Y we get an action of the group i^{−1}(y) on the fibre f^{−1}(y). If γ ∈ i^{−1}(y) and x ∈ f^{−1}(y), then γx = xh(γ). On the left-hand side, (γ, x) ↦ γx denotes the action of i^{−1}(y) on f^{−1}(y). On the right-hand side (x, g) → xg is the G-action on P given by the principal bundle structure. We see again, that the restriction h_{|i^{−1}(y)} : i^{−1}(y) → G is a homomorphism for all y ∈ Y. 2.4.6 Assume that we have chosen an atlas a : A → Y and a trivialization t as in 2.4.3. Let A : A ×_{Y} A ⇒ A be the associated groupoid. Then we get an induced map h_{a} : LA → G. It is equal to the restriction of the cocycle Φ_{a,t} to (LA)^{0} ⊆ A^{1}, i.e. we have the equality

h_{a} = (Φ_{a,t})_{|(LA)^{0}} .  (2.33)

The cocycle h_{a} is closed, i.e. δh_{a} = 0, and it represents the function h ∈ C(LY; G) under the identification H^{0}(LA; G) = C(LY, G). Another interpretation of (2.33) is as the equality h_{a} = tr[Φ_{a,t}], where [Φ_{a,t}] ∈ H^{1}(A; G) is the cohomology class represented by Φ_{a,t}, and tr : C^{•+1}(A; G) → C^{•}(LA; G) is the transgression chain map defined in [ARZ], [LU], [TXb].

2.4.7 Let G be an abelian topological group. In the following Lemma we will assume that for all n ∈ ℕ the subspace of n-torsion points

Tors_{n}(G) := {g ∈ G|g^{n} = 1} ⊆ G

is discrete. This is a non-trivial assumption which, for example, is not true for the topological group ∏_{ℕ} ℤ/nℤ. Let G^{δ} denote the group G with the discrete topology. Let p : Y → BG be a G-principal bundle.

Lemma 2.34 If Y is an orbispace and the subsets Tors_{n}(G) ⊆ G are discrete for all n ∈ ℕ, then the map h : LY → G (defined in (2.32)) factors over G^{δ}.

Proof. We must show that for all spaces T and maps w : T → LY the composition h ∘ w : T → G is locally constant. We choose an orbifold atlas A → Y which gives rise to a very proper separated étale groupoid A : A ×_{Y} A ⇒ A. We consider a point t ∈ T. There exists a neighbourhood t ∈ U ⊆ T which admits a lift

\xymatrix{U \ar@{.>}[r]^{w̃} \ar[d] & A^{0} \ar[d] \\ T \ar@{=>}[ur]^{σ} \ar[r]^{w∘i} & Y} .

By Lemma 2.27 we have the 2-cartesian square in the following diagram:

\xymatrix{U \ar@{.>}[dr]^{v} \ar[drr]^{w} \ar[ddr]_{w̃} \\ & (LA)^{0} \ar[d] \ar[r] & LY \ar[d]^{i} \\ & A^{0} \ar@{=>}[ur]^{σ} \ar[r] & Y} .

We get an induced map v : U → LA^{0} ⊆ A^{1} such that w̃ = s ∘ v. Let a := w̃(t) ∈ A^{0} so that v(t) ∈ A_{a}^{a}. Since the groupoid A is proper the group A_{a}^{a} is finite. Hence there exists an n ∈ ℕ such that v(t)^{n} = id_{a}. The map v^{n} fits into the diagram

\xymatrix{\{t\} \ar[r] \ar[d] & A^{1} \ar[d]^{s} \\ U \ar[ur]^{v^{n}} \ar[r]^{w̃} & A^{0}} .

Note that the map U ∋ u ↦ id_{w̃(u)} ∈ A^{1} would fit into the same diagram in the place of v^{n}. Since s : A^{1} → A^{0} is étale we can shrink U further such that v^{n}(u) = id_{w̃(u)} for all u ∈ U. This implies that h ∘ w_{|U} : U → G factors over the discrete subset Tors_{n}(G) ⊆ G an is therefore locally constant. □

2.4.8 Let G be a topological abelian group such that Tors_{n}(G) ⊂ G is discrete for all n ∈ ℕ. Furthermore, let p : Y → BG be a G-principal bundle over an orbisspace Y and h : LY → G^{δ} as in Lemma 2.34. Then we have a decomposition

LY ≅ ⨆_{g∈G} LY_{g} ,

where LY_{g} := h^{−1}(g) is formally defined by the 2-cartesian square

\xymatrix{LY_{g} \ar[r] \ar[d] & [\{g\}/G] \ar[d] \\ LY \ar@{=>}[ur] \ar[r]^{Lp} & [G^{δ}/G] \ar[r]^{≅} & ⨆_{l∈G}[\{l\}/G]} .

Let f : X → Y be the map of stacks underlying the principal bundle p. It fits into the cartesian diagram

\xymatrix{X \ar[d]^{f} \ar[r] & ∗ \ar[d] \\ Y \ar@{=>}[ur] \ar[r]^{p} & BG} .  (2.35)

Lemma 2.36 The map Lf : LX → LY factors over the G-principal bundle

LX → LY_{1} .

Proof. We apply the loop functor to the 2-cartesian diagram (2.35) and get the 2-cartesian diagram (see Lemma 2.24)

[figure: LX; Lf; L{1}; {1}; LY; p; LBG; [G/G]] .  (2.37)

It follows from the construction of h : LY → G that h ∘ Lf is the constant map with value 1 ∈ G. It remains to show that LX → LY_{1} is a G-principal bundle. To this end we refine the diagram (2.37) to

\xymatrix{LX \ar[d]^{Lf} \ar[r] & \{1\} \ar[d] \\ LY_{1} \ar[r] \ar[d] & [\{1\}/G] \ar[d] \\ LY \ar@{=>}[uur] \ar@{=>}[ur] \ar[r]^{p} & [G/G]} .

By definition of LY_{1} the lower square is 2-cartesian. Since the outer square is the 2-cartesian square (2.37) we conclude that the upper square is 2-cartesian. □

2.4.9 Let Γ be a finite group. The exact segment

\xymatrix{… \ar[r] & H^{1}(Γ; ℝ^{δ}) \ar[r] \ar[d]^{≅} & H^{1}(Γ; U(1)^{δ}) \ar[d]^{≅} \ar[r]_{∂}^{≅} & H^{2}(Γ; ℤ) \ar[r] & H^{2}(Γ; ℝ^{δ}) \ar[d]^{≅} \ar[r] & … \\  & 0 & Γ̂ &  & 0}

of the Bockstein sequence in group cohomology associated to the sequence of coefficients

0 → ℤ → ℝ^{δ} → U(1)^{δ} → 0

gives rise to a natural identification

H^{2}(Γ; ℤ) ≅ Γ̂ ,

where Γ̂ denote the group of U(1)-valued characters of Γ. Let us consider the orbispace [∗/Γ]. Then we have L[∗/Γ] ≅ [Γ/Γ], where Γ acts on itself by conjugation. A character χ ∈ Γ̂ gives rise to a function

χ̄ : L[∗/Γ] → U(1)^{δ} , γ ↦ χ(γ) .

2.4.10 There are various ways to define the integral cohomology of an orbispace B. In order to be able to use results about the classification of U(1)-principal bundles over B we use the definition [BS], where we define

H^{∗}(B; ℤ) := H^{∗}(|A|; ℤ)

using the classifying space |A| of the groupoid A associated to an orbifold atlas a : A → B. Note that by this definition H^{∗}(BΓ; ℤ) ≅ H^{∗}(Γ; ℤ). In fact, if we choose the atlas a : ∗ → BΓ and let A be the associated groupoid, then |A| is the standard model of the classifying space BΓ of Γ. 2.4.11 Let χ ∈ H^{2}(B; ℤ). In this paragraph we generalize the construction 2.4.9 of the map χ ↦ χ̄ to general orbispaces B. We start with describing the values of χ̄ : LB → U(1) at the points of LB. For the moment we do not claim any continuity property, but by Lemma 2.38 we see that it is continuous even if we equip G with the discrete topology. Consider a point u : ∗ → LB. It determines and is determined by a point p_{u} : ∗ →^{u} LB → B in B and an element γ_{u} ∈ Aut(p_{u}) ≅ ∗ ×_{B} ∗. The element γ_{u} generates a finite cyclic group Γ_{u}. We obtain an induced map ũ : [∗/Γ_{u}] → B. We have L[∗/Γ_{u}] ≅ [Γ_{u}/Γ_{u}] and consider γ_{u} ∈ [Γ_{u}/Γ_{u}] (or more formally, as a map γ_{u} : ∗ → [Γ_{u}/Γ_{u}]). We have an induced map Lũ : L[∗/Γ_{u}] → LB such that Lũ(γ_{u}) = u. We can now define

χ̄(u) := \overline{ũ^{∗}χ}(γ_{u}) .

2.4.12 Let B be an orbispace. By [BS, Proposition 4.3]) the class χ ∈ H^{2}(B; ℤ) classifies a U(1)- principal bundle P_{χ} → B. In Lemma 2.38 we will express the corresponding function h_{χ} : LB → U(1)^{δ} (defined in (2.32) directly in terms of χ.

Lemma 2.38 We have the equality h_{χ} = χ̄.

Proof. The constructions of h_{χ} and χ̄ are natural under pull-back. It therefore suffices to show this equality in the case that B ≅ [∗/Γ] for a finite group Γ. In this case we have P_{χ} ≅ [U(1)/_{χ}Γ], where Γ acts on U(1) via the character χ. By construction of h_{χ} we have h_{χ} = χ : [Γ/Γ] → U(1)^{δ}. On the other hand, again by construction, we have χ̄ = χ : [Γ/Γ] → U(1)^{δ}. □

2.4.13 Here is another interpretation. Let a : A → B be a good orbifold atlas. We can choose a trivialization t of the pull-back of the U(1)-bundle to A and get a cocycle Φ_{a,t} ∈ C^{1}(A; U(1)). The definition of an orbifold atlas is in particular made such that H^{i}(A; ℝ_{cont}) = 0 for i ≥ 1.^{13} Hence the boundary operator in cohomology associated to the sequence 0 → ℤ → ℝ_{cont} → U(1)_{cont} → 0 induces an isomorphism ∂ : H^{1}(A; U(1)_{cont}) →^{∼} H^{2}(A; ℤ) ≅ H^{2}(B; ℤ). Under this identification we have χ ≅ ∂[Φ_{a,t}]. Our construction of χ ↦ χ̄ is made such that \overline{∂(ϕ)} ≅ trϕ ∈ H^{0}(LA; U(1)) ≅ C(LB, U(1)) for every class ϕ ∈ H^{1}(A; U(1)). In view of 2.4.6 this assertion is equivalent to Lemma 2.38.

2.5 Gerbes and local systems

2.5.1 We consider stacks in topological spaces StTop. Let H be an abelian topological group and f : G → X be a topological gerbe with band H over some topological stack X We take loops and obtain Lf : LG → LX. We further have a canoical map ĩ : LG → G, and LG/G is a group in StTop/G (see Lemma 2.23). Since i ∘ Lf ≅ f ∘ ĩ we get the dotted arrow

\xymatrix{LG \ar@{.>}[dr]^{π} \ar[ddr]_{Lf} \ar[drr]^{ĩ} \\  & G_{L} \ar[d] \ar[r] & G \ar[d]^{f} \\  & LX \ar@{=>}[ur] \ar[r]^{i} & X} ,  (2.39)

13For a proof see [Cra, Proposition 1] or the corrected version [BS]. In the orginal version an orbifold atlas was characterized by the property that it gives rise to a proper étale groupoid. In order to prove this vanishing of real continuous cohomology we added the assumption of being very proper.

where the gerbe G_{L} → LX is defined by the 2-cartesian square. One way to say that the gerbe G → X has band H is as follows:^{14}

(1) The map π : LG → G_{L} is the underlying map of an H-principal bundle G_{L} → BH.

(2) The sequence of (representable, see 2.28) maps π : LG → G_{L} → G is a central extension of groups

G × H/G → LG/G → G_{L}/G  (2.40)

in StTop/G (the group stack structures of G_{L}/G is induced from that of LX/X. 2.5.2 Proposition 2.41 There exists a canonical central extension

X × H/X → G̃/X → LX/X

of groups in StTop/X whose pull-back along G → X is isomorphic to (2.40). It depends functorially on the datum G → X. Proof. We first go over to topological groupoids by choosing atlases. Then we construct the required extension in topological groupoids. Finally we pass back to stacks.^{15} We choose an atlas a : A → X which admits a lift

[figure: G; f; A; b; a; X; ϕ]  (2.42)

to an atlas of G. We get topological groupoids

X : X^{1} := A ×_{X} A ⇒ X^{0} := A
G : G^{1} := A ×_{G} A ⇒ G^{0} := A ,

and a central H-extension

[figure: X^{0} × H; G^{1}; G^{0}; X^{1}; X^{0}] .

Using the description (2.19) of the objects of LX and LG we get the pull-back of H-principal bundles

\xymatrix{(LG)^{0} \ar[d] \ar[r] & G^{1} \ar[d] \\ (LX)^{0} \ar[r] & X^{1}}

14The definition given in [Hei05, Def. 5.3] expresses these properties using objects.

15This argument is not satisfactory. It would be better to argue directly with stacks. But at the moment we do not know how to do this.

Furthermore, by (2.20) we have the following description of morphisms (LG)^{1} as a pull-back

\xymatrix{(LG)^{1} \ar[d]^{s} \ar[r] & G^{1} \ar[d]^{s} \\ (LG)^{0} \ar[r]^{δ} & G^{0}} .  (2.43)

We see that (LG)^{1} has two commuting H-actions, the first comes from the action on (LG)^{0} (the principal bundle structure of the left lower corner in(2.20)), and the second comes from the action on G^{1}, the right upper corner in (2.20). 2.5.3 We now define the groupoid G_{L} corresponding to the stack G_{L}. The obvious definition would be as LX ×_{X} G, but we consider the simpler equivalent groupoid G_{L} : (G_{L})^{1} ⇒ (LX)^{0} where the morphisms are given by the cartesian diagram

\xymatrix{(G_{L})^{1} \ar[d] \ar[r] & G^{1} \ar[d]^{(r,s)} \\ (LX)^{1} \ar[r]^{i∘r,i∘s} & X^{0} × X^{0}} .  (2.44)

We have a natural homomorphism of groupoids LG → G_{L} which is an H-principal bundle as expected. 2.5.4 Observe that we can define a groupoid G̃: G̃^{1} → G̃^{0} = LG^{0} by taking the quotient of G̃^{1} := (LG)^{1} by the second H-action. In other words, we define G̃^{1} by the cartesian diagram

\xymatrix{G̃^{1} \ar[d] \ar[r] & (LG)^{0} \ar[d] \\ X^{1} \ar[r] & X^{0}} .  (2.45)

With the natural induced map G̃ → LX is an H-principal bundle over LX. We compose this map with LX → X and observe that the groupoid structure on G̃ induces on G̃ → X the structure of a group in groupoids over X. It fits into the central extension

X × H → G̃ → LX .

of groups in gpd(Top)/X. The bundle G̃ → LX fits into a cartesian diagram

\xymatrix{LG \ar[d] \ar[r] & G̃ \ar[d] \\ G_{L} \ar@{=>}[ur] \ar[r] & LX} .

2.5.5 We now pass back to stacks. We interpret the H-principal bundle G̃^{0} → (LX)^{0} as an object (LX)^{0} → BH. The action (LX)^{1} ×_{(LX)^{0}} G̃^{0} → G̃^{0} gives the descend^{16} datum for completing the following

\xymatrix{B^{1} \ar[r]^{s} \ar[d]^{r} & B^{0} \ar[d] \\ B^{0} \ar@{=>}[ur] \ar[r] & U}

16Let B : B^{1} ⇒ B^{0} be a topological groupoid with quotent stack [B^{1}/B^{0}]. Let U be some stack. A descend datum is a diagram which is compatible with the composition in B in the obvious way. We use the equivalence of the category Hom([B^{1}/B^{0}], U)

with the category of descend data.

diagram by the dotted arrows:

\xymatrix{G̃^{0} \ar[r] \ar[d] & G̃ \ar@{.>}[d] \\ (LX)^{0} \ar[dr] \ar[r] & [(LX)^{0}/(LX)^{1}] \ar@{.>}[d] \ar[r]^{≅} & LX \\  & BH}

The H-principal bundle in groupoids G̃ → LX thus gives rise to a H-principal bundle in topological stacks LX → BH with underlying map of stacks G̃ → LX. In fact, it fits into the cartesian diagram

\xymatrix{LG \ar[d] \ar[r] & G̃ \ar[d] \\ G_{L} \ar@{=>}[ur] \ar[r] & LX}

and the central extension

X × H → G̃ → LX

in StTop/X.

In order to answer the question wether G̃ → LX is well-defined up to canonical equivalence we must study how it depends on the choice of the atlas a : A → X and its lift (b, ϕ) (see 2.42). We must show that an automorphism of this datum induces the identity on G̃ → LX. Now observe that the automorphism group of (a, b, ϕ) is the group of automorphisms of b which induce the identity on a (in order not to change ϕ). By the definition of a H-banded gerbe it is given by C(A, H). It acts trivially on G̃ → LX, indeed. Finally observe that the construction of G̃ → LX depends functorially on G → X. We leave the details to the reader. □

2.5.6 We now assume that the stack X is an orbispace. We further assume that Tors_{n}(H) ⊆ H is discrete (compare 2.4.7). Let H^{δ} be the group H equipped with the discrete topology.

Lemma 2.46 The H-bundle ϕ : G̃ → LX admits a natural reduction of structure groups ϕ^{δ} : G̃^{δ} → LX from H to H^{δ}.

Proof. Let T be a space and ∗ ∈ T be a distinguished point. We consider the lifting problem

\xymatrix{∗ \ar[r]^{σ} \ar[d] & G̃ \ar[d] \\ T \ar[r] \ar@{.>}[ur] & LX} .

We must show that this problem has a unique solution after replacing T by some neighbourhood of ∗, if necessary. Using an orbispace atlas A → X we translate to an equivalent lifting problem for topological groupoids

\xymatrix{∗ \ar[r] \ar[d] & G̃ \ar[d] \\ T \ar[r]^{t} \ar@{.>}[ur]^{t̃} & LA \ar[r]^{i} & A} .

Here we consider T as a groupoid T ⇒ T in the canonical way. Let γ := t(∗) ∈ (LA)^{0} ≅ A_{a}^{a}, where a := i(γ) ∈ A^{0} = A. Since A_{a}^{a} is a finite group there exists n ∈ ℕ such that γ^{n} = id_{A_{a}^{a}}. We consider the embedding A^{0} ⊂ A^{1} given by the identities. Using the group structure 2.2.5 of LA → A and the fact that the groupoid A is étale it follows that 1 ≡ t^{n} : T → LA after replacing T by some neighbourhood of ∗, if necessary (see the proof of Lemma 2.34 for a similar argument). It follows that t^{n} : T → LA has a natural lift t̃^{n} given by an H-translate of the identity map such that σ^{n} = t̃^{n}(∗). It remains to find the n‘th root t̃ of t̃^{n}. We now consider the diagram

\xymatrix{ker(…)^{n} \ar[d] \ar[r] & H \ar[d] \ar[r]^{(...)^{n}} \ar[r] & H \ar[d] \\ ker(…)^{n} \ar[r] & G̃ ×_{LA} T \ar[r]^{(...)^{n}}_{c} & G̃ ×_{LA} T} .

The map c : G̃ ×_{LA} T → G̃ ×_{LA} T is étale. Therefore, after replacing T by some neighbourhood of ∗ again, the datum of σ and t̃^{n} give the unique lift t̃. □

2.5.7 For smooth gerbes with band U(1) on orbifolds the analog of Lemma 2.46 was shown e.g. in [TXa] or [LUX]. The argument in these papers uses the existence of a geometric structure (connection and curving) on the gerbe G. This geometry naturally induces a connection on the U(1)-principal bundle G̃ → LX. By a calculation the curvature of this connection vanishes. This gives the reduction of structure groups. 2.5.8 Let g : Y → X be a map of topological stacks and f : G → X be a topological gerbe with band H over X. We consider a 2-cartesian diagram

\xymatrix{K \ar[d] \ar[r] & G \ar[d] \\ Y \ar[r]^{g} \ar@{=>}[ur] & X}.

Lemma 2.47 We have a 2-cartesian diagram

\xymatrix{K̃ \ar[d] \ar[r]^{ϕ} & G̃ \ar[d] \\ LY \ar[r]^{Lg} \ar@{=>}[ur] & LX} .  (2.48)

Under the assumptions of Lemma 2.46 this diagrams refines to a 2-cartesian diagram

\xymatrix{K̃^{δ} \ar[d] \ar[r] & G̃^{δ} \ar[d] \\ LY \ar[r]^{Lg} \ar@{=>}[ur] & LX} .

Proof. We get the square (2.48) from the functoriality part of Proposition 2.41. Since the vertical maps are H-principal bundles it is automatically 2-cartesian. The second statement easily follows from Lemma 2.46. □

□

2.6 The holonomy of G̃^{δ}

2.6.1 Let G → X be a topological gerbe with band U(1) over an orbispace X. In 2.5 we constructed a U(1)^{δ}-principal bundle G^{δ} → LX. It is an instructive exercise to calculate the holonomy of this bundle in terms of the Dixmier-Douady invariant d ∈ H^{3}(X; ℤ) of the gerbe G → X. In the following we consider a special but typical case of this problem. 2.6.2 We consider a U(1)-principal bundle π : E → B in orbispaces and a topological gerbe f : G → E with band U(1). Let h : LB → U(1)^{δ} be the function associated to the principal bundle E → B as in Lemma 2.34 and define LB_{1} := h^{−1}(1). Then by Lemma 2.36 we have an induced U(1)-principal bundle Lπ : LE → LB_{1}. The holonomy of the bundle G̃^{δ} → LE along the fibres of Lπ gives rise to a function

g : LB_{1} → U(1)^{δ}

(see 2.6.3 for a precise construction). The gerbe f : G → B is classified by a Dixmier-Douady class d ∈ H^{3}(E; ℤ). Let π_{!} : H^{3}(E; ℤ) → H^{2}(B; ℤ) be the integration map. According to 2.4.11 the class π_{!}(d) ∈ H^{2}(B; ℤ) gives rise to a function

\overline{π_{!}(d)} : LB → U(1)^{δ} .

The main result of the present subsection is the following proposition.

Proposition 2.49 We have the equality

g = \overline{π_{!}(d)}_{|LB_{1}} .

2.6.3 Here is the precise construction of the function g : LB_{1} → U(1)^{δ}. Let T be a space and T → LB_{1} be a map. The pull-back

\xymatrix{W \ar[r] \ar[d] & G̃^{δ} \ar[d] \\ S \ar@{=>}[ur] \ar[d] \ar[r] & LE \ar[d] \\ T \ar@{=>}[ur] \ar[r] & LB_{1}}

defines a U(1)-principal bundle S → T and a U(1)^{δ}-principal bundle W → S. We chose an open covering (T_{α} → T)_{α∈I} such that for all α ∈ I there exists a section

\xymatrix{ & S \ar[d] \\ T_{α} \ar[r] \ar@{.>}[ur]^{s_{α}} & T}.

The section s_{α} gives rise to a map T_{α} × ℝ → S by (t, x) ↦ s_{α}(t)x, where ℝ acts on S via the covering ℝ → U(1). We can now (after refining the covering (T_{α} → T) if necessary) choose a lift

\xymatrix{ & W \ar[d] \\ T_{α} × ℝ \ar@{.>}[ur]^{w_{α}} \ar[r] & S} .

Then we define a map g_{T_{α}} : T_{α} → U(1)^{δ} such that w_{α}(t, 0) = w_{α}(t, 1)g_{T_{α}}(t). Observe that g_{T_{α}} does not depend on the choices of s_{α} and w_{α}. One easily checks that the family of maps (g_{T_{α}})_{α∈I} determines a map g_{T} : T → U(1)^{δ} which depends functorially on T → LB_{1}. It therefore defines a map g : LB_{1} → U(1)^{δ}.

2.6.4 We now turn to the actual proof of Proposition 2.49. We first consider a special case. Let Γ be a finite cyclic group which we write additively. We let Γ act trivially on U(1) and consider the orbispace E := [U(1)/Γ]. The projection U(1) → ∗ induces a U(1)-principal bundle π : E → B := [∗/Γ]. We calculate H^{3}(E; ℤ) using the Kuenneth formula and the product decomposition E = U(1)×B. Note that H^{∗}(B; ℤ) ≅ H^{∗}(Γ; ℤ). In particular we have H^{3}(B; ℤ) ≅ 0 and a canonical isomorphism H^{2}(B; ℤ) ≅ Γ̂ (see 2.4.9). It follows that

H^{3}(E; ℤ) ≅ H^{1}(U(1); ℤ) ⊗ H^{2}(B; ℤ) ≅ Γ̂

using the canonical orientation H^{1}(U(1); ℤ) ≅ ℤ of U(1). 2.6.5 The group H^{3}(E; ℤ) classifies topological U(1)-gerbes over E. In the following we present a construction which associates to every character ϕ ∈ Γ̂ a U(1)-gerbe G_{ϕ} → E. We construct these gerbes in terms of representing groupoids. The canonical covering ℝ → U(1) induces an atlas ℝ → E. The corresponding topological groupoid is the action groupoid for the action of ℤ × Γ on ℝ by (n, γ)t := t + n. It is given by

ℝ × ℤ × Γ ⇒ ℝ  (2.50)

with range r(t, n, γ) := t + n, source s(t, n, γ) := t, and the composition (t + m, n, γ) • (t, m, γ′) := (t, n + m, γ + γ′). The character ϕ ∈ Γ̂ determines a U(1)-central extension

0 → U(1) → { \atop ℤ × Γ}_{ϕ} → ℤ × Γ → 0 .  (2.51)

If we identify { \atop ℤ × Γ}_{ϕ} ≅ ℤ × Γ × U(1) as sets, then the multiplication is given by (n, γ, z)(n′, γ′, z′) = (n + n′, γ+γ′, ϕ(γ)^{n′}zz′). This central extension acts on ℝ via its projection { \atop ℤ × Γ}_{ϕ} → ℤ×Γ, (n, γ, z) ↦ (n, γ). The gerbe G_{ϕ} → E is then given by

[ℝ/{ \atop ℤ × Γ}_{ϕ}] → [ℝ/ℤ × Γ] .

In terms of groupoids, G_{ϕ} is given as the U(1)-central extension of the groupoid (2.50) which on the level of morphisms is the trivial U(1)-bundle

ℝ × ℤ × Γ × U(1) → ℝ × ℤ × Γ ,

whose source and range maps are

s(t, n, γ, z) := t , r(t, n, γ, z) := t + n ,

and whose composition is given by

(t + m, n, γ, z′)(t, m, γ′, z) := (t, n + m, γ + γ′, ϕ(γ)^{m}z′z) .

2.6.6 We now calculate the bundle G̃_{ϕ}^{δ} → LE. First of all note that

LE ≅ [Γ × U(1)/Γ] ,

where Γ acts trivially on Γ × U(1). The map Γ × ℝ → Γ × U(1) gives an atlas of LE. The associated groupoid is the action groupoid of the action of ℤ × Γ on Γ × ℝ by (n, γ)(σ, t) = (σ, t + n). It is given by

Γ × ℝ × ℤ × Γ ⇒ Γ × ℝ

with range and source given by

r(σ, t, n, γ) := (σ, t + n) , s(σ, t, n, γ) := (σ, t) ,

and with the composition

(σ, t + m, n, γ) ∘ (σ, t, m, γ′) := (σ, t, n + m, γ + γ′) .

We can now read off a groupoid presentation of the U(1)^{δ}-principal bundle G̃_{ϕ}^{δ} → LE. It is is presented by the U(1)^{δ}-principal bundle in groupoids

\xymatrix{Γ × ℝ × U(1)^{δ} × ℤ × Γ \ar[d] \ar@{=>}[r] & Γ × ℝ × U(1)^{δ} \ar[d] \\ Γ × ℝ × ℤ × Γ \ar@{=>}[r] & Γ × ℝ} .

The range and source maps in the upper horizontal line are given by

r(σ, t, z, n, γ) := (σ, t + n, ϕ(σ)^{n}z) , s(σ, t, z, n, γ) := (σ, t, z) ,

and with the composition

(σ, t + m, ϕ(σ)^{m}z, n, γ) ∘ (σ, t, z, n, γ′) := (σ, t, z, n + m, γ + γ′) .

In particular, the holonomy of G̃_{ϕ}^{δ} along the fibre of LE over [{σ}/Γ] is given by ϕ(σ). 2.6.7 In our example we have LB_{1} = [Γ/Γ] = LB, where Γ acts trivially on itself. The function g_{ϕ} : LB_{1} → U(1)^{δ}, which measures the holonomy of G̃_{ϕ} → LE along the fibres of LE → LB_{1}, is given by the calculation above by

g_{ϕ} = ϕ : Γ → U(1)^{δ} .  (2.52)

By the discussion 2.6.4 the character ϕ gives rise to a class d_{ϕ} ∈ H^{3}(E; ℤ) such that

π_{!}(d_{ϕ}) = ϕ

(using the isomorphism Γ̂ ≅ H^{2}(B; ℤ)). Furthermore we have

\overline{π_{!}(d_{ϕ})} = ϕ : Γ → U(1)^{δ} .

2.6.8 In order to finish the proof of Proposition 2.49 in the special case we must show that d_{ϕ} is the Dixmier-Douady class d(G_{ϕ}) of G_{ϕ}. We will use the following two general facts:

(1) Let 1 → U(1) → Ĝ → G → 1 be a U(1)-central extension of a discrete group G classified by e ∈ Ext(G; U(1)) := H^{2}(G; U(1)). Furthermore, let δ : H^{2}(G; U(1)) → H^{3}(G; ℤ) be the boundary operator in the Bockstein sequence in group cohomology associated to the exact sequence of coefficients 0 → ℤ → ℝ → U(1) → 0. Then the Dixmier-Doudady class of the gerbe [∗/Ĝ] → [∗/G] is given by the image of δ(e) ∈ H^{3}(G; ℤ) under the isomorphism H^{3}(G; ℤ) ≅ H^{3}([∗/G]; ℤ).

(2) Let ϕ : G → U(1) be a character of a finite group G. It gives rise to a class ϕ ∈ H^{1}(G; U(1)) and an extension 1 → U(1) → { \atop ℤ × G} → ℤ × G → 1. We can identify { \atop ℤ × G} ≅ ℤ × G × U(1) as sets. Its multipication is then given by (n, g, z)(n′, g′, z′) = (n + n′, gg′, ϕ(g)^{n′}zz′). The class e ∈ Ext(ℤ × G; U(1)) of the extension is then given by image of id_{ℤ} × ϕ ∈ H^{1}(ℤ; ℤ) × H^{1}(G; U(1)) under the product × : H^{1}(ℤ; ℤ) × H^{1}(G; U(1)) → H^{2}(ℤ × G; U(1)), where id_{ℤ} ∈ H^{1}(ℤ; ℤ) is the identity homomorphism.

We now specialize these facts to the present situation. The Kuenneth formula gives an isomorphism

Ext(ℤ × Γ; U(1)) := H^{2}(ℤ × Γ; U(1)) ≅ H^{1}(ℤ; ℤ) ⊗ H^{1}(Γ; U(1)) ≅ ℤ ⊗ Γ̂ ≅ Γ̂ ,  (2.53)

where we use the generator id_{ℤ} ∈ H^{1}(ℤ; ℤ) in order to identify H^{1}(ℤ; ℤ) ≅ ℤ. The class e_{ϕ} ∈ Ext(ℤ × Γ; U(1)) of the extension (2.51) corresponds under this isomorphism to ϕ ∈ Γ̂ (by (2)).

By (1) the Dixmier-Douady class d(ϕ) ∈ H^{3}([∗/ℤ × Γ]; ℤ) of the gerbe

[∗/{ \atop ℤ × Γ}_{ϕ}] → [∗/ℤ × Γ]

corresponds to

δ(e_{ϕ}) ∈ H^{3}(ℤ × Γ; ℤ)

under the identification

H^{3}([∗/ℤ × Γ]; ℤ) ≅ H^{3}(ℤ × Γ; ℤ) .

2.6.9 Let p : B = [ℝ/ℤ × Γ] → [∗/ℤ × Γ] be the canonical projection. Then we have d(G_{ϕ}) = p^{∗}d(ϕ). We now observe that the following diagram commutes,

\xymatrix{d(G_{ϕ}) \ar[ddd] & d(ϕ) \ar[l] & & e_{ϕ} \ar[ll] \ar[dddl]^{(2.53)} \\ H^{3}([ℝ/ℤ × Γ]; ℤ) \ar[d]^{π_{!}} & H^{3}([∗/ℤ × ℝ]; ℤ) \ar[l]^{p^{∗}}_{≅} & H^{3}(ℤ × Γ; ℤ) \ar[l]^{≅} & H^{2}(ℤ × Γ; U(1)) \ar[d]^{≅} \ar[l]_{≅}^{δ} \\ H^{2}([∗/Γ]; ℤ) & H^{2}(Γ; ℤ) \ar[l]^{≅} & H^{1}(Γ; U(1)) \ar[l]^{δ} & H^{1}(ℤ; ℤ) ⊗ H^{1}(Γ; U(1)) \ar[l]^{≅} \\ π_{!}(d_{ϕ}) & & ϕ \ar[ll]_{definition of d_{ϕ}}} ,

and that the elements are mapped as indicated. 2.6.10 We show how the general case of Proposition 2.49 can be reduced to the special case discussed above. The constructions of g and \overline{π_{!}(d)}_{|LB_{1}} are natural with respect to pull-back. Therefore in order to verify Proposition 2.49 it suffices to show the desired equality over each point in LB seperately. As in 2.4.11 a point u ∈ LB is given by a point p ∈ B and an element γ ∈ Aut(p) (in the present subsection we omit the subscript u in order to simplify the notation). Let Γ ⊂ Aut(p) be the cyclic group generated by γ and χ ∈ Γ̂ be the character by which Γ acts on the fibre π^{−1}(p). Note that

χ(γ) = h(u) .  (2.54)

We get a cartesian diagram

\xymatrix{v^{∗}G \ar[d] \ar[r] & G \ar[d]^{f} \\ [U(1)/_{χ}Γ] \ar[r]^{v} \ar[d]^{q} & E \ar[d]^{π} \\ [∗/Γ] \ar[r]^{ũ} & B} .  (2.55)

such that Lũ(γ) = u, where we consider γ ∈ [Γ/Γ] ≅ L[∗/Γ]. In particular, v^{∗}d is a Dixmier-Douady class of the gerbe v^{∗}G → [U(1)/_{χ}Γ] and we have

\overline{π_{!}(d)}(u) = \overline{q_{!}v^{∗}(d)}(γ) .

Observe that L[∗/Γ]_{1} = [ker(χ)/Γ]. Let g_{v^{∗}G} : L[∗/Γ]_{1} → U(1)^{δ} denote the function (2.6.3) which

measures the holonomy of ṽ^{∗}G^{δ} → L[U(1)/_{χ}Γ] along the fibres of q. If u ∈ LB_{1}, then by 2.54 we have χ(γ) = 1 and

g(u) = g_{v^{∗}G}(γ) .

The equation

\overline{π_{!}(d)}(u) = g(u)

now follows from the equation

g_{v^{∗}G}(γ) = \overline{q_{!}v^{∗}(d)}(γ)

which was already shown above. □

3 Delocalized Cohomology of orbispaces and orbifolds

3.1 Definition of delocalized twisted cohomology

3.1.1 A topological stack X gives rise to a site Site(X) = X. The underlying category of X is the subcategory of Top/X of maps (U → X) which are representable and have local sections. The covering families (U_{i} → U) are families of maps U_{i} → U in X which have local sections^{17} and are such that ⊔_{i}U_{i} → U is surjective. One can actually restrict to covering families by open subsets without changing the induced topology (the argument is similar as for [BSSc, Lemma 2.47]). If X is a space, then the small site (X) of X is the category of open subsets of X with the usual notion of covering families. 3.1.2 To the site X we associate categories of presheaves and sheaves PShX and ShX in the usual way. A map p : X → Y of topological stacks induces a pair of adjoint functors

p^{∗} : ShY ⇔ ShX : p_{∗} .

We use this framework of sheaf theory on topological stacks in order to define the delocalized cohomology of an orbispace twisted by a gerbe. For details of the sheaf theory we refer to [BSSc] and [BSSa]. For a site X let i : ShX → PShX denote the canonical embedding of the category of presheaves into the category of sheaves, and let i^{♯} : PShX → ShX denote its left-adjoint, the sheafification functor. We use the same symbols in order to denote the restriction of these functors to the categories PSh_{Ab}X and Sh_{Ab}X of presheaves and sheaves of abelian groups. 3.1.3 Let H be a topological abelian group. We assume that Tors_{n}(H) ⊆ H is discrete for all n ∈ ℕ (see 2.4.7). Let H^{δ} denote the group H with the discrete topology. Furthermore, let Z be a discrete abelian group and λ : H^{δ} → Aut(Z) be a homomorphism. 3.1.4 Let P → X be the underlying map of stacks of an H^{δ}-principal bundle over a topological stack X. If (U → X) ∈ X, then U ×_{X} P → U is an ordinary H^{δ}-principal bundle. We define the abelian group Z_{P,λ}(U) to be the group of continuous sections of the associated bundle (U ×_{X} P) ×_{H^{δ},λ} Z → U under pointwise multiplication. If (U′ → X) → (U → X) is a morphism in X, then we have an induced morphism U′ ×_{X} P → U ×_{X} P of H^{δ}-principal bundles over U′ → U. It induces a homomorphism Z_{P,λ}(U) → Z_{P,λ}(U′). In this way obtain a presheaf of abelian groups Z_{P,λ} ∈ PSh_{Ab}X, U ↦ Z_{P,λ}(U). Note that Z_{P,λ} is actually a sheaf, i.e. we have Z_{P,λ} ∈ Sh_{Ab}X. 3.1.5 Let f : G → X be a gerbe with band H over an orbispace X. Then by Lemma 2.46 we have the H^{δ}-principal bundle G̃^{δ} → LX. By 3.1.4 it gives rise to a the presheaf Z_{G̃^{δ},λ} ∈ PSh_{Ab}LX. 3.1.6 We define a gerbe f_{L} : G_{L} → LX with band H as the pull-back of the gerbe f : G → X along the canonical map i : LX → X (see 2.39). We have a diagram

\xymatrix{∗ & G_{L} \ar[r] \ar[l]^{p} \ar[d]^{f_{L}} & G \ar[d]^{f} \\ & LX \ar@{=>}[ur] \ar[r]^{i} & X} .

We consider f_{L}^{∗}Z_{G̃^{δ},λ} ∈ Sh_{Ab}G_{L}.

17A map of topological spaces f : V → W has local sections if for every w ∈ f(V) there exists an open neighbourhood W_{w} ⊆ W and a map σ : W_{w} → V such that id_{W_{w}} = f ∘ σ

3.1.7 Let ev := ev_{∗→∗} : Sh_{Ab}Site(∗) → Ab be the functor, which evaluates a sheaf of abelian groups on Site(∗) at the object (∗ → ∗) ∈ Site(∗).

Lemma 3.1 The functor ev : Sh_{Ab}Site(∗) → Ab is exact.

Proof. A basic observation lying at the heart of sheaf theory is that evaluation functors are not exact in general. Therefore, a proof of exactness of the evaluation ev is required. First note that Site(∗) is the big site of ∗ which can be identified with the category of all topological spaces. Every non-empty collection of non-empty spaces is a covering family of ∗. The small site (∗) of ∗ has one object ∗ → ∗. In [BSSa] (see also [BSSc, Prop. 2.46], the arguments works equally well in the smooth and topological contexts) we have seen that the restriction functor ν_{∗} : ShSite(∗) → Sh(∗) is exact. Let ẽv : Sh_{Ab}(∗) → Ab denote the corresponding evaluation functor. It is actually an isomorphism of categories, and in particular exact. We have ẽv ∘ ν_{∗} ≅ ev. We see that ev is exact, since it is a composition of exact functors. □

3.1.8 The functor p_{∗} : Sh_{Ab}(G_{L}) → Sh_{Ab}Site(∗) is left-exact and thus admits right-derived functors

Rp_{∗} : D^{+}(Sh_{Ab}G_{L}) → D^{+}(Sh_{Ab}Site(∗))

between the lower bounded derived categories. The functor ev : Sh_{Ab}Site(∗) → Ab is exact and thus descends to the lower-bounded derived catgeories.

Definition 3.2 We define the delocalized G-twisted cohomology of X with coefficients in (Z, λ) by

H_{deloc}^{∗}(X; G, (Z, λ)) := H^{∗}(ev ∘ Rp_{∗}(f_{L}^{∗}Z_{G̃^{δ},λ})) .

3.1.9 The most important example for us is the case where Z := ℂ^{δ} and H := U(1) with λ : H^{δ} → Z → End(Z) being the obvious embedding U(1)^{δ} → End(ℂ^{δ}). In this case we will denote the sheaf ℂ_{G̃^{δ},λ} by L or L_{G}, if a reference to G is necessary.

Definition 3.3 The G-twisted complex delocalized cohomology of X is defined by

H_{deloc}^{∗}(X; G) := H_{deloc}^{∗}(X; G, L) .

3.1.10 Another example related to Spin-structures is the case where Z := ℤ, H := ℤ^{∗} = {1, −1}, and λ : ℤ^{∗} → End(ℤ) is again the canonical embedding. 3.1.11 We now discuss the functial behaviour of the delocalized twisted cohomology. We defined the sheaf Z_{G̃^{δ},λ} down on LX in order to connect with usual conventions in the literature on inner local systems and twisted torsion, and in order to have the formula (3.9) below. This construction depends on descending the H^{δ}-bundle LG → G_{L} to the bundle G̃ → LX. The quite complicated construction was carries out in Proposition 2.41. In the definition of twisted cohomology we then use the pull-back f_{L}^{∗}Z_{G̃^{δ},λ}. It would be much more natural to construct the sheaf Z̃_{LG^{δ},λ} := f_{L}^{∗}Z_{G̃^{δ},λ} directly starting from the H^{δ}- principal bundle LG^{δ} → G_{L}. We can proceed as in the definition of Z_{G̃^{δ},λ}. For an object (U → G_{L}) ∈ G_{L} we define Z̃_{LG^{δ};λ}(U) ∈ Ab as the group of continuous sections of (U ×_{G_{L}} LG^{δ}) ×_{H^{δ},λ} Z under pointwise multiplication. For a morphism U′ → U we then have a natural homomorphism Z̃_{LG^{δ};λ}(U) → Z̃_{LG^{δ};λ}(U′) induced by a corresponding morphism of principal bundles over U′ → U. We have a canonical isomorphism

H_{deloc}(X; G, (Z, λ)) ≅ H^{∗}(ev ∘ Rp_{∗}(Z̃_{LG^{δ},λ})) .

In the case H = U(1) and Z = ℂ^{δ} we set Z̃_{LG^{δ},λ} := L̃.

3.1.12 We consider a 2-cartesian diagram

\xymatrix{G′ \ar[d]^{f′} \ar[r]^{h} & G \ar[d]^{f} \\ X′ \ar@{=>}[ur] \ar[r]^{g} & X} ,

where g is a map of orbispaces, i.e. a representable map.

Lemma 3.4 We have a canonical functorial map

(g, h)^{∗} : H_{deloc}^{∗}(X; G) → H_{deloc}^{∗}(X′; G′) .

Proof. Since the loop functor preserves two-cartesian diagrams we get an induced 2-cartesian diagram

\xymatrix{LG^{′δ} \ar[d] \ar[r]^{Lh} & LG \ar[d] \\ G′_{L} \ar@{=>}[ur] \ar[d]^{f′_{L}} \ar[r]^{h_{L}} & G_{L} \ar[d]^{f_{L}} \\ LX′ \ar@{=>}[ur] \ar[r]^{Lg} & LX} .  (3.5)

Let L̃ = Z̃_{LG^{δ},λ} ∈ Sh_{Ab}G_{L} and L̃′ := Z_{LG^{′δ},λ}Sh_{Ab}G_{L}′ denote the sheaves of abelian groups associated to G and G′ and (Z, λ) as in 3.1.11. The diagram (3.5) induces an isomorphism

h_{L}^{∗}L̃ →^{∼} L̃′  (3.6)

of sheaves on G′_{L}. We now consider the diagram

\xymatrix{G′_{L} \ar[dr]^{p′} \ar[rr]^{h_{L}} & & G_{L} \ar[dl]_{p} \\ & ∗} .

The unit id → R(h_{L})_{∗} ∘ h_{L}^{∗} of the adjoint pair

h_{L}^{∗} : D^{+}(Sh_{Ab}G_{L}) ⇔ D^{+}(Sh_{Ab}G_{L}′) : R(h_{L})_{∗}

induces a natural transformation

Rp_{∗} → Rp_{∗} ∘ R(h_{L})_{∗} ∘ h_{L}^{∗} : D^{+}(Sh_{Ab}G_{L}) → D^{+}(Sh_{Ab}Site(∗)) .  (3.7)

Since p∘h_{L} = p′ and Rp_{∗} ∘R(h_{L})_{∗} ≅ R(p∘h_{L})_{∗} (see [BSSa] and also [BSSc, Lemma 2.26] for an argument in the smooth case) we have an isomorphism

Rp_{∗} ∘ R(h_{L})_{∗} ≅ R(p ∘ h_{L})_{∗} ≅ Rp′_{∗} .

We insert this into (3.7) and get the natural transformation

Rp_{∗} → Rp′_{∗} ∘ h_{L}^{∗} : D^{+}(Sh_{Ab}G_{L}) → D^{+}(Sh_{Ab}Site(∗)) .  (3.8)

We define

(g, h)^{∗} : H^{∗} ∘ ev ∘ Rp_{∗}(L̃) →^{(3.8)} H^{∗} ∘ ev ∘ Rp′_{∗} ∘ h_{L}^{∗}(L̃) ≅^{(3.6)} H^{∗} ∘ ev ∘ Rp′_{∗}(L̃′) .

We leave it to the reader to write out the argument for functoriality. The basic input is the functoriality of the units for a composition f ∘ g which can be expressed as the commutativity of

\xymatrix{id \ar[r] \ar[rrr] & Rf_{∗} ∘ f^{∗} \ar[r] & Rf_{∗} ∘ Rg_{∗} ∘ g^{∗} ∘ f^{∗} \ar[r]^{≅} & R(f ∘ g)_{∗} ∘ (f ∘ g)^{∗}}

(see [BSSa] for a proof). □

3.1.13 ¿From now on we consider the case H := U(1) and Z := ℂ^{δ}. We can decompose p = q ∘ f_{L}, where q : LX → ∗. Since f_{L} has local sections we have an isomorphism

Rp_{∗} ≅ Rq_{∗} ∘ R(f_{L})_{∗}

by [BSSc, 2.26]. We have

Rp_{∗} ∘ f_{L}^{∗}(L) ≅ Rq_{∗} ∘ R(f_{L})_{∗} ∘ f_{L}^{∗}(L)

and the projection formula (see [BSSa])

R(f_{L})_{∗}(L̃) ≅ R(f_{L})_{∗} ∘ f_{L}^{∗}(L) ≅ R(f_{L})_{∗}(ℂ_{G_{L}}) ⊗_{ℂ} L .  (3.9)

Therefore we can write

H_{deloc}^{∗}(X; G) ≅ H^{∗}(ev ∘ Rq_{∗} ∘ (R(f_{L})_{∗}(ℂ_{G_{I}}) ⊗_{ℂ} L)) .

3.2 Twisted de Rham cohomology

3.2.1 The theory developed in the Sections 2.2, 2.3, 2.5 and 3.1 has a counterpart in the world of stacks in smooth manifolds though there is one essential difference. The map LX → X is not representable as a map of stacks in smooth manifolds. Therefore the proof of the fact that LX is a smooth stack is quite different from the topological case^{18}. But note that we have not used representability of LX → X otherwise. In the following we explain the replacements which lead to a precisely analogous theory.

(1) The category of topological spaces Top is replaced by the category of Mf^{∞} of smooth manifolds.

(2) Stacks in topological spaces are replaced by stacks in manifolds.

(3) The condition on a map of having local sections is replaced by the condition of being a submersion (following the conventions from algebraic geometry we will use the term ”smooth” synonymously with ”submersion”).

(4) Topological stacks are replaced by smooth stacks. A stack in smooth manifolds X is called smooth if it admits an atlas a : A → X, i.e. a representable, surjective, and submersive (which replaces the local section condition by the preceding point) map from a manifold A.

(5) The notion of a topological groupoid is replaced by the notion of a Lie groupoid. In particular, we require that range and source maps are submersions.

(6) Orbispaces are replaced by orbifolds. A smooth stack is an orbifold if it admits an orbifold atlas. An orbifold atlas is an atlas which gives rise to a proper and étale groupoid in smooth manifolds. Since manifolds are locally compact and Hausdorff the conditions ”separated” and ”very proper”^{19} hold automatically (see Lemma 2.30).

18One could save our argument by introducing the notion of a smoothly representable map between stacks in smooth manifolds and showing that LX → X is smoothly representable. A map X → Y between stacks in smooth manifolds is called smootly representable, if the fibre product A ×_{Y} X is a manifold for every submersion A → Y.

19The condition ”very proper” is as in 2.3.7 with the difference that the cut-off function must be smooth.

(7) The group H in 3.1.3 must be a Lie group.

(8) For a smooth stack X the site X is the subcategory of Mf^{∞}/X of maps (U → X) which are representable submersions. The covering families are families (U_{i} → U) of submersions such that ⊔_{i}U_{i} → U is surjective.

One problem with the category Mf^{∞} is that fibre products only exist under additional conditions (e.g. if one map is a submersion). We leave it to the interested reader to check that all fibre products used in Sections 2.3, 2.5 and 3.1 exists in manifolds^{20}. Let X be an orbifold and G → X be a smooth gerbe with band U(1). Then by 3.3 we have a well-defined twisted delocalized cohomology

H_{deloc}^{∗}(X; G) .

The main goal of the present section is to calculate this cohomology in terms of a twisted de Rham complex. This generalizes the main result of [BSSc] from smooth manifold X to orbifolds X. 3.2.2 The first goal of the present subsection is to define the de Rham complex associated to a locally constant sheaf of complex vector spaces on an orbifold in two equivalent (according to Lemma 3.11) ways. In the first picture we define a sheaf of de Rham complexes on the big site of the orbifold and then take its global sections. The second picture uses the calculus of differential forms on the orbifold itself. While the first picture belongs to the philosophy of the present paper this second definition is mainly used to compare with other constructions in the literature. In the second part we apply this construction to the local system L ∈ Sh_{Ab}LX associated to an U(1)-gerbe G → X on an orbifold. 3.2.3 Consider a smooth stack X in smooth manifolds. Let E be a locally constant sheaf on X of complex vector spaces. If (U → X) ∈ X, then E_{|U} is the sheaf of parallel sections of a canonically determined complex vector bundle with flat connection (E_{U}, ∇^{E_{U}}). Let Ω^{k}(U, E_{U}) denote the space of global sections of Λ_{ℂ}^{k}T^{∗}U ⊗ E_{U}. The de Rham differential d_{dR} and the connection ∇^{E_{U}} together induce a differential d^{E_{U}} : Ω^{k}(U, E_{U}) → Ω^{k+1}(U, E_{U}). Observe that (Ω^{·}(U, E_{U}), d^{E_{U}}) is a (Ω^{·}(U), d_{dR})-DG-module. If f : (U′ → X) → (U → X) is a morphism in X, then we have a morphism of sheaves f^{∗}E_{|U} → E_{|U′}. This induces a morphism of flat vector bundles f^{∗}E_{U} → E_{U′} and finally a morphism of complexes (Ω^{·}(U, E_{U}), d^{E_{U}}) → (Ω^{·}(U′, E_{U′}), d^{E_{U′}}). We define the sheaf Ω_{X}^{·}(E) of (Ω_{X}^{·}, d_{dR})-DG-modules which associates to (U → X) in X the complex (Ω^{·}(U, E_{U}), d^{E_{U}}).

Lemma 3.10 E → Ω_{X}^{·}(E) is a flabby resolution

Proof. This is shown by adapting the arguments of [BSSc, Sec. 3.1] to differential forms twisted by a flat vector bundle. □

3.2.4 Let p : X → ∗ be the projection. If F ∈ ShX, then we define its global sections by

Γ_{X}F := ev ∘ p_{∗}(F) .

3.2.5 Assume now that X is an orbifold. The sheaf E gives rise to a flat vector bundle E → X in the orbifold sense. We can consider the de Rham complex Ω(X, E) of E-valued forms on X which are smooth in the orbifold sense.

Lemma 3.11 We have a natural isomorphism Γ_{X}Ω_{X}^{·}(E) ≅ Ω(X, E).

Proof. We choose an orbifold atlas A → X, i.e. A is a smooth manifold, A → X is an atlas, and the smooth groupoid A ×_{X} A ⇒ A is very proper, separated and étale. By the definition of smooth forms in the orbifold sense we have the exact sequence

\xymatrix{0 \ar[r] & Ω(X, E) \ar[r] & Ω(A, E_{A}) \ar[r]^{r^{∗}−s^{∗}} & Ω(A ×_{X} A, E_{A×_{X}A})} .

20with the exception that the construction of the simpler model of LX ×_{X} G in 2.5.3 needs different arguments since (2.44) may not be a transversal pull-back.

The composition A → X → ∗ is clearly representable. By [BSSc, Lemma 2.36] we have an exact sequence

\xymatrix{0 \ar[r] & Γ_{X}Ω_{X}^{·}(E) \ar[r] & Ω(A, E_{A}) \ar[r]^{r^{∗}−s^{∗}} & Ω(A ×_{X} A, E_{A×_{X}A})} .

□

3.2.6 In order to indicate that the local system E is the initial datum and the vector bundle E → X is secondary we use the following notation.

Definition 3.12 We define

Ω^{·}(X, E) := Γ_{X}Ω_{X}^{·}(E) .

It is a Ω^{·}(X)-DG-module. Its differential will be denoted by d^{E}.

3.2.7 The twisted de Rham cohomology of X with coefficients in E depends on the choice of a closed form λ ∈ Ω^{3}(X). Let z be a formal variable of degree 2. Then we form the complex Ω^{·}(X, E)[[z]]_{λ} given by

Ω^{·}(X, E)[[z]] , d_{λ} := d^{E} + λT ,

where T := \frac{d}{dz}.

Definition 3.13 The λ-twisted cohomology H^{∗}(X; E, λ) of X with coefficients in E is defined as the cohomology of the complex Ω^{·}(X, E)[[z]]_{λ}.

3.2.8 We can also define a sheaf Ω_{X}^{·}(E)[[z]]_{λ} of (E, λ)-twisted de Rham complexes on X such that for

(U →^{ϕ} X) ∈ X we have Ω_{X}^{·}(E)[[z]]_{λ}(U) := Ω^{·}(U, E_{U})[[z]] with the differential d_{ϕ^{∗}λ}. By Definition 3.12 have an isomorphism of complexes

Ω^{·}(X, E)[[z]]_{λ} ≅ Γ_{X}Ω_{X}^{·}(E)[[z]]_{λ} .

3.2.9 We now take twists into account. Let X be an orbifold and f : G → X be a smooth gerbe with band U(1). Then we can form the orbifold of loops LX → X and the pull-back f_{L} : G_{L} → LX of the gerbe f : G → X. We choose an atlas (A → G_{L}) ∈ G_{L}. It gives rise to a simplicial object A_{G_{L}}^{·} ∈ G_{L}^{Δ^{op}} such that

A_{G_{L}}^{n} := \underbrace{A ×_{G_{L}} ⋯ ×_{G_{L}} A}_{n+1 factors} .

Let Ω_{G_{L}}^{·} denote the de Rham complex (see [BSSc, 3.1.2]) of the smooth stack G_{L}. The associated chain complex of Ω_{G_{L}}^{·}(A_{G_{L}}^{·}) is a double complex with the de Rham differential d_{dR} and the Čech differential δ. Note that A → G_{L} → LX is an atlas. We form the simplicial object A_{LX}^{·} ∈ LX^{Δ^{op}} such that

A_{LX}^{n} := \underbrace{A ×_{LX} ⋯ ×_{LX} A}_{n+1 factors} .

We consider the double complex Ω_{LX}^{·}(A_{LX}^{·}) Note that by [BSSc, Lemma 2.36] we have

Ω^{·}(X) ≅^{Lemma3.11} Γ_{X}Ω_{LX}^{·} ≅ ker(δ : Ω_{LX}^{·}(A_{LX}^{0}) → A_{LX}^{1})) .

The property that G_{L} → LX is a smooth gerbe with band U(1) can be expressed as the fact that the diagram

[figure: A ×_{G_{L}} A; A; A ×_{LX} A; A]

is a central U(1)-extension of smooth groupoids. In particular, we see that the canonical map

ker(δ : Ω_{LX}^{·}(A_{LX}^{0}) → A_{LX}^{1})) → ker(δ : Ω_{LG}^{·}(A_{LG}^{0}) → A_{LG}^{1}))

is an isomorphism, i.e. we see that

Γ_{G_{L}}(Ω_{G_{L}}^{·}) ≅ Γ_{LX}(Ω_{LX}^{·}) ≅ Ω^{·}(LX) .  (3.14)

3.2.10 A connection on the gerbe f_{L} : G_{L} → LX consists of a pair (α, β), where α ∈ Ω^{1}(A ×_{G_{L}} A) is a connection one-form on the U(1)-bundle A ×_{G_{L}} A → A ×_{LX} A, and β ∈ Ω^{2}(A). We consider α ∈ Ω_{G_{L}}^{1}(A_{G_{L}}^{1}) and β ∈ Ω_{G_{L}}^{2}(A_{G_{L}}^{0}). The pair is a connection (α, β) if it satisfies:

(1) δβ = d_{dR}α ,

(2) δα = 0.

Note that δdβ = 0 so that there is a unique λ ∈ Γ_{G_{L}}Ω_{G_{L}}^{3} ≅^{(3.14)} Ω^{3}(LX) which restricts to dβ. We have dλ = 0. 3.2.11 Let use choose a connection (α, β), and let λ ∈ Ω^{3}(LX) be the associated closed three form. In 3.1.9 we have introduced the locally constant sheaf L on LX. The construction 3.2.8 gives the complex of sheaves

(Ω_{LX}^{·}(L)[[z]]_{λ} , d_{λ}) .

Furthermore we set

Ω^{·}(LX, L)[[z]]_{λ} := Γ_{LX}Ω_{LX}^{·}(L)[[z]]_{λ} .

Definition 3.15 The delocalized (G, λ)-twisted de Rham cohomology of X is defined by

H_{dR,deloc}^{∗}(X, (G, λ)) := H^{∗}(Ω^{·}(LX, L)[[z]]_{λ}, d_{λ}) .

In view of Lemma 3.11 this is the definition given in [TXa, 3.10]. Note that H_{dR,deloc}^{∗}(X, (G, λ)) depends on the choice of the connection, through these groups are isomorphic for different choices (see [TXa, 3.11]).

3.3 Comparison

3.3.1 In this subsection we prove

Theorem 3.16 There is an isomorphism

H_{deloc}^{∗}(X; G) ≅ H_{dR,deloc}^{∗}(X, (G, λ)) .

Actually, this theorem follows from the following stronger statement. Recall that f_{L} : G_{L} → LX is the pull-back of f : G → X via the canonical map LX → X. Let ℝ_{G_{L}} ∈ Sh_{Ab}G_{L} denote the constant sheaf with value ℝ.

Theorem 3.17 There is an isomorphism in D^{+}(Sh_{Ab}LX)

R(f_{L})_{∗}(ℝ_{G_{L}}) ⊗_{ℝ} L ≅ Ω_{LX}^{·}(L)[[z]]_{λ} .

The remainder of the present subsection is devoted to the proofs of Theorems 3.16 and 3.17.

3.3.2 Let us prove Theorem 3.17. First observe that Ω_{LX}^{·}(L)[[z]]_{λ} ≅ Ω_{LX}^{·}[[z]]_{λ} ⊗_{ℝ} L. Therefore it suffices to show that

R(f_{L})_{∗}(ℝ_{G_{L}}) ≅ Ω_{LX}^{·}[[z]]_{λ} .

This is exactly the assertion of [BSSc, Theorem 1.1], with the difference, that now LX is an orbifold instead of a smooth manifold. We repeat the proof of [BSSc, Theorem 1.1] given by [BSSc, Subsection 3.2] with the following modifications (the numbers refer to the paragraphs in [BSSc, Subsection 3.2]:

(1) 3.2.1 : The manifold X is replaced by the orbifold LX. The gerbe G → X is replaced by the gerbe G_{L} → LX. Furthermore, A → G_{L} is some atlas. It induces an atlas A → G_{L} → LX. The U(1)-central extension of groupoids (A ×_{G_{L}} A ⇒ A) → (A ×_{LX} A → A) represents a gerbe in the language of groupoids, but we can no longer refer to the paper [Hit01]. For existence of a connection we now refer to [TXa, Prop. 3.6].

(2) 3.2.2 : We use the notation Ω_{G_{L}} instead of Ω(G_{L}) for the de Rham complex of the smooth stack G_{L}. Ω^{·}(LX) must be interpreted as in 3.2.5. For the existence of connections we refer to [TXa]. The construction of the three-form associated to a connection (α, β) was explained in 3.2.10.

(3) 3.2.6 : We must show that the map ϕ : Ω^{·}[[z]]_{λ} → i^{♯}C_{A}(Ω^{·}(G_{L})) is a quasi-isomorphism. This can be shown locally. Since we can cover LX by smooth manifolds the local isomorphism immediately follows from the result proved in [BSSc]. This argument avoids repeating the proof of [BSSc, Proposition 3.4].

□

3.3.3 We now show Theorem 3.16. We need the following well-known fact. Let X be an orbispace or orbifold and p : X → ∗ be the projection. Recall that ev ∘ p_{∗} = Γ_{X} : Sh_{Ab}X → Ab. This functor is left exact and can thus be derived. Let O_{X} be the sheaf of continuous or smooth real functions on X, i.e. O_{X} = Ω_{X}^{0} in the smooth case.

Lemma 3.18 If F ∈ Sh_{Ab}X is a flabby sheaf and a sheaf of O_{X}-modules, then R^{i}Γ_{X}(F) = 0 for i ≥ 1.

Proof. Let A → X be an orbispace (orbifold) atlas. Then A ×_{X} A ⇒ A is a very proper, separated, and étale groupoid. Let A^{·} be the associated simplicial space (manifold). The complex F (A^{·}) represents RΓ_{X}(F) by [BSSc, Lemma 2.41]. We now employ the method of [BS, Section 4.1] in order to show that H^{i}(F (A^{·})) = 0 for i ≥ 1. We use the O_{X}-module structure in order to multiply by cut-off function.□

3.3.4 We first observe that Ω_{LX}^{·}(L)[[z]]_{λ} is a complex of flabby sheaves and of O_{LX} = Ω_{LX}^{0}-modules. Therefore by Lemmas 3.18 and 3.11 we have (see [Beh04, Cor. 25] for a related result)

RΓ_{LX}(Ω_{LX}^{·}(L)[[z]]_{λ}) ≅ Ω^{·}(LX, L)[[z]]_{λ} .

By Definition 3.15 the cohomology of the right-hand side is H_{deloc,dR}^{∗}(X, (G, λ)). On the other hand by Theorem 3.16

RΓ_{LX}(Ω_{LX}^{·}(L)[[z]]_{λ}) ≅ ev ∘ Rp_{∗} (R(f_{I})_{∗}(ℝ_{G_{L}}) ⊗_{ℝ} L) .

Its cohomology is by 3.1.13 isomorphic to H_{deloc}^{∗}(X; G). □

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