Principal \(\infty \)bundles: general theory
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Abstract
The theory of principal bundles makes sense in any \(\infty \)topos, such as the \(\infty \)topos of topological, of smooth, or of otherwise geometric \(\infty \)groupoids/\(\infty \)stacks, and more generally in slices of these. It provides a natural geometric model for structured higher nonabelian cohomology and controls general fiber bundles in terms of associated bundles. For suitable choices of structure \(\infty \)group \(G\) these \(G\)principal \(\infty \)bundles reproduce various higher structures that have been considered in the literature and further generalize these to a full geometric model for twisted higher nonabelian sheaf cohomology. We discuss here this general abstract theory of principal \(\infty \)bundles, observing that it is intimately related to the axioms that characterize \(\infty \)toposes. A central result is a natural equivalence between principal \(\infty \)bundles and intrinsic nonabelian cocycles, implying the classification of principal \(\infty \)bundles by nonabelian sheaf hypercohomology. We observe that the theory of geometric fiber \(\infty \)bundles associated to principal \(\infty \)bundles subsumes a theory of \(\infty \)gerbes and of twisted \(\infty \)bundles, with twists deriving from local coefficient \(\infty \)bundles, which we define, relate to extensions of principal \(\infty \)bundles and show to be classified by a corresponding notion of twisted cohomology, identified with the cohomology of a corresponding slice \(\infty \)topos.
Keywords
Nonabelian cohomology Higher topos theory Principal bundles1 Overview
The concept of a \(G\)principal bundle for a topological or Lie group \(G\) is fundamental in classical topology and differential geometry, e.g. [10]. More generally, for \(G\) a geometric group in the sense of a sheaf of groups over some site, the notion of \(G\)principal bundle or Gtorsor is fundamental in topos theory [12, 19]. Its relevance rests in the fact that \(G\)principal bundles constitute natural geometric representatives of cocycles in degree 1 nonabelian cohomology \(H^1(,G)\) and that general fiber bundles are associated to principal bundles.
In recent years it has become clear that various applications, notably in “Stringgeometry” [27, 28], involve a notion of principal bundles where geometric groups \(G\) are generalized to geometric grouplike \(A_\infty \)spaces, in other words geometric \(\infty \)groups: geometric objects that are equipped with a group structure up to higher coherent homotopy. The resulting principal \(\infty \)bundles should be natural geometric representatives of geometric nonabelian hypercohomology: Čech cohomology with coefficients in arbitrary positive degree.
In the absence of geometry, these principal \(\infty \)bundles are essentially just the classical simplicial principal bundles of simplicial sets [16] (this we discuss in Section 4.1 of [21]). However, in the presence of nontrivial geometry the situation is both more subtle and richer, and plain simplicial principal bundles can only serve as a specific presentation for the general notion (section 3.7.2 of [21]).
For the case of principal 2bundles, which is the first step after ordinary principal bundles, aspects of a geometric definition and theory have been proposed and developed by various authors, see section 1 of [21] for references and see [22] for a comprehensive discussion. Notably the notion of a bundle gerbe [20] is, when regarded as an extension of a Čechgroupoid, almost manifestly that of a principal 2bundle, even though this perspective is not prominent in the respective literature.
The oldest definition of geometric 2bundles is conceptually different, but closely related: Giraud’s Ggerbes [9] are by definition not principal 2bundles but are fiber 2bundles associated to \(\mathbf {Aut}(\mathbf {B}G)\)principal 2bundles, where \(\mathbf {B}G\) is the geometric moduli stack of \(G\)principal bundles. This means that \(G\)gerbes provide the universal local coefficients, in the sense of twisted cohomology, for \(G\)principal bundles.
From the definition of principal 2bundles/bundle gerbes it is fairly clear that these ought to be just the first step (or second step) in an infinite tower of higher analogs. Accordingly, definitions of principal 3bundles have been considered in the literature, mostly in the guise of bundle 2gerbes [31]. The older notion of Breen’s \(G\) 2gerbes [6] (also discussed by BrylinskiMacLaughlin), is, as before, not that of a principal 3bundle, but that of a fiber 3bundle which is associated to an \(\mathbf {Aut}(\mathbf {B}G)\)principal 3bundle, where now \(\mathbf {B}G\) is the geometric moduli 2stack of \(G\)principal 2bundles .
Generally, for every \(n \in \mathbb {N}\) and every geometric \(n\)group \(G\), it is natural to consider the theory of \(G\)principal \(n\)bundles twisted by an \(\mathbf {Aut}(\mathbf {B}G)\)principal \((n+1)\)bundle, hence by the associated Gngerbe. A complete theory of principal bundles therefore needs to involve the notion of principal \(n\)bundles and also that of twisted principal \(n\)bundles in the limit as \(n \rightarrow \infty \).
As \(n\) increases, the piecemeal conceptualization of principal \(n\)bundles quickly becomes tedious and their structure opaque, without a general theory of higher geometric structures. In recent years such a theory—long conjectured and with many precursors—has materialized in a comprehensive and elegant form, now known as \(\infty \)topos theory [13, 25, 32]. Whereas an ordinary topos is a category of sheaves over some site,^{1} an \(\infty \)topos is an \(\infty \)category of \(\infty \)sheaves or equivalently of \(\infty \)stacks over some \(\infty \)site, where the prefix “\(\infty \)” indicates that all these notions are generalized to structures up to coherent higher homotopy (as in the older terminology of \(A_\infty \), \(C_\infty \), \(E_\infty \) and \(L_\infty \)algebras, all of which reappear as algebraic structures in \(\infty \)topos theory). In as far as an ordinary topos is a context for general geometry, an \(\infty \)topos is a context for what is called higher geometry or derived geometry: the pairing of the notion of geometry with that of homotopy. (Here “derived” alludes to “derived category” and “derived functor” in homological algebra, but refers in fact to a nonabelian generalization of these concepts.) Therefore we may refer to objects of an \(\infty \)topos also as geometric homotopy types.
Topos theory is renowned for providing a general convenient context for the development of geometric structures. In some sense, \(\infty \)topos theory provides an even more convenient context, due to the fact that \(\infty \) (co)limits or homotopy (co)limits in an \(\infty \)topos exist, and refine the corresponding naive (co)limits. This convenience manifests itself in the central definition of principal \(\infty \)bundles (Definition 3.4 below): whereas the traditional definition of a \(G\)principal bundle over \(X\) as a quotient map \(P \rightarrow P/G \simeq X\) requires the additional clause that the quotient be locally trivial, \(\infty \)topos theory comes preinstalled with the correct homotopy quotient for higher geometry, and as a result the local triviality of \(P \rightarrow P/\!/G =: X\) is automatic; we discuss this in more detail in Sect. 3.1 below. Hence conceptually, \(G\)principal \(\infty \)bundles are in fact simpler than their traditional analogs, and so their theory is stronger.
When the internal Postnikov tower of a coefficient object is regarded as a sequence of local coefficient bundles as above, the induced twisted \(\infty \)bundles are decompositions of nonabelian principal \(\infty \)bundles into ordinary principal bundles together with equivariant abelian hypercohomology cocycles on their total spaces. This construction identifies much of equivariant cohomology theory as a special case of higher nonabelian cohomology. Specifically, when applied to a Postnikov stage of the delooping of an \(\infty \)group of internal automorphisms, the corresponding twisted cohomology reproduces the notion of Breen Ggerbes with band (Giraud’s liens); and the corresponding twisted \(\infty \)bundles are their incarnation as equivariant bundle gerbes over principal bundles.
The classification statements for principal and fiber \(\infty \)bundles in this article, Theorems 3.17 and 4.11 are not surprising, they say exactly what one would hope for. It is however useful to see how they flow naturally from the abstract axioms of \(\infty \)topos theory, and to observe that they immediately imply a series of classical as well as recent theorems as special cases, see Remark 4.12. Also the corresponding long exact sequences in (nonabelian) cohomology, Theorem 2.27, reproduce classical theorems, see Remark 2.28. Similarly the definition and classification of lifting of principal \(\infty \)bundles, Theorem 4.35, and of twisted principal \(\infty \)bundles in Theorem 4.39 flows naturally from the \(\infty \)topos theory and yet it immediately implies various constructions and results in the literature as special cases, see Remark 4.36 and Remark 4.40, respectively. In particular the notion of nonabelian twisted cohomology itself is elementary in \(\infty \)topos theory, Sect. 4.2, and yet it sheds light on a wealth of applications, see Remark 4.22.
This should serve to indicate that the theory of (twisted) principal \(\infty \)bundles is rich and interesting. The present article is intentionally written in general abstraction only, aiming to present the general theory of (twisted) principal \(\infty \)bundles as elegantly as possible, true to its roots in abstract higher topos theory. We believe that this serves to usefully make transparent the overall picture. In the companion article [21] we give a complementary discussion and construct explicit presentations of the structures appearing here that lend themselves to explicit computations.
2 Preliminaries
The discussion of principal \(\infty \)bundles in Sect. 3 below builds on the concept of an \(\infty \)topos and on a handful of basic structures and notions that are present in any \(\infty \)topos, in particular the notion of group objects and of cohomology with coefficients in these group objects. The relevant theory has been developed in [13, 15, 25, 32]. The purpose of this section is to recall the main aspects of this theory that we need, to establish our notation, and to highlight some aspects of the general theory that are relevant to our discussion and which have perhaps not been highlighted in this way in the existing literature.
One may reason about \(\infty \)categories in a modelindependent way, using the universal constructions that hold equivalently in all models—but the reader wishing to do so is invited to think specifically of the model given by quasicategories, due to Joyal and laid out in detail in [13]. Ordinary categories naturally embed into \(\infty \)categories; and in terms of quasicategories this embedding is given by sending a category to its simplicial nerve. In view of this we will freely regard 1categories as \(\infty \)categories—such as for instance the simplex category \(\Delta \). This allows us to define a simplicial object in an \(\infty \)category \(\mathcal {C}\) in a direct generalization of the usual notion as an \(\infty \)functor \(\Delta ^{\mathrm{op}} \rightarrow \mathcal {C}\) (note that this is now automatically a simplicial object “up to coherent higher homotopy”).
We are particularly concerned with those \(\infty \)categories that are \(\infty \)toposes. For many purposes the notion of \(\infty \)topos is best thought of as a generalization of the notion of a sheaf topos—the category of sheaves over some site is replaced by an \(\infty \)category of \(\infty \)stacks/ \(\infty \) sheaves over some \(\infty \)site (there is also supposed to be a more general notion of an elementary \(\infty \)topos, which however we do not consider here). In this context the \(\infty \)topos \(\mathrm{Gpd}_\infty \) of \(\infty \)groupoids is the natural generalization of the punctual topos \(\mathrm{Set}\) to \(\infty \)topos theory. A major achievement of [13, 25, 32] and was to provide a more intrinsic characterization of \(\infty \)toposes, which generalizes the classical characterization of sheaf toposes (Grothendieck toposes) originally given by Giraud. We will show that the theory of principal \(\infty \)bundles is naturally expressed in terms of these intrinsic properties, and therefore we here take these GiraudToënVezzosiRezkLurie axioms to be the very definition of an \(\infty \)topos ([13], Theorem 6.1.0.6, the main ingredients will be recalled below):
Definition 2.1
 1.Coproducts are disjoint. For every two objects \(A, B \in \mathbf {H}\), the intersection of \(A\) and \(B\) in their coproduct is the initial object: in other words the diagram is a pullback.
 2.Colimits are preserved by pullback. For all morphisms \(f:X\rightarrow B\) in \(\mathbf {H}\) and all small diagrams \(A:I\rightarrow \mathbf {H}_{/B}\), there is an equivalencebetween the pullback of the colimit and the colimit over the pullbacks of its components.$$\begin{aligned} \lim _{\mathop i\limits ^{\longrightarrow }} f^*A_i \simeq f^*(\lim _{\mathop i\limits ^{\longrightarrow }} A_i) \end{aligned}$$
 3.Quotient maps are effective epimorphisms. Every simplicial object \(A_\bullet : \Delta ^{\mathrm{op}} \rightarrow \mathbf {H}\) that satisfies the groupoidal Segal property (Definition 2.13) is the Čech nerve of its quotient projection:$$\begin{aligned} A_n \simeq A_0 \times _{\lim _{\mathop n\limits ^{\longrightarrow }} A_n} A_0 \times _{\lim _{\mathop n\limits ^{\longrightarrow }} A_n} \cdots \times _{\lim _{\mathop n\limits ^{\longrightarrow }} A_n} A_0 \;\;\; (n \text{ factors }). \end{aligned}$$
Repeated application of the second and third axiom provides the proof of the classification of principal \(\infty \)bundles, Theorem 3.17 and the universality of the universal associated \(\infty \)bundle, Proposition 4.6.
An ordinary topos is famously characterized by the existence of a classifier object for monomorphisms, the subobject classifier. With hindsight, this statement already carries in it the seed of the close relation between topos theory and bundle theory, for we may think of a monomorphism \(E \hookrightarrow X\) as being a bundle of (1)truncated fibers over \(X\). The following axiomatizes the existence of arbitrary universal bundles, providing a different but equivalent definition of \(\infty \)toposes.
Proposition 2.2
 1.
Colimits are preserved by pullback.
 2.There are universal \(\kappa \)small bundles. For every sufficiently large regular cardinal \(\kappa \), there exists a morphism \(\widehat{\mathrm{Obj}}_\kappa \rightarrow \mathrm{Obj}_\kappa \) in \(\mathbf {H}\), such that for every other object \(X\), pullback along morphisms \(X \rightarrow \mathrm{Obj}\) constitutes an equivalence^{2}between the \(\infty \)groupoid of bundles (morphisms) \(E \rightarrow X\) which are \(\kappa \)small over \(X\) and the \(\infty \)groupoid of morphisms from \(X\) into \(\mathrm{Obj}_\kappa \).$$\begin{aligned} \mathrm{Core}(\mathbf {H}_{/_{\kappa }X}) \simeq \mathbf {H}(X, \mathrm{Obj}_\kappa ) \end{aligned}$$
This is due to Rezk and Lurie, appearing as Theorem 6.1.6.8 in [13]. We find that this second version of the axioms naturally gives the equivalence between \(V\)fiber bundles and \(\mathbf {Aut}(V)\)principal \(\infty \)bundles in Proposition 4.10.
In addition to these axioms, a basic property of \(\infty \)toposes (and generally of \(\infty \)categories with pullbacks) which we will repeatedly invoke, is the following.
Proposition 2.3

all square diagrams are filled by a 2cell, even if we do not indicate this notationally;

all limits are \(\infty \)limits/homotopy limits (hence all pullbacks are \(\infty \)pullbacks/homotopy pullbacks), and so on;
Definition 2.4
Observation 2.5
Let \(f:Y\rightarrow Z\) in \(\mathbf {H}\) be as above. Suppose that \(Y\) is pointed and \(f\) is a morphism of pointed objects. Then the \(\infty \)fiber of an \(\infty \)fiber is the loop object of the base.
2.1 Epimorphisms and monomorphisms
In an \(\infty \)topos there is an infinite tower of notions of epimorphisms and monomorphisms: the \(n\)connected and \(n\)truncated morphisms for all \(2 \le n \le \infty \) [13, 25].
Definition 2.6
For \(n \in \mathbb {N}\) an \(\infty \)groupoid is called ntruncated (or: an \(n\)type) if all its homotopy groups in degree greater than \(n\) are trivial. It is called (1)truncated if it is either empty or contractible and (2)truncated if it is nonempty and contractible. An object in an arbitrary \(\infty \)category is \(n\)truncated for \(2 \le n < \infty \) if all hom\(\infty \)groupoids into it are \(n\)truncated. A morphism of an \(\infty \)category is called \(n\)truncated if it is so as an object in the slice over its codomain (which means internally that its homotopy fibers are \(n\)truncated). A \((1)\)truncated morphism is also called a monomorphism. The full embedding of the \(n\)truncated objects of an \(\infty \)topos is reflective, and the reflector \(\tau _{\le n}\) is called the \(n\)truncation operation.
This is the topic of section 5.5.6 in [13].
Remark 2.7
In a general \(\infty \)topos every object has (groups of) homotopy sheaves generalizing the homotopy groups for bare \(\infty \)groupoids. If one knows that an object \(X\) in an \(\infty \)topos is truncated at all (for some possibly large truncation degree) then it is \(n\)truncated if all its homotopy sheaves \(\pi _k(X)\) vanish in degree \(k > n\).
This is the content of Proposition 6.5.1.7 in [13].
Definition 2.8
Definition 2.9
For \(n \in \mathbb {N}\) a morphism in an \(\infty \)topos is called \(n\)connected if it is an effective epimorphism and all its homotopy sheaves are trivial in degree greater than \(n\) when it is regarded as an object in the slice \(\infty \)topos over its codomain. Any effective epimorphism is called \((1)\)connected. An object \(X\) is called \(n\)connected if the canonical morphism \(X \rightarrow *\) is \(n\)connected.
This is the topic of section 6.5.1 in [13].
Proposition 2.10
A morphism \(f : X \rightarrow Y\) in the \(\infty \)topos \(\mathbf {H}\) is an effective epimorphism precisely if its 0truncation \(\tau _0 f : \tau _0 X \rightarrow \tau _0 Y\) is an epimorphism (necessarily effective) in the 1topos \(\tau _{\le 0} \mathbf {H}\).
This is Proposition 7.2.1.14 in [13].
Proposition 2.11
The classes \(( \mathrm{Epi}(\mathbf {H}), \mathrm{Mono}(\mathbf {H}) )\) constitute an orthogonal factorization system.
This is Proposition 8.5 in [25] and Example 5.2.8.16 in [13].
Definition 2.12
2.2 Groupoids and groups
In any \(\infty \)topos \(\mathbf {H}\) we may consider groupoids internal to \(\mathbf {H}\), in the sense of internal category theory (as exposed for instance in the introduction of [14]).
Such a groupoid object \(\mathcal {G}\) in \(\mathbf {H}\) is an \(\mathbf {H}\)object \(\mathcal {G}_0\) “of \(\mathcal {G}\)objects” together with an \(\mathbf {H}\)object \(\mathcal {G}_1\) “of \(\mathcal {G}\)morphisms” equipped with source and target assigning morphisms \(s,t : \mathcal {G}_1 \rightarrow \mathcal {G}_0\), an identityassigning morphism \(i : \mathcal {G}_0 \rightarrow \mathcal {G}_1\) and a composition morphism \(\mathcal {G}_1 \times _{\mathcal {G}_0} \mathcal {G}_1 \rightarrow \mathcal {G}_1\) which together satisfy all the axioms of a groupoid (unitality, associativity, existence of inverses) up to coherent homotopy in \(\mathbf {H}\). One way to formalize what it means for these axioms to hold up to coherent homotopy is as follows.
After this preliminary discussion we state the following definition of groupoid object in an \(\infty \)topos (this definition appears in [13] as Definition 6.1.2.7, using Proposition 6.1.2.6).
Definition 2.13
The following example is fundamental. In fact the third \(\infty \)Giraud axiom says that up to equivalence, all groupoid objects are of this form.
Example 2.14
For \(X \rightarrow Y\) any morphism in \(\mathbf {H}\), the Čech nerve \(\check{C}(X\rightarrow Y)\) of \(X\rightarrow Y\) (Definition 2.8) is a groupoid object. This example appears in [13] as Proposition 6.1.2.11.
The following statement refines the third \(\infty \)Giraud axiom, Definition 2.1.
Theorem 2.15
This appears below Corollary 6.2.3.5 in [13].
In addition, every \(\infty \)topos \(\mathbf {H}\) comes with a notion of \(\infty \)group objects that generalize both the ordinary notion of group objects in a topos as well as that of grouplike \(A_\infty \)spaces in \(\mathrm{Grpd}_{\infty }\).
There is an evident definition of what an \(\infty \)group object in \(\mathbf {H}\) should be, and then there is a theorem saying that this is equivalent to a certain kind of simplicial object in \(\mathbf {H}\). This theorem is part of what, we find, makes the theory of groups, group actions and principal bundles in an \(\infty \)topos be so well behaved, and we will mostly work with this simplicial incarnation of group objects. But the evident definition that the reasoning starts with is of course this: a group object is an object which is equipped with an associative and unital product operation such that for each element there is an inverse. Now in the homotopytheoretic context of \(\infty \)topos theory an associative unital structure means an associative unital structure up to coherent homotopy and the technical term for this is \(A_\infty \)structure, famous from the theory of loop spaces, see [15] for a comprehensive modern account. Moreover, statements about elements here are supposed to be statements about connected components, and hence we ask for such \(A_\infty \) structures such that on connected components the product operation is invertible (such \(A_\infty \)structures are traditionally also called “groupal” or “grouplike”).
Therefore the manifest definition of \(\infty \)group objects in \(\mathbf {H}\) is the following (this appears as Definition 5.1.3.2 together with Remark 5.1.3.3 in [15]).
Definition 2.16
An \(\infty \)group in \(\mathbf {H}\) is an \(A_\infty \)algebra \(G\) in \(\mathbf {H}\) such that the sheaf of connected components \(\pi _0(G)\) is a group object in \(\tau _{\le 0} \mathbf {H}\). Write \(\mathrm{Grp}(\mathbf {H})\) for the \(\infty \)category of \(\infty \)groups in \(\mathbf {H}\).
As in classical algebraic topology, the fundamental examples of such \(\infty \)groups arise from forming loops, and there is a central delooping theorem saying that, up to equivalence, in fact all \(\infty \)groups arise this way:
Definition 2.17

\(\mathbf {H}^{*/}\) for the \(\infty \)category of pointed objects in \(\mathbf {H}\);

\(\mathbf {H}_{\ge 1}\) for the full sub\(\infty \)category of \(\mathbf {H}\) on the connected objects;

\(\mathbf {H}^{*/}_{\ge 1}\) for the full sub\(\infty \)category of the pointed objects on the connected objects.
Definition 2.18
Theorem 2.19
This is Lemma 7.2.2.1 in [13]. (See also Theorem 5.1.3.6 of [15] where this is the equivalence denoted \(\phi _0\) in the proof.) For \(\mathbf {H} = \mathrm{Grpd}_{\infty }\) this reduces to various classical theorems in homotopy theory, for instance the construction of classifying spaces (Kan and Milnor) and delooping theorems (May and Segal).
Definition 2.20
We call the inverse \(\mathbf {B} : \mathrm{Grp}(\mathbf {H}) \rightarrow \mathbf {H}^{*/}_{\ge 1}\) in Theorem 2.19 above the delooping functor of \(\mathbf {H}\). By convenient abuse of notation we write \(\mathbf {B}\) also for the composite \(\mathbf {B} : \mathrm{Grpd}(\mathbf {H}) \rightarrow \mathbf {H}^{*/}_{\ge 1} \rightarrow \mathbf {H}\) with the functor that forgets the basepoint and the connectivity.
Remark 2.21
Even if the connected objects involved admit an essentially unique point, the homotopy type of the full hom\(\infty \)groupoid \(\mathbf {H}^{*/}(\mathbf {B}G, \mathbf {B}H)\) of pointed objects in general differs from the hom \(\infty \)groupoid \(\mathbf {H}(\mathbf {B}G, \mathbf {B}H)\) of the underlying unpointed objects. For instance let \(\mathbf {H} := \mathrm{Grpd}_{\infty }\) and let \(G\) be an ordinary group, regarded as a group object in \(\mathrm{Grpd}_{\infty }\). Then \(\mathbf {H}^{*/}(\mathbf {B}G, \mathbf {B}G) \simeq \mathrm{Aut}(G)\) is the ordinary automorphism group of \(G\), but \(\mathbf {H}(\mathbf {B}G, \mathbf {B}G) = \mathrm{Aut}(\mathbf {B}G)\) is the automorphism 2group of \(G\), we discuss this further around Example 4.50 below.
Now observe that for \(X\) a pointed connected object, then the point inclusion \(*\rightarrow X\) is an effective epimorphism and the loop space object \(\Omega X\) in def. 2.18 is the first stage of the corresponding Čech nerve, as in the discussion of groupoid objects above in 2.2. This suggests that, moreover, group objects in \(\mathbf {H}\) should be equivalent to those groupoid objects whose degree0 piece is equivalent to the point. This is indeed the case, and this is central to the development of our discussion:
Proposition 2.22
(Lurie). \(\infty \)groups \(G\) in \(\mathbf {H}\) are equivalently those groupoid objects \(\mathcal {G}\) in \(\mathbf {H}\) (Definition 2.13) for which \(\mathcal {G}_0 \simeq *\).
This is the statement of the compound equivalence \(\phi _3\phi _2\phi _1\) in the proof of Theorem 5.1.3.6 in [15].
Remark 2.23
2.3 Cohomology
There is an intrinsic notion of cohomology in every \(\infty \)topos \(\mathbf {H}\): it is simply given by the connected components of mapping spaces. Of course such mapping spaces exist in every \(\infty \)category, but we need some extra conditions on \(\mathbf {H}\) in order for them to behave like cohomology sets. For instance, if \(\mathbf {H}\) has pullbacks then there is a notion of long exact sequences in cohomology. Our main theorem (Theorem 3.17 below) will show that the second and third \(\infty \)Giraud axioms imply that this intrinsic notion of cohomology has the property that it classifies certain geometric structures in the \(\infty \)topos.
Definition 2.24

the homspace \(\mathbf {H}(X,A)\) is the cocycle \(\infty \)groupoid;

an object \(g : X \rightarrow A\) in \(\mathbf {H}(X,A)\) is a cocycle;

a morphism: \(g \Rightarrow h\) in \(\mathbf {H}(X,A)\) is a coboundary between cocycles.
 a morphism \(c : A \rightarrow B\) in \(\mathbf {H}\) represents the universal characteristic class (cohomology operation)$$\begin{aligned}{}[c] : H^0(,A) \rightarrow H^0(,B). \end{aligned}$$
Remark 2.25
Definition 2.26
We have the following basic fact.
Theorem 2.27
 1.
In the long fiber sequence to the left of \(c : \mathbf {B}G \rightarrow \mathbf {B}H\) after \(n\) iterations all terms are equivalent to the point if \(H\) and \(G\) are \(n\)truncated.
 2.For every object \(X \in \mathbf {H}\) we have a long exact sequence of pointed cohomology sets$$\begin{aligned} \cdots \rightarrow H^0(X,G) \rightarrow H^0(X,H) \rightarrow H^1(X,F) \rightarrow H^1(X,G) \rightarrow H^1(X,H). \end{aligned}$$
Remark 2.28
3 Principal bundles
We define here \(G\)principal \(\infty \)bundles in any \(\infty \)topos \(\mathbf {H}\), discuss their basic properties and show that they are classified by the intrinsic \(G\)cohomology in \(\mathbf {H}\), as discussed in Definition 2.24.
3.1 Introduction and survey
As motivation for this, notice that if a Lie group \(G\) acts properly, but not freely, then the quotient \(P \rightarrow X := P/G\) differs from the homotopy quotient, which instead is locally the quotient stack by the nonfree part of the group action (an orbifold, if the stabilizers are finite).
Conversely this means that in the context of higher geometry a nonfree action may also be principal: with respect not to a base space, but with respect to a base groupoid/stack. In the example just discussed, we have that the projection \(P \rightarrow X/\!/ G_{\mathrm{stab}}\) exhibits \(P\) as a \(G\)principal bundle over the action groupoid \(P /\!/ G \simeq X/\!/ G_{\mathrm{stab}}\). For instance if \(P = V\) is a vector space equipped with a \(G\)representation, then \(V \rightarrow V/\!/ G\) is a \(G\)principal bundle over a groupoid/stack. In other words, the traditional requirement of freeness in a principal action is not so much a characterization of principality as such, as rather a condition that ensures that the base of a principal action is a 0truncated object in higher geometry.
Beyond this specific class of 0truncated examples, this means that we have the following noteworthy general statement: in higher geometry every \(\infty \)action is principal with respect to some base, namely with respect to its \(\infty \)quotient.
More is true: in the context of an \(\infty \)topos every \(\infty \)quotient projection of an \(\infty \)group action is locally trivial, with respect to the canonical intrinsic notion of cover, hence of locality. Therefore also the condition of local triviality in the classical definition of principality becomes automatic. In fact, from the \(\infty \)Giraud axioms, we see that the projection map \(P \rightarrow P /\!/ G\) is always a cover (an effective epimorphism) and so, since every \(G\)principal \(\infty \)bundle trivializes over itself, it exhibits a local trivialization of itself; even without explicitly requiring it to be locally trivial.
As before, this means that the local triviality clause appearing in the traditional definition of principal bundles is not so much a characteristic of principality as such, as rather a condition that ensures that a given quotient taken in a category of geometric spaces coincides with the “refined” quotient obtained when regarding the situation in the ambient \(\infty \)topos.
\(\infty \)Giraud axioms  Principal \(\infty \)bundle theory 

quotients are effective  every \(\infty \)quotient \(P \rightarrow X := P/\!/ G\) 
is principal  
colimits are preserved by pullback  \(G\)principal \(\infty \)bundles 
are classified by \(\mathbf {H}(X,\mathbf {B}G)\) 
3.2 Definition and classification
Definition 3.1
Remark 3.2
Remark 3.3
We list examples of \(\infty \)actions below as Example 4.13. This is most conveniently done after establishing the theory of principal \(\infty \)actions, to which we now turn.
Definition 3.4
 1.
a morphism \(P \rightarrow X\) in \(\mathbf {H}\);
 2.
together with a \(G\)action on \(P\);
Remark 3.5
By the third \(\infty \)Giraud axiom (Definition 2.1) this means in particular that a \(G\)principal \(\infty \)bundle \(P \rightarrow X\) is an effective epimorphism in \(\mathbf {H}\).
Remark 3.6
Even though \(G \mathrm{Bund}(X)\) is by definition a priori an \(\infty \)category, Proposition 3.16 below says that in fact it happens to be an \(\infty \)groupoid: all its morphisms are invertible.
Proposition 3.7
Proof
By the third \(\infty \)Giraud axiom (Definition 2.1) the groupoid object \(P/\!/G\) is effective, which means that it is equivalent to the Čech nerve of \(P \rightarrow X\). In first degree this implies a canonical equivalence \(P \times G \rightarrow P \times _X P\). Since the two face maps \(d_0, d_1 : P \times _X P \rightarrow P\) in the Čech nerve are simply the projections out of the fiber product, it follows that the two components of this canonical equivalence are the two face maps \(d_0, d_1 : P \times G \rightarrow P\) of \(P/\!/G\). By definition, these are the projection onto the first factor and the action itself. \(\square \)
Proposition 3.8
For \(g : X \rightarrow \mathbf {B}G\) any morphism, its homotopy fiber \(P \rightarrow X\) canonically carries the structure of a \(G\)principal \(\infty \)bundle over \(X\).
Proof
Lemma 3.9
Proof
Definition 3.10
The trivial \(G\)principal \(\infty \)bundle \((P \rightarrow X) \simeq (X \times G \rightarrow X)\) is, up to equivalence, the one obtained via Proposition 3.8 from the morphism \(X \rightarrow * \rightarrow \mathbf {B}G\).
Proposition 3.11
For \(P \rightarrow X\) a \(G\)principal \(\infty \)bundle and \(Y \rightarrow X\) any morphism, the \(\infty \)pullback \(Y \times _X P\) naturally inherits the structure of a \(G\)principal \(\infty \)bundle.
Proof
This uses the same kind of argument as in Proposition 3.8. \(\square \)
In fact this is the special case of the pullback of what we will see below in Proposition 3.13 is the universal \(G\)principal \(\infty \)bundle \(*\rightarrow \mathbf {B}G\).
Proposition 3.12
Proof
For \(P \rightarrow X\) a \(G\)principal \(\infty \)bundle, it is, by Remark 3.5, itself an effective epimorphism. The pullback of the \(G\)bundle to its own total space along this morphism is trivial, by the principality condition (Proposition 3.7). Hence setting \(U := P\) proves the claim. \(\square \)
Proposition 3.13
Proof
Example 3.14
Below in Theorem 4.39 this relation is strengthened: every automorphism of a \(G\)principal \(\infty \)bundle, and in fact its full automorphism \(\infty \)group arises from pullback of the above universal \(G\)principal \(\infty \)bundle: \(\mathbf {B}G\) is the fine moduli \(\infty \)stack of \(G\)principal \(\infty \)bundles.
The traditional definition of universal \(G\)principal bundles in terms of contractible objects equipped with a free \(G\)action has no intrinsic meaning in higher topos theory. Instead this appears in presentations of the general theory in model categories (or categories of fibrant objects) as fibrant representatives \(\mathbf {E}G \rightarrow \mathbf {B}G\) of the above point inclusion. This we discuss in [21].
The main classification Theorem 3.17 below implies in particular that every morphism in \(G\mathrm{Bund}(X)\) is an equivalence. For emphasis we note how this also follows directly:
Lemma 3.15
Let \(\mathbf {H}\) be an \(\infty \)topos and let \(X\) be an object of \(\mathbf {H}\). A morphism \(f:A\rightarrow B\) in \(\mathbf {H}_{/X}\) is an equivalence if and only if \(p^*f\) is an equivalence in \(\mathbf {H}_{/Y}\) for any effective epimorphism \(p:Y\rightarrow X\) in \(\mathbf {H}\).
Proof
Proposition 3.16
Every morphism between \(G\)actions over \(X\) that are \(G\)principal \(\infty \)bundles over \(X\) is an equivalence.
Proof
Since a morphism of \(G\)principal bundles \(P_1 \rightarrow P_2\) is a morphism of Čech nerves that fixes their \(\infty \)colimit \(X\), up to equivalence, and since \(* \rightarrow \mathbf {B}G\) is an effective epimorphism, we are, by Proposition 3.13, in the situation of Lemma 3.15. \(\square \)
Theorem 3.17

\(\mathbf {B}G\) is the classifying object or moduli \(\infty \)stack for \(G\)principal \(\infty \)bundles;

a morphism \(c : X \rightarrow \mathbf {B}G\) is a cocycle for the corresponding \(G\)principal \(\infty \)bundle and its class \([c] \in \mathrm{H}^1(X,G)\) is its characteristic class.
Proof
Corollary 3.18
Proof
By Definition 2.24 this is the restriction of the equivalence \(G \mathrm{Bund}(X) \simeq \mathbf {H}(X, \mathbf {B}G)\) to connected components. \(\square \)
4 Twisted bundles and twisted cohomology
We show here how the general notion of cohomology in an \(\infty \)topos, considered above in Sect. 2.3, subsumes the notion of twisted cohomology and we discuss the corresponding geometric structures classified by twisted cohomology: extensions of principal \(\infty \)bundles and twisted \(\infty \)bundles.
Whereas ordinary cohomology is given by a derived hom\(\infty \)groupoid, twisted cohomology is given by the \(\infty \)groupoid of sections of a local coefficient bundle in an \(\infty \)topos, which in turn is an associated \(\infty \)bundle induced via a representation of an \(\infty \)group \(G\) from a \(G\)principal \(\infty \)bundle (this is a geometric and unstable variant of the picture of twisted cohomology developed in [1, 17]).
It is fairly immediate that, given a universal local coefficient bundle associated to a universal principal \(\infty \)bundle, the induced twisted cohomology is equivalently ordinary cohomology in the corresponding slice \(\infty \)topos. This identification provides a clean formulation of the contravariance of twisted cocycles. However, a universal coefficient bundle is a pointed connected object in the slice \(\infty \)topos only when it is a trivial bundle, so that twisted cohomology does not classify principal \(\infty \)bundles in the slice. We show below that instead it classifies twisted principal \(\infty \)bundles, which are natural structures that generalize the twisted bundles familiar from twisted Ktheory. Finally, we observe that twisted cohomology in an \(\infty \)topos equivalently classifies extensions of structure groups of principal \(\infty \)bundles.
A wealth of structures turn out to be special cases of nonabelian twisted cohomology and of twisted principal \(\infty \)bundles and their study benefits from the general theory of twisted cohomology.
4.1 Actions and associated \(\infty \)bundles
Let \(\mathbf {H}\) be an \(\infty \)topos, \(G \in \mathrm{Grp}(\mathbf {H})\) an \(\infty \)group. Fix an action \(\rho : V \times G \rightarrow V\) on an object \(V\in \mathbf {H}\) as in Definition 3.1. We discuss the induced notion of \(\rho \)associated \(V\) fiber \(\infty \)bundles. We show that there is a universal \(\rho \)associated \(V\)fiber bundle over \(\mathbf {B}G\) and observe that under Theorem 3.17 this is effectively identified with the action itself. Accordingly, we also further discuss \(\infty \)actions as such.
Definition 4.1
We say that \(E \rightarrow X\) locally trivializes with respect to \(U\). As usual, we often say \(V\)bundle for short.
Definition 4.2
Remark 4.3
Example 4.4
Lemma 4.5
Proof
Proposition 4.6
Proof
Remark 4.7
This says that Open image in new window is both, the \(V\)fiber \(\infty \)bundle \(\rho \)associated to the universal \(G\)principal \(\infty \)bundle, Example 4.4, as well as the universal \(\infty \)bundle for \(\rho \)associated \(\infty \)bundles.
Proposition 4.8
Every \(\rho \)associated \(\infty \)bundle is a \(V\)fiber \(\infty \)bundle, Definition 4.1.
Proof
So far this shows that every \(\rho \)associated \(\infty \)bundle is a \(V\)fiber bundle. We want to show that, conversely, every \(V\)fiber bundle is associated to a principal \(\infty \)bundle.
Definition 4.9
Proposition 4.10
Every \(V\)fiber \(\infty \)bundle is \(\rho _{\mathbf {Aut}(V)}\)associated to an \(\mathbf {Aut}(V)\)principal \(\infty \)bundle.
Proof
Theorem 4.11
\(V\)fiber \(\infty \)bundles over \(X \in \mathbf {H}\) are classified by \(H^1(X, \mathbf {Aut}(V))\).
Under this classification, the \(V\)fiber \(\infty \)bundle corresponding to \([g] \in H^1(X, \mathbf {Aut}(V))\) is identified, up to equivalence, with the \(\rho _{\mathbf {Aut}(V)}\)associated \(\infty \)bundle (as in Definition 4.2) to the \(\mathbf {Aut}(V)\)principal \(\infty \)bundle corresponding to \([g]\) by Theorem 3.17.
Proof
Remark 4.12
In the special case that \(\mathbf {H} = \mathrm{Grpd}_{\infty }\), the classification Theorem 4.11 is classical [16, 30], traditionally stated in (what in modern terminology is) the presentation of \(\mathrm{Grpd}_{\infty }\) by simplicial sets or by topological spaces. Recent discussions include [3]. For \(\mathbf {H}\) a general 1localic \(\infty \)topos (meaning: with a 1site of definition), the statement of Theorem 4.11 appears in [34], formulated there in terms of the presentation of \(\mathbf {H}\) by simplicial presheaves. (We discuss the relation of these presentations to the above general abstract result in [21].) Finally, one finds that the classification of \(G\)gerbes [9] and \(G\)2gerbes in [6] is the special case of the general statement, for \(V = \mathbf {B}G\) and \(G\) a 1truncated \(\infty \)group. This we discuss below in Sect. 4.4.
We close this section with a list of some fundamental classes of examples of \(\infty \)actions, or equivalently, by Remark 4.7, of universal associated \(\infty \)bundles. For doing so we use again that, by Theorem 3.17, to give an \(\infty \)action of \(G\) on \(V\) is equivalent to giving a fiber sequence of the form \(V \rightarrow V/\!/G \rightarrow \mathbf {B}G\). Therefore the following list mainly serves to associate a traditional name with a given \(\infty \)action.
Example 4.13
 1.For every \(V \in \mathbf {H}\), the fiber sequence is the trivial \(\infty \)action of \(G\) on \(V\).
 2.For every \(G \in \mathrm{Grp}(\mathbf {H})\), the fiber sequence which defines \(\mathbf {B}G\) by Theorem 2.19 induces the right action of \(G\) on itselfAt the same time this sequence, but now regarded as a bundle over \(\mathbf {B}G\), is the universal \(G\)principal \(\infty \)bundle, Remark 3.14.$$\begin{aligned} * \simeq G/\!/G. \end{aligned}$$
 3.For every object \(X \in \mathbf {H}\) writefor its free loop space object, the \(\infty \)fiber product of the diagonal on \(X\) along itself For every \(G \in \mathrm{Grp}(\mathbf {H})\) there is a fiber sequence This exhibits the adjoint action of G on itself$$\begin{aligned} \mathbf {L}X := X \times _{X \times X} X \end{aligned}$$$$\begin{aligned} \mathbf {L}\mathbf {B}G \simeq G/\!/_{\mathrm{ad}} G. \end{aligned}$$
 4.For every \(V \in \mathbf {H}\) there is the canonical \(\infty \)action of the automorphism \(\infty \)group introduced in Definition 4.9, this exhibits the automorphism action.
 5.For \(\rho _1, \rho _2 \in \mathbf {H}_{/\mathbf {B}G}\) two \(G\)\(\infty \)actions on objects \(V_1, V_2 \in \mathbf {H}\), respectively, their internal hom \([\rho _1, \rho _2] \in \mathbf {H}_{/\mathbf {B}G}\) in the slice over \(\mathbf {B}G\) is a \(G\)\(\infty \)action on the internal hom \([V_1, V_2] \in \mathbf {H}\): hence \([V_1, V_2]/\!/G \simeq \sum _{\mathbf {B}G}[\rho _1, \rho _2]\), where \(\sum _{\mathbf {B}G} : \mathbf {H}_{/\mathbf {B}G} \rightarrow \mathbf {H}\) is the left adjoint to pullback along the terminal map. (This follows by the fact that the inverse image of base change along \(\mathrm{pt}_{\mathbf {B}G} : * \rightarrow \mathbf {B}G\) is a cartesian closed \(\infty \)functor and hence preserves internal homs.^{4}) This is the conjugation \(\infty \)action of \(G\) on morphisms \(V_1 \rightarrow V_2\) by pre and postcomposition with the action of \(G\) on \(V_1\) and \(V_2\), respectively.
4.2 Sections and twisted cohomology
We discuss a general notion of twisted cohomology or cohomology with local coefficients in any \(\infty \)topos \(\mathbf {H}\), where the local coefficient \(\infty \)bundles are associated \(\infty \)bundles as discussed above, and where the cocycles are sections of these local coefficient bundles.
Definition 4.14
We record two elementary but important lemmas about spaces of sections.
Lemma 4.15
Proof
For instance by Proposition 5.5.5.12 in [13]. \(\square \)
Lemma 4.16
Proof
Proposition 4.17
Proof
By Lemma 4.15 and Lemma 4.16. \(\square \)
Corollary 4.18
Proof
Proposition 4.19
Proof
Remark 4.20
Since by Proposition 3.12 every cocycle \(g_X\) trivializes locally over some cover Open image in new window and equivalently, by Proposition 4.8, every \(\infty \)bundle \(P \times _G V\) trivializes locally, Proposition 4.19 says that elements \( \sigma \in \Gamma _X(P \times _G V) \simeq \mathbf {H}_{/\mathbf {B}G}(g_X, \mathbf {c})\) locally are morphisms \(\sigma _U : U \rightarrow V\) with values in \(V\). They fail to be so globally to the extent that \([g_X] \in H^1(X, G)\) is nontrivial, hence to the extent that \(P \times _G V \rightarrow X\) is nontrivial.
This motivates the following definition.
Definition 4.21
We say that the \(\infty \)groupoid \(\Gamma _X(P \times _G V) \simeq \mathbf {H}_{/\mathbf {B}G}(g_X, \mathbf {c})\) from Proposition 4.17 is the \(\infty \)groupoid of \([g_X]\)twisted cocycles with values in \(V\), with respect to the local coefficient \(\infty \)bundle \(V/\!/G \mathop {\rightarrow }\limits ^{\mathbf {c}} \mathbf {B}G\).
Remark 4.22
The perspective that twisted cohomology is the theory of sections of associated bundles whose fibers are classifying spaces is maybe most famous for the case of twisted Ktheory, where it was described in this form in [26]. But already the old theory of ordinary cohomology with local coefficients is of this form, as is made manifest in [8].
A proposal for a comprehensive theory in terms of bundles of topological spaces is in [17] and a systematic formulation in \(\infty \)category theory and for the case of multiplicative generalized cohomology theories is in [1]. The formulation above refines this, unstably, to geometric cohomology theories/(nonabelian) sheaf hypercohomology, hence from bundles of classifying spaces to \(\infty \)bundles of moduli \(\infty \)stacks.
A wealth of examples and applications of such geometric nonabelian twisted cohomology of relevance in quantum field theory and in string theory is discussed in [27, 28].
Example 4.23
Remark 4.24
In this notation the local coefficient bundle \(\mathbf {c}\) is left implicit. This convenient abuse of notation is justified to some extent by the fact that there is a universal local coefficient bundle:
Example 4.25
4.3 Extensions and twisted bundles
We discuss the notion of extensions of \(\infty \)groups (see Sect. 2.2), generalizing the traditional notion of group extensions. This is in fact a special case of the notion of principal \(\infty \)bundle, Definition 3.4, for base space objects that are themselves deloopings of \(\infty \)groups. For every extension of \(\infty \)groups, there is the corresponding notion of lifts of structure \(\infty \)groups of principal \(\infty \)bundles. These are classified equivalently by trivializations of an obstruction class and by the twisted cohomology with coefficients in the extension itself, regarded as a local coefficient \(\infty \)bundle.
Moreover, we show that principal \(\infty \)bundles with an extended structure \(\infty \)group are equivalent to principal \(\infty \)bundles with unextended structure \(\infty \)group but carrying a principal \(\infty \)bundle for the extending \(\infty \)group on their total space, which on fibers restricts to the given \(\infty \)group extension. We formalize these twisted (principal) \(\infty \)bundles and observe that they are classified by twisted cohomology, Definition 4.21.
Definition 4.26
Remark 4.27
Definition 4.28
 1.
a lift of the defining groupal \(A_\infty \simeq E_1\)action to an \(E_2\)action;
 2.
a group structure on the delooping \(\mathbf {B}A\);
 3.
a double delooping \(\mathbf {B}^2 A\).
Remark 4.29
The equivalence of the items in Definition 4.28 is essentially the content of theorem 5.1.3.6 in [15].
Definition 4.30
Example 4.31
An ordinary group (1group) \(A\) that is braided is already abelian (by the EckmannHilton argument). In this case a braidedcentral extension as above of a 1group \(G\) is a central extension of \(G\) in the traditional sense.
Definition 4.32
Observation 4.33
This is implied by Theorem 4.35, to which we turn after introducing the following terminology.
Definition 4.34
In the above situation, we call \([\mathbf {c}(g_X)]\) the obstruction class to the extension; and we call \([\mathbf {c}] \in H^2(\mathbf {B}G, A)\) the universal obstruction class of extensions through \(\mathbf {p}\).
We give now three different characterizations of spaces of extensions of \(\infty \)bundles. The first two, by spaces of twisted cocycles and by spaces of trivializations of the obstruction class, are immediate consequences of the previous discussion:
Theorem 4.35
 1.There is a natural equivalencebetween the \(\infty \)groupoid of lifts of \(P\) through \(\mathbf {p}\), Definition 4.32, and the \(\infty \)groupoid of trivializations of the obstruction class, Definition 4.34.$$\begin{aligned} \mathrm{Lift}(P, \mathbf {p}) \simeq \mathrm{Triv}(\mathbf {c}(g_X)) \end{aligned}$$
 2.There is a natural equivalence \(\mathrm{Lift}(P, \mathbf {p}) \simeq \mathbf {H}_{/\mathbf {B}G}(g_X, \mathbf {p})\) between the \(\infty \)groupoid of lifts and the \(\infty \)groupoid of \(g_X\)twisted cocycles relative to \(\mathbf {p}\), Definition 4.21, hence a classificationof equivalence classses of lifts by the \([g_X]\)twisted \(A\)cohomology of \(X\) relative to the local coefficient bundle$$\begin{aligned} \pi _0 \mathrm{Lift}(P, \mathbf {P}) \simeq H^{1,[g_X]}(X, A) \end{aligned}$$
Proof
The first statement is the special case of Lemma 4.16 where the \(\infty \)pullback \(E_1 \simeq f^* E_2\) in the notation there is identified with \(\mathbf {B}\hat{G} \simeq \mathbf {c}^* {*}\). The second is evident after unwinding the definitions. \(\square \)
Remark 4.36
For the special case that \(A\) is 0truncated, we may, by the discussion in [22] identify \(\mathbf {B}A\)principal \(\infty \)bundles with \(A\)bundle gerbes, [20]. Under this identification the \(\infty \)bundle classified by the obstruction class \([\mathbf {c}(g_X)]\) above is what is called the lifting bundle gerbe of the lifting problem, see for instance [4] for a review. In this case the first item of Theorem 4.35 reduces to Theorem 2.1 in [33] and Theorem A (5.2.3) in [23]. The reduction of this statement to connected components, hence the special case of Observation 4.33, was shown in [5].
While, therefore, the discussion of extensions of \(\infty \)groups and of lifts of structure \(\infty \)groups is just a special case of the discussion in the previous sections, this special case admits geometric representatives of cocycles in the corresponding twisted cohomology by twisted principal \(\infty \)bundles. This we turn to now.
Definition 4.37
Given an extension of \(\infty \)groups Open image in new window and given a \(G\)principal \(\infty \)bundle \(P \rightarrow X\), with class \([g_X] \in H^1(X,G)\), a \([g_X]\)twisted Aprincipal \(\infty \)bundle on \(X\) is an \(A\)principal \(\infty \)bundle \(\hat{P} \rightarrow P\) such that the cocycle \(q : P \rightarrow \mathbf {B}A\) corresponding to it under Theorem 3.17 is a morphism of \(G\)\(\infty \)actions.
Proposition 4.38
 1.
\(\hat{P} \rightarrow P\) is a \([g_X]\)twisted \(A\)principal bundle, Definition 4.37;
 2.
for all points \(x : * \rightarrow X\) the restriction of \(\hat{P} \rightarrow P\) to the fiber \(P_x\) is canonically equivalent to the \(\infty \)group extension \(\hat{G} \rightarrow G\).
Proof
This follows from repeated application of the pasting law for \(\infty \)pullbacks, Proposition 2.3.
The bottom composite \(g : X \rightarrow \mathbf {B}G\) is a cocycle for the given \(G\)principal \(\infty \)bundle \(P \rightarrow X\) and it factors through \(\hat{g} : X \rightarrow \mathbf {B}\hat{G}\) by assumption of the existence of the extension \(\hat{P} \rightarrow P\).
Since also the bottom right square is an \(\infty \)pullback by the given \(\infty \)group extension, the pasting law asserts that the square over \(\hat{g}\) is also an \(\infty \)pullback, and then that so is the square over \(q\). This exhibits \(\hat{P}\) as an \(A\)principal \(\infty \)bundle over \(P\) classified by the cocycle \(q\) on \(P\). By Corollary 4.18 this \(\hat{P} \rightarrow P\) is twisted \(G\)equivariant.
Now choose any point \(x : {*} \rightarrow X\) of the base space as on the left of the diagram. Pulling this back upwards through the diagram and using the pasting law and the definition of loop space objects \(G \simeq \Omega \mathbf {B}G \simeq * \times _{\mathbf {B}G} *\) the diagram completes by \(\infty \)pullback squares on the left as indicated, which proves the claim. \(\square \)
Theorem 4.39
Proof
Remark 4.40
Aspects of special cases of this theorem can be identified in the literature. For the special case of ordinary extensions of ordinary Lie groups, the equivalence of the corresponding extensions of a principal bundle with certain equivariant structures on its total space is essentially the content of [2, 18]. In particular the twisted unitary bundles or gerbe modules of twisted Ktheory [4] are equivalent to such structures.
For the case of \(\mathbf {B}U(1)\)extensions of Lie groups, such as the \(\mathrm{String}\)2group, the equivalence of the corresponding \(\mathrm{String}\)principal 2bundles, by the above theorem, to certain bundle gerbes on the total spaces of principal bundles underlies constructions such as in [24]. Similarly the bundle gerbes on double covers considered in [29] are \(\mathbf {B}U(1)\)principal 2bundles on \(\mathbb {Z}_2\)principal bundles arising by the above theorem from the extension \(\mathbf {B}U(1) \rightarrow \mathbf {Aut}(\mathbf {B}U(1)) \rightarrow \mathbb {Z}_2\), a special case of the extensions that we consider in the next Sect. 4.4.
4.4 Gerbes
Recall from Remark 4.24 above that in an \(\infty \)topos \(\mathbf {H}\), those \(V\)fiber \(\infty \)bundles \(E\rightarrow X\) whose typical fiber \(V\) is pointed connected are of special relevance. Recall that such a \(V\) is the moduli \(\infty \)stack \(V = \mathbf {B}G\) of \(G\)principal \(\infty \)bundles for some \(\infty \)group \(G\). Due to their local triviality, when regarded as objects in the slice \(\infty \)topos \(\mathbf {H}_{/X}\), these \(\mathbf {B}G\)fiber \(\infty \)bundles are themselves connected objects. Generally, for \(\mathcal {X}\) an \(\infty \)topos regarded as an \(\infty \)topos of \(\infty \)stacks over a given space \(X\), it makes sense to consider its connected objects as \(\infty \)bundles over \(X\). Here we discuss these \(\infty \)gerbes.
 1.
an “ambient” \(\infty \)topos \(\mathbf {H}\) as before, to be thought of as an \(\infty \)topos “of all geometric homotopy types” for a given notion of geometry (recall the discussion in Sect. 1), in which \(\infty \)bundles are given by morphisms and the terminal object plays the role of the geometric point \(*\);
 2.
an \(\infty \)topos \(\mathcal {X}\), to be thought of as the topostheoretic incarnation of a single geometric homotopy type (space) \(X\), hence as an \(\infty \)topos of “geometric homotopy types étale over \(X\)”, in which an \(\infty \)bundle over \(X\) is given by an object and the terminal object plays the role of the base space \(X\). In practice, \(\mathcal {X}\) is the slice \(\mathbf {H}_{/X}\) of the previous ambient \(\infty \)topos over \(X \in \mathbf {H}\), or the smaller \(\infty \)topos \(\mathcal {X} = \mathrm{Sh}_\infty (X)\) of (internal) \(\infty \)stacks over \(X\) (hence étale objects over \(X\), see section 3.10.7 of [28]).
The original definition of a gerbe on \(X\) as given by [9] is: a stack \(E\) (i.e. a 1truncated \(\infty \)stack) over \(X\) that is 1. locally nonempty and 2. locally connected. In the more intrinsic language of higher topos theory, these two conditions simply say that \(E\) is a connected object (Definition 6.5.1.10 in [13]): 1. the terminal morphism \(E \rightarrow *\) is an effective epimorphism and 2. the 0th homotopy sheaf is trivial, \(\pi _0(E) \simeq *\). This reformulation is made explicit in the literature for instance in Section 5 of [11] and in Section 7.2.2 of [13]. Therefore:
Definition 4.41
 1.
connected;
 2.
1truncated.
Remark 4.42
Notice that conceptually this is different from the notion of bundle gerbe introduced in [20] (see [22] for a review). Bundle gerbes are presentations of principal \(\infty \)bundles (Definition 3.4). But gerbes—at least the Ggerbes considered in a moment in Definition 4.48—are \(V\)fiber \(\infty \)bundles (Definition 4.1) hence associated to principal \(\infty \)bundles (Proposition 4.10) with the special property of having pointed connected fibers. By Theorem 4.11 \(V\)fiber \(\infty \)bundles may be identified with their underlying \(\mathbf {Aut}(V)\)principal \(\infty \)bundles and so one may identify \(G\)gerbes with nonabelian \(\mathrm{Aut}(\mathbf {B}G)\)bundle gerbes (see also around Corollary 4.51 below), but considered generally, neither of these two notions is a special case of the other. Therefore the terminology is slightly unfortunate, but it is standard.
Definition 4.41 has various obvious generalizations. The following is considered in [13].
Definition 4.43
 1.
\((n1)\)connected;
 2.
\(n\)truncated.
Remark 4.44
This is almost the definition of an EilenbergMac Lane object in \(\mathcal {X}\), only that the condition requiring a global section \(* \rightarrow E\) (hence \(X \rightarrow E\)) is missing. Indeed, the EilenbergMac Lane objects of degree \(n\) in \(\mathcal {X}\) are precisely the EM \(n\)gerbes of trivial class, according to Corollary 4.51 below.
There is also an earlier established definition of 2gerbes in the literature [6], which is more general than EM 2gerbes. Stated in the above fashion it reads as follows.
Definition 4.45
 1.
connected;
 2.
2truncated.
This definition has an evident generalization to arbitrary degree, which we adopt here.
Definition 4.46
 1.
connected;
 2.
\(n\)truncated.
The real interest is in those \(\infty \)gerbes which have a prescribed typical fiber:
Remark 4.47
By the above, \(\infty \)gerbes (and hence EM \(n\)gerbes and 2gerbes and hence gerbes) are much like deloopings of \(\infty \)groups (Theorem 2.19) only that there is no requirement that there exists a global section. An \(\infty \)gerbe for which there exists a global section \(X \rightarrow E\) is called trivializable. By Theorem 2.19 trivializable \(\infty \)gerbes are equivalent to \(\infty \)group objects in \(\mathcal {X}\) (and the \(\infty \)groupoids of all of these are equivalent when transformations are required to preserve the canonical global section).
Definition 4.48
 1.
an effective epimorphism Open image in new window (onto the terminal object \(X\) of \(\mathcal {X}\));
 2.
an equivalence \(E_U \simeq \mathbf {B} G_U\).
In words this says that a \(G\)\(\infty \)gerbe is one that locally looks like the moduli \(\infty \)stack of \(G\)principal \(\infty \)bundles.
Example 4.49

a 0group object \(G \in \tau _{0}\mathrm{Grp}(\mathcal {X}) \subset \mathrm{Grp}(\mathcal {X})\) is a sheaf of groups on \(X\) (here \(\tau _0\mathrm{Grp}(\mathcal {X})\) denotes the 0truncation of \(\mathrm{Grp}(\mathcal {X})\));

for \(\{U_i \rightarrow X\}\) any open cover, the canonical morphism \(\coprod _i U_i \rightarrow X\) is an effective epimorphism to the terminal object;

\((\mathbf {B}G)_{U_i}\) is the stack of \(G_{U_i}\)principal bundles (\(G_{U_i}\)torsors).
It is clear that one way to construct a \(G\)\(\infty \)gerbe should be to start with an \(\mathbf {Aut}(\mathbf {B}G)\)principal \(\infty \)bundle, Remark 4.25, and then canonically associate a fiber \(\infty \)bundle to it.
Example 4.50
Corollary 4.51
Proof
This is the special case of Theorem 4.11 for \(V = \mathbf {B}G\). \(\square \)
For the case that \(G\) is 0truncated (an ordinary group object) this is the content of Theorem 23 in [11].
Example 4.52
For \(G \in \tau _{\le 0}\mathrm{Grp}(\mathcal {X}) \subset \mathrm{Grp}(\mathcal {X})\) an ordinary 1group object, this reproduces the classical result of [9], which originally motivated the whole subject: by Example 4.50 in this case \(\mathbf {Aut}(\mathbf {B}G)\) is the traditional automorphism 2group and \(H^1(X, \mathbf {Aut}(\mathbf {B}G))\) is Giraud’s nonabelian \(G\)cohomology that classifies \(G\)gerbes (for arbitrary band, see Definition 4.59 below).
For \(G \in \tau _{\le 1}\mathrm{Grp}(\mathcal {X}) \subset \mathrm{Grp}(\mathcal {X})\) a 2group, we recover the classification of 2gerbes as in [6, 7].
Remark 4.53
We now discuss how the \(\infty \)group extensions (Definition 4.26) given by the Postnikov stages of \(\mathbf {Aut}(\mathbf {B}G)\), induce the notion of band of a gerbe, and how the corresponding twisted cohomology, according to Remark 4.35, reproduces the original definition of nonabelian cohomology in [9] and generalizes it to higher degree.
Definition 4.54
Example 4.55
Let \(G \in \tau _0\mathrm{Grp}(\mathrm{Grpd}_{\infty })\) be a 0truncated group object, an ordinary group. Then by Example 4.50, \(\mathbf {Out}(G)\) is the coimage of \(\mathrm{Ad} : G \rightarrow \mathrm{Aut}(G)\), which is the traditional group of outer automorphisms of \(G\).
Definition 4.56
Remark 4.57
To see that the fiber of \(\Omega \mathbf {c}\) here is indeed the delooping of a group, notice that by theorem 2.19 one has to see that it is connected and pointed. Now the fiber of \(\Omega \mathbf {c}\) is connected due to definition of \(\mathbf {c}\) as a truncation map and the induced long exact sequence of (sheaves of) homotopy groups. It is moreover pointed since \(\Omega \mathbf {c}\), being a morphism of groups, is a pointed morphism (the point being the neutral element) and using the universal property of the homotopy fiber.
Example 4.58
For \(G\) an ordinary group, so that \(\mathbf {Aut}(\mathbf {B}G)\) is the automorphism 2group from Example 4.50, \(\mathbf {Z}(G)\) is the center of \(G\) in the traditional sense.
Definition 4.59
For \(E \in G \mathrm{Gerbe}\) we call \(\mathrm{Band}(E)\) the band of \(E\).
Remark 4.60
The original definition of gerbe with band in [9] is slightly more general than that of Ggerbe (with band) in [6]: in the former the local sheaf of groups whose delooping is locally equivalent to the gerbe need not descend to the base. These more general Giraud gerbes are 1gerbes in the sense of Definition 4.46, but only the slightly more restrictive \(G\)gerbes of Breen have the good property of being connected fiber \(\infty \)bundles. From our perspective this is the decisive property of gerbes, and the notion of band is relevant only in this case.
Footnotes
 1.
Throughout topos here stands for Grothendieck topos, as opposed to the more general notion of elementary topos.
 2.
Here \(\mathrm{Core}\) denotes the maximal \(\infty \)groupoid inside an \(\infty \)category.
 3.
The concept of (fine) moduli stacks is historically most commonly associated with algebraic geometry, but the problem which they solve, namely the classification of structures including their (auto)equivalences, is universal. Specifically, if \(\mathbf {H}\) is the \(\infty \)topos over a site of schemes then it contains the moduli stacks as they appear in algebraic geometry.
 4.
U.S. thanks Mike Shulman for discussion of this point.
Notes
Acknowledgments
The writeup of this article and the companion [21] was initiated during a visit by the first two authors to the third author’s institution, University of Glasgow, in summer 2011. It was completed in summer 2012 when all three authors were guests at the Erwin Schrödinger Institute in Vienna. The authors gratefully acknowledge the support of the Engineering and Physical Sciences Research Council grant number EP/I010610/1 and the support of the ESI; D.S. gratefully acknowledges the support of the Australian Research Council (grant number DP120100106); U.S. acknowledges the support of the Dutch Research Organization NWO (project number 613.000.802). U.S. thanks Domenico Fiorenza for inspiring discussion about twisted cohomology.
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