Principal $\infty$-Bundles and Smooth String Group Models

We provide a general, homotopy-theoretic definition of string group models within an $\infty$-category of smooth spaces, and we present new smooth models for the string group. Here, a smooth space is a presheaf of $\infty$-groupoids on the category of cartesian spaces. The key to our definition and construction of smooth string group models is a version of the singular complex functor, which assigns to a smooth space an underlying ordinary space. We provide new characterisations of principal $\infty$-bundles and group extensions in $\infty$-topoi, building on work of Nikolaus, Schreiber, and Stevenson. These insights allow us to transfer the definition of string group extensions from the $\infty$-category of spaces to the $\infty$-category of smooth spaces. Finally, we consider smooth higher-categorical group extensions that arise as obstructions to the existence of equivariant structures on gerbes. We show that these extensions give rise to new smooth models for the string group, as recently conjectured in joint work with M\"uller and Szabo.


Introduction and overview
The most direct way to define the string group is via the Whitehead tower of O(n), (1.1)By this approach, String(n) is defined as a 3-connected topological space with a continuous map String(n) → Spin(n) which induces an isomorphism on all homotopy groups except for in degree three.So far, this defines String(n) only as a space, but in [Sto96] Stolz constructed String(n) as a topological group and the map String(n) → Spin(n) as a morphism of topological groups.In fact, he presented a construction that produces, for any compact, simple, and simply connected Lie group H, a morphism String(H) → H of topological groups whose underlying continuous map is a three-connected covering.A covering of this type is also called a string group extension of H.In these conventions, we write String(n) := String(Spin(n)).
The string group is important in geometry and topology in several ways.Originally, Killingback [Kil87] and Witten [Wit] investigated the two-dimensional supersymmetric σ-model on background manifolds M and found that this is well-defined only if the free loop space LM admits a spin structure.Witten, moreover, computed the index of a hypothetical Dirac operator on LM based on physical arguments, leading to the definition of the Witten genus.By now, it has been understood that the Witten genus is related to the cohomology theory of topological modular forms (TMF).The string group enters in this story, for example by defining orientations in TMF [AHR,DHH11], analogously to how the spin group underlies orientations in real K-theory.
Since the free loop space LM is less tractable than the manifold M itself, it is an important question whether the condition that LM admit a spin structure can be recast as a condition on the manifold M itself.This is indeed the case: spin structures on LM correspond to string structures on M [ST, ST04,Wal15].Topologically, a string structure on M is a lift of the classifying map M → BO(n) of the tangent bundle T M → M to a map M → BString(n).That is, a string structure is a reduction of the structure group of T M to String(n).From a geometric perspective, the interest ultimately is in identifying consequences and constructions that are facilitated by a string structure on a manifold.Concrete examples include the Höhn-Stolz conjecture [Höh,Sto96] that the Witten genus is trivial for any Riemannian 4k-manifold with positive Ricci curvature which admits a string structure, or the long-standing goal to define a Dirac operator on the loop space LM .
In order to study the differential geometric, rather than topological, implications of string structures, it is paramount to have models for String(n) not just as a topological group, but as a group object in some geometric category.For instance, given a Riemannian manifold M , the construction of the Dirac operator associated with a spin structure on M depends on the ability to glue the tangent bundle T M from smooth Spin(n)-valued functions.Technically, one also needs to find local frames for T M in which the Levi-Civita connection of M is represented by 1-forms valued in the Lie algebra spin(n) rather than so(n); however, since the fibre of the map Spin(n) → SO(n) is discrete, these Lie algebras happen to be canonically isomorphic (for more background on spin geometry and Dirac operators, see, for instance, [LM89]).Analogously to how spin structures on LM stem from string structures on M , a hypothetical Dirac operator on LM may well stem from a geometric operator on M itself (e.g. via some transgression procedure), obtained from a further lift of the Levi-Civita connection to the Lie algebra string(n).However, for this to make sense, one must work with a smooth, rather than topological, model for String(n).
Classical results on cohomology readily imply that it is impossible to construct String(H) as a finitedimensional Lie group (for any compact, simple, simply connected Lie group H).Thus, to find geometric models for String(H), one needs to look beyond the category of smooth, finite-dimensional manifolds.Indeed, a number of models for String(H) have been found in (higher) categories of smooth spaces that generalise the notion of a manifold in various ways [BSCS07,Hen08,SP11,Wal12,NSW13,FRS16].
In each of these constructions, an extension of a compact, simple, simply connected Lie group H is constructed within the chosen ambient category of smooth spaces.It is then argued that on the underlying ordinary spaces (meaning topological spaces or simplicial sets) one obtains a string group extension in the sense of (1.1).However, so far there is no general definition of String(H) in a smooth context that formalises this procedure.Consequently, in geometric models for String(H) the extending group A currently has to be chosen ad hoc as an explicit delooping of the Lie group U(1) in a rather strict sense.This obscures the homotopy-theoretic nature of String(H), since from a homotopical point of view, not A is fixed, but only its homotopy type.
In [BMS], studying symmetries of gerbes, we came across extensions of Lie groups H not by a delooping of the Lie group U(1), but by the delooping of the diffeological group U(1) H of smooth maps from H to U(1).However, if H is simply connected, then the smooth group U(1) H is homotopy equivalent to U(1).Therefore, extensions of H by the delooping B(U(1) H ) potentially have the correct homotopy type to produce smooth string group extensions of H. Nevertheless, we could not make this rigorous due to the lack of a homotopy-theoretic notion of smooth string group extensions that does not fix the extending group, but only its homotopy type.
Here, we provide such a general definition of smooth string group extensions, and we prove that the string group models proposed in [BMS] fit within this definition.Let Mfd denote the category of manifolds and smooth maps, and let Cart ⊂ Mfd be the full subcategory on those manifolds that are diffeomorphic to R n for any n ∈ N 0 .We denote the ∞-category of spaces by S. As our ambient ∞-category of smooth spaces, we choose the ∞-category H ∞ := Fun(Cart op , S) of presheaves of spaces on Cart.This provides a very general notion of smooth space: for instance, H ∞ contains the categories of manifolds, diffeological spaces, and Lie groupoids.We write M for the image of a manifold M under the fully faithful inclusion Mfd ֒→ H ∞ .
The ∞-category H ∞ is even an ∞-topos.There exists an established theory of group objects in ∞topoi [Lur09].Moreover, there exists a notion of principal ∞-bundles and extensions of group objects in ∞-topoi, due to [NSS15].A large part of this paper is devoted to developing this theory further.In particular, we show that group actions in ∞-topoi automatically form groupoid objects (Theorem 3.19) and that principal ∞-bundles essentially consists of an effective epimorphism and a principal group action (Theorem 3.31); this is analogous to the definition of principal bundles of topological spaces as a locally trivial map and a principal group action.A group object in an ∞-topos H is a simplicial object A ∈ Fun(∆ op , H) satisfying certain properties (see Definitions 2.6 and 3.2).We provide the following characterisation of extensions of group objects: (3) A 1 − → H 1 is a fibre sequence in H and p 1 : G 1 → H 1 is an effective epimorphism.(4) The morphism p 1 : G 1 → H 1 together with the action of A on G 1 induced by ι define a principal A-bundle over H 1 .
In order to give a general homotopy-theoretic definition of string group extensions within H ∞ , we need to associate an underlying space to an object in H ∞ .In [Buna] we investigated (a model categorical presentation of) a functor S e : H ∞ → S from H ∞ to the ∞-category S of spaces.It evaluates a smooth space B ∈ H ∞ on the extended affine simplices ∆ k e ∈ Cart and then takes the geometric realisation of the resulting simplicial object in S. One can think of S e is a version of the singular complex functor for smooth spaces.Here, we give further interpretation and context to this functor.Consider the adjunction c ⊣ Γ, where Γ : H ∞ → S is the global-section functor and c is the constant-presheaf functor.This fits into a triple adjunction Π ⊣ c ⊣ Γ ⊣ codisc, where codisc is fully faithful and where Π preserves finite products.That is, the ∞-topos H ∞ is cohesive.
This has already been argued in [BEBdBP] and proven on the level of model categories of simplicial presheaves in [Buna]; here we provide an ∞-categorical proof based on findings from [Buna].Let L : H → H ′ be a functor between ∞-topoi which preserves finite products and geometric realisations of simplicial objects.We show that L maps principal ∞-bundles in H to principal ∞-bundles in H ′ and group extensions in H to group extensions in H ′ (this relies on Theorem 3.19).In particular, the functor S e : H ∞ → S has these properties.In S, a string group extension of a compact, simple, simply connected Lie group H can be defined as usual: it is an extension A → String(H) → H of group objects in S such that String(H) is 3-connected and such that the morphism String(H) → H induces an isomorphism on all homotopy groups of the underlying spaces except for in degree three.Using that S e M ≃ M for any manifold M (see [Buna] for a proof of this classical fact using the present technology) and that S e preserves principal ∞-bundles and group extensions, we can now transfer this definition to H ∞ : Definition 1.4 Let H be a compact, simple, and simply connected Lie group, and let H denote the induced group object in H ∞ .An extension of the group object H in H ∞ is called a smooth string group extension of H if its image under S e is a string group extension in S.
We show that the string group models conjectured in [BMS] fit within Definition 1.4.Let M be a manifold endowed with a bundle gerbe G (a categorified hermitean line bundle).In [BMS], we addressed the question of when an action of a Lie group H on M lifts to an equivariant structure on G.We found that the obstruction to such a lift is captured by an extension of H by the smooth 2-group HLB M of hermitean line bundles on M .Each of the above objects can be interpreted as a group object in H ∞ via the nerve functor N , and so the sequence (1.5) enhances to an extension of H as a group object in H ∞ .The case relevant for string group extensions is M = H, where H is a compact, simple and simply connected Lie group, acting on itself via left multiplication.Since H is 2-connected, there is an objectwise equivalence HLB H ≃ B(U(1) H ), and since H is 1-connected, there is a smooth homotopy equivalence U(1) H ≃ U(1).Therefore, the extending group in (1.5) has the correct homotopy type for a string group extension.We prove: Theorem 1.6 Let H be a compact, simple, simply connected Lie group, and let N be the nerve functor.Consider the left-action of H on itself via left multiplication.Let G ∈ Grb(H) be a gerbe on H whose class in H 3 (H; Z) ∼ = Z is a generator.The sequence is a smooth string group extension of H.
This string group model is somewhat similar to the model in [FRS16], which is obtained by studying symmetries of gerbes with connection.However, here the presence of connections forces the extending group to be the delooping BU(1).It is interesting that the connection does not change the homotopy type of the extension.In [BMS,Def. 5.33], we also constructed a second extension of H with a connection on the gerbe G acting as crucial auxiliary data.We showed that this extension is equivalent to the one in (1.5) [BMS,Thm. 5.36], it gives rise to a second smooth string group extension of H. Finally, we expect that most (or possibly all) of the aforementioned smooth string group models fit within Definition 1.4.Checking this in full detail in each case would go beyond the scope of this article, but we outline the relevant arguments here: for the models in [FRS16] and [Wal12] the methods we use here should adapt in a straightforward manner.The models [Hen08,SP11] should fit into the present framework via the presentation of sheaves of ∞-groupoids on Cart as ∞-Lie groupoids (see, for instance, [Nuia, Nuib, Sch]).For the infinite-dimensional models in [Sto96] and [BSCS07], one needs to be able to compute homotopy types of infinite-dimensional manifolds, such as gauge groups and loop spaces, via the functor S e .In the case of loop spaces, this is facilitated by the Smooth Oka Principle from [SS].For gauge groups one needs that the homotopy type induced from the infinite-dimensional manifold structure agrees with the homotopy type extracted by using S e .This should follow from general results on the relation between diffeological spaces and infinite-dimensional manifolds in [Kih].
Outline.In Section 2 we investigate the functor S e : H ∞ → S. Further, we recall some basic notions and facts about ∞-topoi and prove Theorem 1.3.Section 3 is devoted to the theory of group objects, group extensions, and principal ∞-bundles in ∞-topoi.We recall the definitions of these notions from [NSS15] and provide new characterisations of principal ∞-bundles and group extensions.In particular, we prove Theorem 1.2.
In Section 4, we use the results obtained thus far to transfer the definition of string group extensions in S to the ∞-topos H ∞ .After recalling from [BMS] the smooth 2-group extensions which control equivariant structures on gerbes, we show that these extensions give rise to new smooth models for string group, thus proving Theorem 1.6.
Finally, in Appendix A we prove Theorem 3.19: we show that group actions in an ∞-topos give rise to groupoid objects.
Notation.We usually make no notational distinction between ordinary categories and ∞-categories; the nerve functor will be used implicitly where necessary.
We write ∆ for the simplex category, and Set ∆ for the category of simplicial sets.In a simplicial category C, we denote the simplicially enriched hom-functor by C(−, −) : We write |−| = colim C ∆ op for the colimit of simplicial objects in an ∞-category C.Moreover, we also refer to |X| (if it exists) as the geometric realisation of a simplicial object X in C.
Usually, we denote ∞-categories by letters C, D, . . ., but for ∞-topoi we use bold-face letters H.In particular, the ∞-topos of spaces is denoted by S. We write D(−, −) : D op × D → S for the mapping spaces in an ∞-category D. We model ∞-categories by quasi-categories.Given an ∞-category C and a simplicial set We let ∆ + denote the augmented simplex category, i.e. the category ∆ with an initial object adjoined.We usually do not distinguish notationally between augmented simplicial objects X ∈ Fun(∆ op + , C) in an ∞-category C and their underlying simplicial objects.If we wish to make this distinction explicit for clarity, we will denote the latter by the restriction X |∆ op .
If M is a simplicial model category, then M • is the full simplicial subcategory on the cofibrant-fibrant objects of M. Recall from [Lur09] that the coherent nerve If C is a (small) ∞-category, we write P(C) = Fun(C op , S) for the ∞-category of presheaves of spaces on C.

Presheaves on cartesian spaces and the smooth singular complex
We let Cart denote the (small) category whose objects are submanifolds of R ∞ that are diffeomorphic to R n for any n ∈ N 0 , and whose morphisms are the smooth maps between these manifolds.We let denote the ∞-category of presheaves of spaces on Cart.The ∞-category H ∞ is presented by several model categories of simplicial presheaves on Cart-for example, there is a canonical equivalence [Lur09] ∞ is the category of simplicial presheaves on Cart, endowed with the injective model structure.Let I := {c × R → c | c ∈ Cart} denote the set of morphisms in Cart of the form 1 c × c R , where c R : R → * is the map that collapses the real line to the point.We can localise both H i ∞ and H ∞ at this set of morphisms (or rather at its image under the Yoneda embedding), and there is still a canonical equivalence between the localisations [Lur09], were the subject of [Buna].On the level of their underlying ∞-categories, one of the main results of that paper can be phrased as follows.For k ∈ N 0 , we let ∆ k e := {t ∈ R k+1 | k i=0 t i = 1} denote the extended (affine) k-simplex.This is a k-dimensional affine subspace of R k+1 , and hence forms a cartesian space.The face and degeneracy maps of the standard topological simplices |∆ k | extend to the extended affine simplices, turning them into a functor We let S e : H ∞ → S denote the composition of functors We refer to this functor as the smooth singular complex functor ; viewing the ∞-category H ∞ as an ∞-category of smooth spaces, S e thus assigns an underlying ordinary space to a smooth space.
Theorem 2.2 [Buna] There exist adjunctions of ∞-categories where S I e is the restriction of S e to L I H ∞ ⊂ H ∞ .Furthermore, the following statements hold true: (1) The functor S e : H ∞ → S preserves and reflects I-local equivalences.
(2) The morphism ι is fully faithful, i.e.Loc is a reflective localisation.
(3) The three right-hand vertical functors are equivalences of ∞-categories.
(4) The diagram obtained by omitting the morphism L I e is (weakly) commutative.Proof.The first claim follows readily from Proposition 3.6, Corollary 3.12, and Corollary 3.37 of [Buna].(Note that model categorical presentations of H ∞ , L I H ∞ , and S are used in [Buna], and the functors in the statement are presented by Quillen functors.)Further, claim (1) follows readily from [Buna,Cor. 3.15].Claim (2) follows from general properties of ∞-categories underlying simplicial model categories and their Bousfield localisations [Lur09].Claim (3) is the version on the underlying ∞-categories of Theorems 3.14 and 3.40 of [Buna].Claim (4) holds true because the diagram of the right-adjoints clearly commutes (ι is an inclusion, and R e simply factors through L I H ∞ ⊂ H ∞ [Buna]).
Remark 2.3 There is a fully faithful embedding Mfd ֒→ H ∞ from the category of manifolds into H ∞ : it sends a manifold M to the presheaf M of discrete spaces that maps a cartesian space c to the set Mfd(c, M ) of smooth maps from c to M .By [Buna,Thm. 5 This completes the proof.

Background on ∞-topoi
In this section, we briefly recall some background on ∞-topoi.Most of the material in this section can be found in [Lur09,Sch,NSS15].For n ∈ N 0 and a subset S ⊂ [n], let ∆ S ⊂ ∆ n be the full ∞-subcategory on the vertices that lie in S.There is a canonical isomorphism ∆ S ∼ = ∆ |S| as simplicial sets, where |S| is the cardinality of S. The simplicial set ∆ S can equivalently be seen as the image of an inclusion ∆ |S| ֒→ ∆ n that sends the i-th vertex of ∆ |S| to the vertex of ∆ n which corresponds to the i-th element of S (with the order induced from the inclusion S ⊂ [n]).Given an ∞-category C and a simplicial object X ∈ Fun(∆ op , C), we set X(S) := X(∆ |S| ).This comes with a canonical morphism X n → X(S), induced by the inclusion S ⊂ [n].
Definition 2.6 Let C be an ∞-category.A groupoid object in C is a simplicial object X ∈ Fun(∆ op , C) such that, for every n ∈ N 0 and every partition [n] = S ∪ S ′ (as finite sets) with S ∩ S ′ ∼ = { * } consisting of a single element, the diagram In particular, any groupoid object is a category object (see also Definition A.1): for every n ≥ 1, the spine decomposition We denote the full subcategory of Fun(∆ op , C) on the groupoid objects by Let ∆ + denote the simplex category with an initial object .
(2) The underlying simplicial object X |∆ op is a groupoid object in C and the diagram Definition 2.9 Let C be an ∞-category, and let p : P → B be a morphism in C.Then, p is an effective epimorphism if the augmented Čech nerve Čp ∈ Fun(∆ op + , C) ∼ = Fun((∆ op ) ⊲ , C) is a colimiting cocone in C. In other words, the morphism p : P → B is an effective epimorphism precisely if the colimit | Čp| exists in C and the induced morphism | Čp| → B is an equivalence.
is an equivalence (on the left-hand side, cB, cC : K → H are the constant diagrams with values B and C, respectively, and the pullback is formed in Fun(K, H)).
(3) Coproducts in H are disjoint: for every pair of objects B, C ∈ H, the pushout diagram (4) Groupoids in H are effective: given any groupoid object X ∈ Gpd(H), let p : X 0 → |X| denote the canonical morphism which is part of the colimiting cocone.Then, the comparison morphism η : X → Čp constructed in (2.10) is an equivalence of simplicial objects in H.In particular, p is an effective epimorphism.
Example 2.12 We list some examples of ∞-topoi; we will mostly be using the first two of these.
(2) Any ∞-category P(C) of presheaves of spaces on a (small) ∞-category C is an ∞-topos.
(3) Any accessible, left-exact, reflective localisation of an ∞-category P(C) of presheaves on a small ∞-category C is an ∞-topos; in fact, every ∞-topos is equivalent to an ∞-topos of this form [Lur09, Thm.6.1.0.6,Prop.6.1.5.3].⊳ We will later need the following properties of effective epimorphisms in an ∞-topos: An important notion of morphism between ∞-topoi is that of a geometric morphism, which is more adapted to the additional structure on ∞-topoi than a mere functor of ∞-categories: One can show that the ∞-category S of spaces is final in the ∞-category of ∞-topoi and geometric morphisms [Lur09, Prop.6.3.4.1].That is, for every ∞-topos H there exists an essentially unique geometric morphism H → S. We will denote the corresponding adjunction by c : S ⇄ H : Γ and refer to Γ as the global-section functor of H.
Example 2.15 Consider a Grothendieck ∞-site, i.e. a small ∞-category C with a Grothendieck coverage.Suppose C additionally has a final object.If H is the ∞-category of sheaves of spaces on C, then the global section functor Γ of H agrees with the evaluation of sheaves at the final object of C.
In particular, this applies to H ∞ , the ∞-topos of presheaves of spaces on Cart from Section 2.1.⊳ Definition 2.16 An ∞-topos H is called cohesive if the adjunction c : H ⇄ S : Γ can be extended to a triple adjunction Π ⊣ c ⊣ Γ ⊣ codisc, in which the left adjoint Π preserves finite products and the right adjoint codisc is fully faithful.
Cohesive ∞-topoi have been studied extensively in [Sch] and related works.
Theorem 2.17 The ∞-topos H ∞ is cohesive, i.e. there exists a triple adjunction Π ⊣ c ⊣ Γ ⊣ codisc as in Definition 2.16, and there is a canonical equivalence Remark 2.18 The fact that H ∞ is cohesive is not new, see [Sch].The second statement has been proven in a model categorical presentation in [Buna], and a different argument has been given in [BEBdBP].Here, we give an ∞-categorical proof of this fact for completeness.⊳ Proof.The ∞-topos H ∞ = P(Cart) admits a right-adjoint to its global-section functor Γ by abstract arguments: evaluation of a presheaf at any object preserves colimits, and since both H ∞ and S are presentable, Γ must admit a further right adjoint [Cis19, Prop.7.11.8].It is well-known that this can in fact be extended into a triple adjunction which establishes that H ∞ is cohesive [Sch].For the second part of the statement, we show that S e is left-adjoint to the functor c.Recall from Section 2.1 that here c simply sends a space K ∈ S to the constant presheaf on Cart with value K. ⊥ By the definition (2.1) of S e , for any space K ∈ S we have a canonical natural equivalence because left-hand side is the colimit of a constant diagram over an indexing category whose nerve is contractible in the Kan-Quillen model structure on Set ∆ (see Lemma A.7, Example A.8).In other words, there is a canonical natural equivalence S e • c ≃ 1 S .Consequently, there is also a canonical equivalence on the right adjoints, Γ • R e ≃ 1 S .We obtain natural equivalences In the second equivalence we have used that R e takes values in L I H ∞ ⊂ H ∞ and that on objects in L I H ∞ the morphism c • Γ → 1 H∞ is an equivalence.From the equivalence R e ≃ c and the adjunction S e ⊣ R e we infer that S e is a further left adjoint to c. Hence, it is equivalent to the functor Π. Theorem 2.17 shows that the smooth singular complex functor S e : H ∞ → S has a deep homotopical meaning for assigning homotopy types to objects in H ∞ and for studying these homotopy types.It also provides an additional, refined, perspective on the good homotopical properties of the functor S e that were found and studied in [Buna].Finally, note that we also obtain from this a natural equivalence That is, we see that S e computes the ∞-categorical colimit of Cart op -shaped diagrams of spaces, and thus, by [Cis19, Thm.6.4.5], that N ∆ e : N ∆ → N Cart is a cofinal morphism in Set ∆ .
3 Principal ∞-bundles and group extensions in ∞-topoi In this section, starting from the theory introduced in [NSS15], we develop characterisations of principal ∞-bundles and extensions of group objects in ∞-topoi.These characterisations are interesting already in their own right.In Section 4 they will also allow us to transfer the definition of string group extensions from S to H ∞ and to construct explicit smooth models for the string group.

Groups and group extensions
Here we recall the definitions of group objects and their extensions in ∞-topoi [NSS15].We investigate how to compute limits of group and groupoid objects in ∞-topoi, and how group objects and their classifying objects behave under functors between ∞-topoi that preserve finite products and geometric realisations.
Let H be an ∞-topos, and let Gpd(H) be the ∞-category of groupoid objects in H. Further, let EEpi(H) ⊂ Fun(∆ 1 , H) denote the full ∞-subcategory on the effective epimorphisms in H. Recall that by Definition 2.11(4) and Proposition 2.8 there is a canonical equivalence given by forming colimits and Čech nerves, respectively.
Definition 3.2 Let C be an ∞-category.Let Grp(C) ⊂ Gpd(C) denote the full ∞-subcategory on those groupoid objects X in C for which X 0 is a final object of C. We call Grp(C) the ∞-category of group objects in C.
Proposition 3.3 For any ∞-topos H, there are reflective localisations Proof.First, the right adjoints in the above sequence of adjunctions are fully faithful by definition.
The first morphism has a left adjoint by [Lur09, Prop.6.1.2.9].For the second left adjoint, we use the equivalence (3.1)2 : this equivalence induces a commutative square where EEpi * (H) ⊂ EEpi(H) is the full ∞-subcategory on those effective epimorphisms f : X 0 → X −1 where X 0 is a final object.A left adjoint to the bottom morphism is given by the functor that sends an effective epimorphism f : X 0 → X −1 to the morphism g : * → X −1 ⊔ X 0 * induced by the pushout.Since f is an effective epimorphism, Lemma 2.13 implies that so is g.
For a group object G ∈ Grp(H) in an ∞-topos H, we set Note that in an ∞-topos H, for any groupoid object X ∈ Gpd(H) the map X 0 → colim H ∆ op X is an effective epimorphism.Hence, given a group object G in H, the morphism * ≃ G 0 → BG is an effective epimorphism.Moreover, the functor B is part of an equivalence [Lur09, Lemma 7.2.2.11] (see also [NSS15, Thm.2.19]) where H * / ≥1 is the ∞-category of pointed, connected objects in H.Note that, for each T ∈ H * / ≥1 , we view ΩT as a group object in H, i.e.ΩT ∈ Grp(H).Its underlying object in H is (ΩT ) 1 =: Ω 1 T .Unravelling the definition, we obtain that a group object in an ∞-category C with a final object * ∈ C is equivalently a simplicial object G in C such that G 0 ≃ * and, for any [n] ∈ ∆ and any partition In particular, iterating this for the spine partition (1) If L preserves finite products, then it preserves group objects.
(2) If L additionally preserves geometric realisations, then, for any group object G in H, there is a canonical equivalence B(LG) ≃ L(BG) .Proof.The first part of the Proposition is known [Lur09]; we include its proof only for completeness.Any functor F : C → D between ∞-categories preserves simplicial objects, i.e. it induces a functor Fun(∆ op , C) −→ Fun(∆ op , D). Suppose that G ∈ Fun(∆ op , H) is a group object in H. Since L preserves finite products, it preserves final objects, so that (LG) 0 ≃ * is final in H ′ .For n = 0 and any partition The top morphism is an equivalence since G is a group object in H and the vertical morphism is an equivalence since L preserves products.This proves claim (1).Using that BG = colim H ∆ op G = |G|, the second part is now immediate.
Remark 3.6 We will prove a number of statements about functors as in Proposition 3.5(2), i.e. functors between ∞-topoi which preserve geometric realisations and finite products.An important class of such functors is given by the additional left-adjoints of cohesive ∞-topoi-see Definition 2.16.In particular, the functor S e : H ∞ → S from Section 2.1 is of this type by Theorem 2.17.
Proof.Proposition 3.3 implies that an equivalence of groupoid objects X ≃ −→ Y in H is the same as an objectwise equivalence of the underlying simplicial objects in H: X and Y are local objects in Fun(∆ op , H) with respect to the localisation Gpd(H) ⊂ Fun(∆ op , H), so that the local equivalences between them are precisely the original, i.e. the levelwise, equivalences.In particular, this implies the 'only if' part of claim (1).
Conversely, if we are given a morphism It then follows that the morphism X → Y is also an equivalence in Gpd(H).The same line of argument shows the second claim.
Lemma 3.8 Let H be an ∞-topos, and let K ∈ Set ∆ be a simplicial set.
(1) A diagram X : K ⊳ → Gpd(H) of groupoid objects in H is a limit diagram if and only if the composition ιX : (2) A diagram X : K ⊳ → Gpd(H) of groupoid objects in H is a limit diagram if and only if the induced diagrams X i : K ⊳ → H are limit diagrams for i = 0, 1.
Proof.One can see the 'only if' direction of claims (1) and (2) as follows: we first note that since the inclusion Gpd(H) ⊂ Fun(∆ op , H) is a right adjoint, we have that if X : For the converse direction in claim (1), we first show that limits of diagrams in Gpd(H) can be computed in Fun(∆ op , H).More precisely, a functor X : K ⊳ → Gpd(H) is a limit diagram whenever its composition with the inclusion ι : Gpd(H) ֒→ Fun(∆ op , H) is so, i.e. the inclusion reflects limits.Equivalently, the ∞-subcategory Gpd(H) ֒→ Fun(∆ op , H) is closed under limits in Fun(∆ op , H).This is seen as follows: consider a functor X : K ⊳ → Gpd(H) and a decomposition [n] = S ∪ S ′ with S ∩ S ′ = { * }.This induces an equivalence (ιX) and using that limits commute with limits [Lur09, Lemma 5.5.2.3], we have which shows that Y ∈ Fun(∆ op , H) is local with respect to the localisation Gpd(H) ֒→ Fun(∆ op , H), i.e. that Y ∈ Gpd(H).Since the inclusion ι : Gpd(H) ֒→ Fun(∆ op , H) is fully faithful, Y is also a limit of the diagram X : K → Gpd(H).Consequently, if the composition ιX : For the converse direction in claim (2), suppose that X : K ⊳ → Gpd(H) is a diagram such that the functors X i : K ⊳ → H are limit diagrams for i = 0, 1.By part (1) it suffices to show that the composition ιX : Since X : K ⊳ → Fun(∆ op , H) is valued in groupoid objects, and since limits in H commute with limits, it follows from (3.9) that for every The proof of claim (3) proceeds along the exact same line as the proof of part (2): the key insight is the fact that if G : Nevertheless, it appears that there should be a simpler definition that more directly generalises extensions of groups in Set to the ∞-categorical setting.For group objects in Set, a group extension is a sequence A → G → H of group homomorphisms such that G → H is surjective and A is the fibre of the morphism G → H at the identity element of H.We will prove in Theorem 3.48 that one can indeed generalise this view on group extensions to group objects in ∞-topoi.⊳

Group actions in ∞-categories
We now investigate actions of group objects in ∞-topoi.Let Set ∆J denote the category of simplicial sets endowed with the Joyal model structure.For K ∈ Set ∆ , we let obj(K) be the set K 0 of vertices of K, seen as a discrete simplicial set.Let J := ∆ 1 [f −1 ] be the localisation of ∆ 1 at its non-trivial edge (see e.g.[Cis19, Sec.3.3]).
Lemma 3.12 Let C be an ∞-category, and let K be a simplicial set.
(1) The inclusion ι : obj(K) ֒→ K induces a morphism of simplicial sets, which is a fibration between fibrant objects in the Joyal model structure.
Proof.Part (1) follows since obj(K) ֒→ K is a cofibration in the Joyal model category Set ∆J , C is a fibrant object in Set ∆J , and Set ∆J is a (closed) symmetric monoidal model category.
For part (2), we apply [Cis19, Cor.3.6.4]to the categorical anodyne extension ∆ {i} ֒→ J = ∆ 1 [f −1 ] and the Joyal fibration (i.e.isofibration) from part (1).Part (3) is obtained by taking the fibre of the morphism from part (2), which is a contractible Kan complex since it is the fibre of a trivial Kan fibration.This fibre is equivalently described as the space of lifts in the commutative diagram which is precisely the space of pairs (ĝ ′ , η) of lifts ĝ′ : K → C of g ′ and equivalences η : Example 3.13 Let G be a group object in an ∞-category C with a final object.This is, in particular, a simplicial object G : ∆ op → C (we suppress the canonical inclusion functors Grp(C) ֒→ Gpd(C) ֒→ Fun(∆ op , C)).Consider the functor where ⋆ denotes the join of categories (and where we view partially ordered sets as categories).The induced pullback functor is also called the decalage functor; see [Ste12] for more background.For each n ≥ 1, the partition [n] = {0, 1} ⊔ {1} {1, . . ., n} induces an equivalence We can phrase this as an equivalence of functors From Lemma 3.12 we obtain that there exists an essentially unique way to lift these data to a functor ∆ op → C, which we denote by G 1 //G, and an equivalence γ : Given a group action P //G, we set a := d 0 : P × G 1 → P .It follows by the pasting law for pullbacks that there are canonical equivalences of morphisms between d 0 : , and similarly between and the projection onto the first n factors.
Remark 3.16 Definition 3.15 is taken from [NSS15, Def.3.1] almost verbatim, but it differs from that source in that we do not require group actions to be groupoid objects.Instead, we show in Theorem 3.19 that this is a consequence of the axioms in Definition 3.15.A second (minor) difference is that we also fix the level-zero degeneracy map s 0 : , we also denote a G-action over B by and (2) the collapse morphisms P → * and Q → * induce a (weakly) commutative diagram of simplicial objects in H.The ∞-category of G-actions over B ∈ H is the full ∞-subcategory of Fun(∆ op , H) /(B×( * / /G)) on those objects whose underlying simplicial object is a G-action.
An ordinary G-action is equivalent to a G-action over the final object * ∈ H.
Example 3.22 For a group object G ∈ Grp(H) and an action P //G of G on an object P ∈ H, let q : (∆ op ) ⊲ → H be a colimiting cocone of the simplicial diagram P //G in H. Observing that (∆ op ) ⊲ ∼ = (∆ ⊳ ) op ∼ = (∆ + ) op , this defines an augmented simplicial object in H, which we denote as Therefore, the data P //G → |P //G| form a G-action over |P //G|.In particular, the canonical morphism * //G → BG is of this form.⊳ Another example of a morphism of this type is the collapse morphism G 1 //G → * , as we show now: Any morphism A → G of group objects induces an action of A on G 1 by the following construction: Proposition 3.24 Let f : A → G be a morphism in Grp(H).Define a simplicial object G 1 //A as the pullback Proof.We check the axioms in Definition 3.15: axiom (1) follows from the pasting law for pullbacks and the diagram in which the right-hand square is a pullback for any n ∈ N 0 by construction of G 1 //G.Axiom (2) is readily seen from applying the maps d 1 and s 0 to the diagram (3.25), for n = 0, 1. Axiom (3) follows since the morphism G 1 //A −→ * //A induced by the above diagram agrees with the morphism obtained by collapsing the first factor G 1 .

Principal ∞-bundles
In this subsection, we characterise principal ∞-bundles and group extensions in ∞-topoi.Throughout this section, let H be an ∞-topos and let G ∈ Grp(H) be a group object in H.
Definition 3.26 [NSS15, Def.3.4] A G-principal ∞-bundle on an object B ∈ H is a G-action P //G → B over B such that the augmented simplicial object P //G → B is a colimiting cocone for the simplicial diagram P //G ∈ Fun(∆ op , H).In other words, the augmenting map p : P → B induces an equivalence Observe that, by construction, the restriction of α along ı is η.We will not distinguish notationally between α as defined here and its restriction along the inclusion This is an ∞-categorical version of the principality condition for a group action.It is, in fact, equivalent to the usual principality condition-that the action morphism P × G 1 → P × B P is an equivalence-in the following sense (in particular, this implies the converse to [NSS15, Prop.Proof. (1) implies (2) since the action P //G p −→ B is principal precisely if it is equivalent, as an augmented simplicial object in H, to the Čech nerve Čp = Ran ι {p}.Thus, the implication follows from Proposition 2.8.Conversely, (2) also implies (1): we know from Theorem 3.19 that P //G is a groupoid object.If we additionally have that (3.30) is a pullback diagram, then we can again apply Proposition 2.8 to obtain the claim.
We can use Lemma 3.29 to give a characterisation of principal ∞-bundles which can be understood as encoding directly the classical criteria for principal bundles: a locally trivial map p : P → B and a principal G-action over B. (2) The morphism p is an effective epimorphism and the action P //G is principal.Proof.To see that (1) implies (2), first observe that since P //G is a groupoid object in H, and since by assumption the canonical morphism |P //G| → B is an equivalence, it follows from Definition 2.11(4) that the canonical morphism α : P //G → Čp is an equivalence in Fun(∆ op , H).In particular, p is an effective epimorphism.Further, it has been shown in [NSS15, Prop.3.7] that if P //G → B is a principal bundle, then the action P //G satisfies condition (2) of Lemma 3.29, and so the action is principal.To see the other direction, consider the commutative diagram In this case, both the top and the right-hand morphisms in diagram are equivalences.It thus follows that also the left-hand morphism is an equivalence, which amounts to the fact that P //G → B is a principal G-bundle in the sense of Definition 3.26.
Theorem 3.32 Let L : H → H ′ be a functor between ∞-topoi which preserves geometric realisations and finite products.Suppose G is a group object in H.
( in Fun(∆ op , H) (or, equivalently, in Gpd(H)).For any [n] ∈ ∆ there exists a canonical equivalence We use Lemma 3.12 to obtain from these equivalences a canonical pair (up to contractible choices) of an object (B × C P )//G ∈ Fun(∆ op , H), with ((B × C P )//G) n = (B × C P ) × G n 1 for all n ∈ N 0 , together with an equivalence of simplicial objects in H.By a slight abuse of notation, we also denote the composition We are hence left to show that it is a principal ∞-bundle.
To that end, we will show that the morphism This establishes that the morphism f * p in diagram (3.37) is the pullback of an equivalence in H, and hence that f * p is an equivalence itself.
One can now show that every G-principal ∞-bundle arises as a pullback of the bundle ( * //G) → BG.This insight is not new, but has been observed in [NSS15, Prop.3.13, Thm.3.17] already.However, in Section 4.3 it will be important to have a good understanding the classifying map of a principal ∞-bundle, and so we include a brief treatment of these maps.We start with two short technical lemmas, before constructing for each G-principal ∞-bundle in H its classifying map.
there is an equivalence of simplicial objects in H /B between P //G and B × (G//G) that commutes with the canonical morphisms to * //G.Lemma 3.40 [NSS15, Prop.3.12] For every G-principal ∞-bundle P //G → B in H, there exists an effective epimorphism U → B such that the pullback bundle U × B (P //G) is trivial.
Proof.We give an alternative proof to [NSS15].Given a G-principal ∞-bundle P //G → B in H, consider the effective epimorphism P → B and the pullback bundle P × B (P //G).We have a commutative diagram This is a refinement of [NSS15, Prop.3.13] to a statement on the level of simplicial objects, rather than only on their zeroth level.

Proof. Consider the diagram
in Fun(∆ op , H).Here, the front and rear squares are pullbacks (by Lemmas 3.40 and 3.38, respectively), and the diagram is obtained as a morphism of pullback diagrams.We need to show that the right-hand face is a pullback square in Fun(∆ op , H).First, we show that the top square of (3.43) is a pullback.By Lemma 3.8 it suffices to check this level-wise: at simplicial level n = 0, it is trivial.For n ∈ N, the square consists of the the image under the functor This a pullback diagram: there is a commutative diagram in which the dashed morphism is given by the composition where ∆ G 1 : G 1 → G 2 1 is the diagonal morphism, and where inv : G 1 → G 1 is the choice of an inverse in G: since the group object G ∈ Grp(H) is in particular a groupoid object, we have a diagram where we have used the characterisation of groupoid objects as certain category objects from Proposition A.5. Choosing an inverse for the right-facing morphism defines the morphism inv.
The right-hand side is precisely the pullback described by the top square in diagram (3.43).Since we already know that the top square of (3.43) is cartesian, we obtain an equivalence in Fun(∆ op , H).Thus, it follows that The last equivalence can be seen either by combining Proposition 3.
That is, ϕ ′ is the image under Ran ι (compare Definition 2.7) of the square This ϕ ′ 0 is an equivalence since the bottom square of (3.43) is a pullback.Consequently, the morphism ϕ is an equivalence in Fun(∆ op , H), and thus the right-hand face in (3.43) is a pullback.We now state several alternative characterisations of group extensions in ∞-topoi.These clarify the relation between the original notion of an extension of group objects from Definition 3.10 and more direct categorifications of several perspectives on group extensions in Set.The last of these alternative characterisations will be important in Section 4.3.(2) (4) The morphism p 1 : G 1 → H 1 together with the action G 1 //A of A on G 1 induced by ι define a principal A-bundle over H 1 . Proof.
(1) ⇒ (3): This implication was proven in [NSS15] already.We import the proof for completeness: consider the diagram ) is a pullback square (this assumes (1)).It thus follows that the sequence − → H 1 is a fibre sequence in H.By this construction, the morphisms ι ′ and p ′ coincide with the morphisms Ω•B(ι) and Ω•B(p), respectively.The equivalence (3.4) then yields that also A 1 Observe that each vertical morphism in diagram (3.49) is an effective epimorphism since * → BG is an effective epimorphism for every group object G ∈ Grp(H) and since effective epimorphisms are stable under pullback.In particular, p 1 is an effective epimorphism.
(3) ⇒ (4): By Proposition 3.31 it suffices to show that the action of A on G 1 is principal (in the sense of Definition 3.28).We will make use of Lemma 3.29.By assumption, the diagram be the multiplication on G 1 (it can be identified with the morphism d 1 : G 2 → G 1 ).Further, let inv : G 1 → G 1 be a choice of inverse for the group object G (compare the proof of Proposition 3.41).There is a commutative diagram Using that A 1 ≃ * × H 1 G 1 , the universal property of pullbacks thus provides an essentially unique morphism ϕ : The canonical morphism * → H 1 and the morphism ι : A → G induce a morphism of simplicial objects ψ : * //A −→ H 1 //G.Its colimit is a morphism |ψ| : BA → |H 1 //G|.In particular, we obtain from this an augmented simplicial object * //A −→ |H 1 //G| in H. Let q : * → |H 1 //G| denote the restriction of this augmented simplicial object to ∆ +,≤0 .We claim that * //A −→ |H 1 //G| is equivalent to the Čech nerve of q.By Definition 2.7, Proposition 2.8 and the fact that * //A is a groupoid object, it suffices to show that the diagram is a pullback diagram.We can see this as follows: with the G-action induced by p : G → H, the morphism H 1 → |H 1 //G| becomes a principal G-bundle in H. Since the sequence of morphisms A 1 → G 1 → H 1 is a fibre sequence in H by Corollary 3.45, we obtain a double pullback diagram We have thus shown that the augmented simplicial object * //A −→ |H 1 //G| is equivalent to the Čech nerve of q : * → |H 1 //G|.As the latter describes the loop object Ω|H 1 //G| as a group object in H, we infer that i.e. as group objects in H.
The claim will now follow from the equivalence (3.4), provided we can additionally show that |H 1 //G| is a connected object in H.One way of proving this is by showing that the morphism * → |H 1 //G| is an effective epimorphism.Consider the morphism of simplicial objects induced by p.In each simplicial level, this is an effective epimorphism by assumption on p. Since effective epimorphisms are stable under colimits (see Lemma 2.13), it follows that the morphism induced on colimits, * ≃ |G 1 //G| −→ |H 1 //G| is an effective epimorphism as well.Alternatively, one can see that the left-hand morphism in (3.50) is 0-connected: the morphism Bp : BG → BH is a colimit of effective epimorphisms.Thus, it is an effective epimorphism itself.It is also a morphism between connected objects in H and therefore necessarily induces an isomorphism on zeroth homotopy groups (i.e. it is a connected morphism).The claim then follows since n-connected morphisms in ∞-topoi are stable under pullback [Lur09, Prop.6.5.1.16(6)].

Corollary 3.51 Suppose
This is the ∞-categorical analogue of the canonical isomorphism G/A ∼ = H for ordinary (set-theoretic) group extensions A → G → H. Proof.This statement now follows from combining Theorem 3.32 and Theorem 3.48.

Homotopy-theoretic smooth string group models
In this section, we present a definition of string group extensions within the ∞-category H ∞ of smooth spaces.It relies on the singular complex functor S e : H ∞ → S for smooth spaces from Section 2 and the theory of group extensions in ∞-topoi from Section 3. We begin by recalling the definition of a string group extension in the ∞-category S of spaces.Then, we use our results thus far to transfer this definition to H ∞ along the functor S e , leading to a homotopy-theoretic definition of smooth string group extensions (Definition 4.2).
After recalling some background on bundle gerbes in Section 4.2, we provide new smooth models for the string group in Section 4.3, building on recent constructions of smooth 2-group extensions in [BMS].(There already, evidence was given that these smooth 2-group extensions can model the string group; here we provide a full formal framework and proof for that conjecture.)

The definition of smooth string groups
The definition of a string group via the Whitehead tower (see Section 1) is purely homotopy-theoretic.In particular, in a string group extension A → String(H) → H the extending group A is not fixed, but only its underlying homotopy type.So far, to our knowledge there does not exist a definition of string group extensions in a smooth context that contains this flexibility-the extending group A is usually chosen ad hoc to be some smooth version of BU(1).Here, we provide a smooth version of the original homotopy-theoretic definition (see Definition 4.2).In particular, only the underlying homotopy type of the extending smooth group A is fixed in this definition.To avoid clashes of notation, we denote a Lie group by a triple (H 1 , •, e H ), where H 1 ∈ Mfd is the underlying manifold, (−)•(−) denotes the multiplication on H 1 and e H : * → H 1 is the neutral element in H 1 .Recall that each compact, simple and simply connected Lie group (H 1 , •, e H ) is also 2-connected and satisfies •, e H ) canonically defines a group object H in the ∞-topos of spaces S. We start by reformulating the definition of a string group extension of topological groups within the ∞-category of spaces: •, e H ) be a compact, simple and simply connected Lie group, and denote by H ∈ Grp(S) its associated group object in S. A string group extension of (H 1 , •, e H ) is an extension of group objects in S such that (1) A 1 is an Eilenberg-MacLane space K(Z, 2), and (2) under the isomorphism .46 and Theorem 3.48(4)) represents a generator of Z.
Given condition (1), condition (2) is equivalent to saying that the map String(H) 1 → H 1 of spaces induces an isomorphism π i (String(H) 1 ) → π i (H 1 ) for i = 3 and that π 3 (String(H) 1 ) ∼ = 0.This is a consequence of the Hurewicz Theorem, the Universal Coefficient Theorem, and the long exact sequence of homotopy groups associated to a (homotopy) fibre sequence of spaces.That is, Recall the ∞-topos H ∞ = P(Cart) from Section 2.1.There we also introduced the localisation In [FRS16] string group extension of H are defined via the pullback Here, 1 2 p 1 denotes the fractional first Pontryagin class, which is a generator of H 4 (BH; Z) ∼ = Z.However, this definition of String(H) is considerably stricter than the original perception of String(H) as a 3connected covering of H 1 by another group object (Definition 4.1).For instance, the definition of a string group extension based on (4.4) enforces that the morphism String(H) 1 → H 1 is a BU(1)-principal ∞-bundle (note that if H is an ∞-topos and A ∈ Grp(H) is a group object whose multiplication lifts to an E 2 -algebra structure, then BA is canonically the underlying object of a group object in H [NSS15]).However, from the purely homotopy-theoretic point of view, it is not the actual fibre of this map in H that should be fixed, but only the homotopy type of its underlying space in S (which must be a K(Z; 2)).Definition 4.2 emphasises this latter, homotopy-theoretic aspect of string group extensions.

Bundle gerbes and their symmetries
Before we can present our smooth string group extension, we need to recall some background on bundle gerbes.We will not give full definitions or details here, but refer the reader to [Wal07b,Wal07a,Bun17,BSS18] for technical background and [BS17,Bun21] for an introduction to the topic.Bundle gerbes provide an explicit, geometric model for categorified line bundles.We point out that there also exists a notion of connection on a bundle gerbe, but here we will only be working with bundle gerbes without connection.(This is the main technical cause for the distinction between our smooth string group model and that in [FRS16].) To any manifold M , we can assign a symmetric monoidal 2-groupoid (Grb(M ), ⊗) of bundle gerbes on M .Given a bundle gerbe G ∈ Grb(M ), the monoidal groupoid Grb(M )(G, G) of automorphisms of G is canonically equivalent to the symmetric monoidal groupoid (HLB(M ), ⊗) of hermitean line bundles on M with the usual tensor product (which we also denote by ⊗).Note that (HLB(M ), ⊗) is even a 2-group; that is, it is a symmetric monoidal groupoid in which every object has an inverse with respect to the monoidal product.Every smooth map f : N → M of manifolds induces a monoidal 2-functor Isomorphism classes of gerbes are in canonical bijection with the third integer cohomology of M : there is an isomorphism of abelian groups The class associated to a gerbe G under this isomorphism is called the Dixmier-Douady class of G.
We let H ≤1 denote the following 2-category: its objects are functors π : C → Cart that are Grothendieck fibrations in groupoids (that is, π is a Grothendieck fibration and all its fibres are groupoids).Its morphisms (π : and its 2morphisms are natural transformations η : F → F ′ such that π ′ η is the identity natural transformation 1 Cart → 1 Cart .Note that the 2-category H ≤1 is canonically equivalent to the 2-category of pseudofunctors Cart op → Gpd from Cart op to the 2-category of groupoids via the Grothendieck construction.We make the following definitions; for more background, see [BMS,SP11].
Definition 4.8 [BMS] The 2-category of smooth 2-groups is the 2-category of group objects in the 2-category H ≤1 .
Example 4.9 Let (H 1 , •, e H ) be a Lie group.We associate to it the following category, denoted by H 1 : its objects are pairs (c, h) of a cartesian space c ∈ Cart and a smooth map The category H 1 comes with a canonical projection functor H 1 → Cart.The product on H 1 -valued maps turns H 1 into a smooth 2-group in the sense of Definition 4.8; we denote this smooth 2-group by H.Note that H 1 is simply the Grothendieck construction of the presheaf of sets H 1 on Cart.⊳ Example 4.10 For a manifold M , let HLB(M ) denote the groupoid of hermitean line bundles on M .For fixed manifold M , we define a category (HLB M ) 1 as follows: its objects are pairs (c, L) of a cartesian space c ∈ Cart and a hermitean line bundle L ∈ HLB(c×M ).A morphism (c, L) → (c ′ , L ′ ) is a pair (f, ψ) of a smooth map f : c → c ′ and an isomorphism ψ : L → (f ×1 M ) * L ′ of hermitean line bundles over c.This category comes with a projection functor (HLB M ) 1 → Cart.The tensor product of hermitean line bundles turns (HLB M ) 1 into a smooth 2-group, which we denote by HLB M .⊳ Let M be a manifold, and let G ∈ Grb(M ) be a gerbe on M .Further, let (H 1 , •, e H ) be a connected Lie group acting smoothly on M from the left; we denote the action by Φ : H 1 × M → M .Given these data, we define a category Sym(G) 1 as follows: an object in Sym(G) 1 is a triple (c, h, A), where c ∈ Cart is a cartesian space and where h : c → H 1 is a smooth map.These give rise to a smooth map Φ h : c × M → c × M , defined as the composition where ∆ : c → c × c is the diagonal map.Then, A is a 1-isomorphism and where ψ is a 2-isomorphism ψ : A −→ (f × 1 M ) * A ′ (where we have implicitly used that there is a canonical 1-isomorphism Observe that there is a projection functor p 1 : Sym(G) 1 → H 1 , acting as (c, h, A) → (c, h) and (f, ψ) → f .Remark 4.11 In this set-up, the following statements hold true: (1) There is a canonical inclusion ι 1 : (HLB M ) 1 ֒→ Sym(G) 1 in H ≤1 .
(2) The connectedness of H 1 ensures that the functor p 1 is surjective on objects.Further, p 1 is an essentially surjective Grothendieck fibration in groupoids; it is even strictly surjective on objects [BMS,Thm. 5.27].
(2) The functors ι 1 and p 1 canonically lift to morphisms of smooth 2-groups and induce a sequence of smooth 2-groups.
The nerve functor N : Cat → Cat ∞ induces a functor N : H ≤1 → H ∞ (where we have used the canonical equivalence between H ≤1 and the 2-category of pseudo-functors Cart op → Gpd from Cart op to the 2category of groupoids).This functor, in particular, preserves final objects and products, so that it maps smooth 2-groups to group objects in H ∞ .Our smooth string group model will be obtained by applying this functor to the sequence (4.13).

A smooth string group model
We To that end, we first show the following lemma: Lemma 4.17 Let M be a connected manifold with H 2 (M ; Z) ∼ = 0.
(1) The object (N HLB M ) 1 = N (HLB M ) 1 ∈ H ∞ is equivalent to B(U(1) M ). ( Both equivalences are even established by morphisms of group objects in H ∞ . Since S e maps I-local equivalences in H ∞ to equivalences of spaces, applying Lemma 4.17 to M = H 1 establishes axiom (1) of Definition 4.1 for the image of the sequence (4.15) under the functor S e .Note that for a simplicial object X ∈ Fun(∆ op , H) in an ∞-topos H and an object B ∈ H, we can form the level-wise exponential X B ∈ Fun(∆ op , H).There are canonical equivalences (X B ) n ≃ (X n ) B .Thus, we can simply write X B n .If X is a groupoid or group object, then so is X B .Proof.We proceed in parallel to the proof of [BMS, Thm.8.7]: since any c ∈ Cart is contractible and since H 2 (M ; Z) ∼ = 0, it follows that any hermitean line bundle on c × M is trivialisable.Consequently, the groupoid HLB M (c) is equivalent to the groupoid with one object and morphisms given by the group U(1) M (c) of smooth maps from c × M to U(1).This induces an equivalence (N HLB M ) 1 ≃ B(U(1) M ) in H ∞ , which extends to a morphism of group objects in H ∞ .This proves (1).
Next, since π 1 (M ) is trivial, there exists a smooth homotopy equivalence ev x,1 : U(1) M 1 → U(1) 1 , given by restricting a smooth map c × M → U(1) to c × {x}, where x ∈ M is any point.A homotopy inverse to ev x,1 is given by pulling a smooth map c → U(1) back along the projection c × M → c [BMS, Lemma 8.9].In particular, ev x,1 is an I-local equivalence [Buna,Cor. 3.16].
Since ev x,1 is an I-local equivalence in H ∞ and I-local equivalences are closed under finite products (Proposition 2.4), the morphism ev x is a levelwise I-local equivalence of simplicial objects in H ∞ .Further, the class W I of I-local equivalences in H ∞ is strongly saturated [Lur09, Lemma 5.5.4.11].In particular, the full subcategory of Fun(∆ 1 , H ∞ ) on the I-local equivalences is stable under colimits.Therefore, taking the colimit in H ∞ of simplicial objects (i.e.taking geometric realisations), we obtain an I-local equivalence (2) The presheaf B k (U(1) M ) satisfies descent with respect to the Grothendieck coverage τ of differentiably good open coverings on Cart.
Proof.Ad (1): This is an iteration of the argument in the proof of Lemma 4.17 is an equivalence of spaces.Here, ČU ∈ Fun(∆ op , H ∞ ) is the Čech nerve of the covering U.
We first show that q * is essentially surjective; that is, it induces a bijection on isomorphism classes of objects.Since limits and colimits in H ∞ = Fun(Cart op , S) are computed pointwise, we have isomorphisms On the other hand, we have that where on the right-hand side we have the usual Čech cohomology group with respect to the covering U of the sheaf of abelian groups on c given by .
By a slight abuse of notation, we also denote this sheaf by U(1) M .We claim that the right-hand side of (4.20) is further isomorphic to H k+2 (c; Z) ∼ = * .First, the Čech cohomology groups Ȟk+1 (U; U(1) M ) are isomorphic to the sheaf cohomology groups of U(1) M : since the covering U of c is differentiably good, it follows from [Bry08, Thm.1.3.6]that there is a canonical isomorphism Ȟk U; U(1) M ∼ = H k c; U(1) M .
Next, we observe that since M is simply connected, there is a short exact sequence We further observe that the sheaf R M is fine (it admits partitions of unity, for instance those induced from the canonical map R → R M ) when seen as a sheaf on the open subsets of a manifold.Therefore, for any manifold Y , we have a canonical isomorphism for every k ≥ 1.We thus arrive at for each cartesian space c ∈ Cart and each k > 0. This completes the proof that the morphism q * from (4.19) is bijective on connected components.It remains to check that the morphism q * from (4.19) is an isomorphism on all higher homotopy groups in H ∞ = P(Cart).We will achieve this by comparing the automorphisms of the unique object in the source and target space of q * .On the source side, this automorphism space is given as the pullback of spaces and there is a canonical equivalence in H ∞ , On the target side of q * , the automorphism space of the (essentially) unique object is the pullback Since limits in H ∞ are computed objectwise, there are canonical equivalences However, by the induction hypothesis, the presheaf B k (U(1) M ) is a sheaf, so that q * induces an equivalence between the automorphism spaces.This proves that q * is indeed an equivalence.The first morphism is applying the functor S e .For the second morphism we have used [Buna,Thm. 5.1]: for every manifold Y ∈ Mfd, there is a canonical equivalence S e Y ≃ Y in S. Further, here we have used that S e commutes with B (Proposition 3.5).For the third morphism, we have used that the inclusion U(1) ֒→ U(1) H 1 is an I-local equivalence in H ∞ : since H 1 is connected and simply connected, this morphism is a smooth homotopy equivalence by [BMS,Lemma 8.9], and by [Buna,Cor. 3.16] any smooth homotopy equivalence is an I-local equivalence.The last morphism again uses [Buna, Thm.5.1].Since S e preserves finite products, the equivalence S e U(1) 1 ≃ U(1) 1 in S is compatible with the group structure3 on U(1).

An application of
We can describe the map (4.23) more explicitly as follows: we have already seen above that any element in π 0 H ∞ (H 1 , B n (U(1) H 1 )) can be described as a smooth U(1) H 1 -valued Čech cocycle with respect to a (differentiably good) open cover V of H 1 .Under the map (4.23), these data are sent first to the same Čech cocycle, but seen as a map of spaces, and then this resulting Čech cocycle is composed with the evaluation U(1) H 1 1 → U(1) 1 at the unit element in H. Therefore, using the canonical isomorphism π 0 S(H 1 , B n U(1)) ∼ = Ȟn (H 1 ; U(1)) and combining this with the maps (4.In this diagram, the left-hand vertical morphism is invertible as argued before (4.21).The bottom morphism is an isomorphism by [BMS,Prop. 8.11] and the fact that Čech cohomology and abelian sheaf cohomology are isomorphic on manifolds.The right-hand vertical morphism is invertible as a consequence of the isomorphisms in (4.22) and the fact that the map ev e,1 : U(1) (Again, one can alternatively see the coincidence of the cohomology classes more explicitly on the level of Čech cocycles in the presentation of H ∞ by the simplicial model category H p ∞ : a smooth bundle represented by a smooth U(1) H 1 -valued cocycle on H gets sent to the topological bundle represented by the same Čech cocycle interpreted as a collection of continuous maps.)It thus remains to compute the cohomology class associated to these bundles.In [BMS,Sec. 8] it has been shown that the class in H 3 (H 1 ; Z) of the bundle (4.25) agrees with the class in H 3 (H 1 ; Z) that classifies the gerbe G under the isomorphism (4.7).Since we started our construction from a so-called basic gerbe, i.e. one whose Dixmier-Douady class is a generator of H 3 (H 1 ; Z), this completes the proof of Theorem 4.14.
Remark 4.26 We conclude with the following remarks: (1) In [BMS], we suggested the smooth 2-group extension (4.15) as a model for the string group extension of (H 1 , •, e H ).However, the necessary formalism to make this precise was not available then-its development was the main goal of the present paper.
(2) Moreover, in [BMS,Sec. 5.5] we also presented a second smooth 2-group extension of (H 1 , •, e H ); its construction uses a connection on G as auxiliary data and relies heavily on a notion of parallel transport on a gerbe G with connection, as developed in [BMS].The extension (4.27) is then obtained via a homotopy-coherent version of an associated bundle construction.By [BMS, Thm.5.33], there is an equivalence (in H ≤1 ) between the smooth 2-group extension in (4.27) and (4.13), so that we automatically obtain an equivalence between the group objects in H ∞ they induce under the nerve functor.Hence, given the input of a basic gerbe G on H 1 , by Theorem 4.14 the extension (4.27) also gives rise to a second, equivalent smooth string group extension •, e H ), for any compact, simple and simply connected Lie group (H 1 , •, e H ). ⊳ (3) A category object X ∈ Fun(∆ op , C) in C is a groupoid object precisely if the diagram Proof.For claim (1), note that an object of ∆ /Λ 2 0 is a pair ([n], ϕ) of an object [n] ∈ ∆ and a morphism of simplicial sets ϕ : ∆ n → Λ 2 0 .We show that, for each object ([n], ϕ) of ∆ /Λ 2 0 , the slice category ([n], ϕ) /D is contractible.Since the horn Λ 2 0 fits into a pushout diagram in Set ∆ , the morphism ϕ : ∆ n → Λ 2 0 is either the constant map at the apex of the horn, i.e. ϕ factors as ϕ : ∆ n → ∆ {0} ֒→ Λ 2 0 , or it factors through a unique map ∆ n → ∆ {0,i} , for i = 0 or i = 2, but not through the apex ∆ {0} ֒→ Λ 2 0 .(One can see this either by writing ∆ n = N [n] and Λ 2 0 = N I and using that the nerve functor is fully faithful, or by using that Set ∆ (∆ n , −) = (−) n preserves colimits.)In the first case, the slice category ([n], ϕ) /D is the category describing spans; in other words, it is isomorphic to I, and we have In the other cases, the slice category ([n], ϕ) /D is the final category, and hence contractible as well.Claim (2) now follows directly from the definition of X(K), for K ∈ Set ∆ , together with part (1) (after taking opposites), and claim (3) then follows by combining claim (2) with Proposition A.5. Example A.8 We need the following two specific cases in which Lemma A.7 applies: (1) The nerve N I ∈ Set ∆ is contractible, as already seen in the proof of Lemma A.6.
(2) The inclusion {[0]} ֒→ ∆ is the inclusion of a final object.Thus, the nerve N ∆ ∈ Set ∆ is contractible in Set ∆ .⊳ Lemma A.9 Let K ∈ Set ∆ be contractible (in the Kan-Quillen model structure) and let C be an ∞-category admitting limits of shape K. Let P ∈ C be any object.

Theorem 1. 2
Let H be an ∞-topos.Let A ι − → G p − → H be a sequence of morphisms of group objects in H.The following are equivalent: an extension of group objects in H in the sense of [NSS15], i.e. the sequence BA → BG → BH is a fibre sequence in H. (2) A ι − → G p − → H is a fibre sequence of group objects in H and p 1 : G 1 → H 1 is an effective epimorphism (the subscript 1 denotes evaluation at [1] ∈ ∆).
Further, recall from the proof of Proposition 2.4 (and [Buna, Prop.2.13]) that the I-local objects in H ∞ are precisely the essentially constant presheaves, i.e. those F ∈ H ∞ for which the canonical morphism F ( * ) → F (c) is an equivalence for every c ∈ Cart.Equivalently, F is I-local if and only if the canonical morphism c • Γ(F ) → F is an equivalence in H ∞ .Further, by Theorem 2.2 the right adjoint R e to S e factors through the localisation L I H ∞ ⊂ H ∞ ; this is precisely the full ∞-subcategory of H ∞ on the I-local objects.Consider the two adjunctions S e : H ∞ ⇄ S : R e and c : S ⇄ H ∞ : Γ.They induce an adjunction S e • c : S S : Γ • R e .

0 →
whose components are exactly the equivalences γ n from (3.14).One can now check that G 1 //G is the simplicial object in C that describes the right action of G on itself via the group multiplication in G. ⊳ Definition 3.15 Let C be an ∞-category with pullbacks and a final object.Let G be a group object in C, and let P ∈ C.An action of G on P is a simplicial object P //G ∈ Fun(∆ op , C) such that (1) for each n ∈ N 0 , we have (P //G) n = P × G n 1 , (2) the morphism d 1 : P × G 1 → P is the canonical projection onto P , the morphism s 0 : P → P × G 1 agrees with the morphism 1 P × ( * s G 1 ), and (3) the collapse morphism P → * induces a morphism P //G → G in Fun(∆ op , C).
3.7]): Lemma 3.29 Let P //G → B be a G-action over B ∈ H.The following are equivalent: (1) The G-action is principal.diagram in H.
Proposition 3.31 Let P //G p − → B be a G-action over an object B ∈ H.The following are equivalent: (1) P //G p −→ B is a principal ∞-bundle (in the sense of Definition 3.26).
1) L maps G-actions P //G p − → B over B ∈ H to LG-actions LP //LG Lp −→ LB over LB ∈ H ′ .(2) If the action P //G → B is a principal G-bundle, then the action LP //LG Lp −→ LB is a principal LG-bundle.Proof.Since L preserves finite products, the first claim follows readily from Definition 3.15.For the second claim, recall that P //G → B is a principal G-bundle precisely if the map |P //G| → B is an equivalence.Applying the functor L to this morphism, we obtain an equivalence L|P //G| ≃ −→ LB.Since L preserves geometric realisations, and using claim (1), we obtain further canonical equivalences |LP //LG| ≃ −→ L|P //G| ≃ −→ LB , which establishes the action LP //LG Lp − → LB as a principal LG-bundle over B. Proposition 3.33 Let G be a group object in H, and let P //G → C be a G-principal ∞-bundle in H.For any morphism f : B → C in H, there is a canonical G-action over B on the pullback Q := B × C P that makes Q//G → B into a G-principal ∞-bundle on B. Proof.Let c : H −→ Fun(∆ op , H) be the constant-diagram functor.Consider the pullback diagram cB × cC (P //G) P Composing with the functor colim H ∆ op = |−| : Fun(∆ op , H) −→ H we obtain a diagram |(B × C P )//G| |P //G| B C | f | f * p p ≃ f (3.36) in H.The right-hand morphism is an equivalence since P //G → C is assumed to be a principal ∞-bundle.Using the equivalence (3.35), diagram (3.36) is equivalent to the diagram |cB × cC (P //G)| |P /of colimits in H, we have a canonical equivalence |cB × cC (P //G)| ≃ B × C |P //G| .
Lemma 3.38 Let G be a group object in H, and let f : * → BG be the base point of BG.The pullback of the canonical bundle ( * //G) → BG along f agrees with the bundle G 1 //G → * .Proof.Consider the commutative square G 1 //G * //G * cBG in Fun(∆ op , H).By the canonical equivalence G ≃ ΩBG in Grp(H) (see (3.4)), this diagram is level-wise a pullback, i.e. it is a pullback diagram in Fun(∆ op , H).That proves the claim by Proposition 3.33.
23 with the fact that |−| preserves finite products (because ∆ op is sifted [Lur09, Lemma 5.5.8.4]), or simply by recalling that P ×(G 1 //G) → P is a G-principal ∞-bundle on P .This shows that the bottom square in (3.43) is a pullback.Finally, we prove that the right-hand square in (3.43) is a pullback as well.Consider the commutative diagram of solid arrows essentially unique morphism ϕ of simplicial objects in H.By the commutativity of the right-hand triangle in this diagram, ϕ is even a morphism of G-actions.By the commutativity of the left-hand triangle it is even a morphism of G-actions over B. Since its source and target are G-principal ∞-bundles, ϕ is equivalent to a morphism of Čech nerves

Theorem 3 .
48 Let H be an ∞-topos, and let A ι − → G p − → H be a sequence of morphisms in Grp(H).The following are equivalent: (1) A ι − → G p − → H is an extension of group objects in H, i.e. the induced sequence BA → BG → BH is a fibre sequence in H (see Definition 3.10).

Corollary 3 .
52 Let L : H → H ′ be a functor between ∞-topoi which preserves geometric realisations and finite products.Suppose A ι − → G p − → H is an extension of group objects in H.Then, the sequence LA Lι − → LG Lp −→ LH is an extension of group objects in H ′ .

B
ev x : B U(1) M −→ BU(1) in H ∞ .Composing with the morphism constructed in part (1), we now obtain the desired I-local equivalence N HLB M −→ BU(1) in H ∞ .We are thus left to show that the sequence of group objects in S obtained by applying the functor S e to the sequence (4.15) of group objects in H ∞ satisfies axiom (2) of Definition 4.1.That is, we have to show that the principal ∞-bundle of spaces S e N Sym(G) 1 // S e N HLB H −→ S e H 1 represents a generator of H 3 (H; Z) ∼ = Z.This is best checked using Čech cohomology.Recall the Grothendieck coverage τ of differentiably good open coverings on Cart from Remark 4.3.Lemma 4.18 Let M ∈ Mfd be a simply connected manifold, and let k ∈ N 0 .Then, (1) In H ∞ , there is an I-local equivalence B k (U(1) M ) ≃ B k U(1) .
(2).Ad (2): We prove this claim by induction.For k = 0, we have to check that the functor Cart op → S, c → Mfd(c × M, U(1)) satisfies descent with respect to good open coverings of c.However, this follows directly from the fact that, for any manifold Y , the functor Op(Y ) op → Set , U → Mfd U, U(1) defines a sheaf on Y , where Op(Y ) is the category of open subsets of Y and their inclusions.Suppose that B l (U(1) M ) is a sheaf on Cart for all l = 0, . . ., k.Let c ∈ Cart, and let U = {c a ֒→ c} a∈Λ be a differentiably good open covering of c.We have to show that the canonical morphism q [Bunb, Thm.1.1] now implies that, for each simply connected manifold M , the presheaf of spaces H ∞ −, B n U(1) M : Mfd op −→ S satisfies descent with respect to open coverings (and even surjective submersions).Consequently, given any open covering V = {c a ֒→ M } a∈Λ , whose Čech nerve we denote by ČV → M , the canonical morphismH ∞ M , B n (U(1) M ) ∼ = lim ∆ S H ∞ ČV, B n (U(1) M )is an equivalence of spaces.Therefore, there is an isomorphismπ 0 H ∞ M , B n (U(1) M ) ∼ = Ȟn M ; U(1) M , (4.21) which can be represented explicitly by composing a morphism M → B n (U(1) M ) with any Čech nerve ČV → M of an open covering of M .(Alternatively, this can be seen directly in the presentation of H ∞ by the projective model structure on simplicial presheaves on Cart.)Now let us return to the case where M = H 1 is the manifold underlying the compact, simple, simply connected Lie group (H 1 , •, e H ). Let ev e,1 : U(1) H 11 −→ U(1) 1 be the morphism induced by pullback along the base-point inclusion e H : * ֒→ H 1 .It induces a morphism of group objects ev e : U(1) H 1 −→ U(1).We obtain a commutative diagramπ 0 H ∞ H 1 , B n (U(1) H 1 ) π 0 H ∞ H 1 , B n U(1)Ȟn H 1 ; U(1) H 1 Ȟn H; U(1) in[BMS, Prop.8.11] that the bottom horizontal morphism is an isomorphism for all n ∈ N (with n > 0); thus, so is the top horizontal morphism.Consider the morphismsπ 0 H ∞ H 1 , B n (U(1) H 1 ) −→ π 0 S Se H 1 , S e B n (U(1) H 1 ) (4.23) ∼ = π 0 S H 1 , B n S e (U(1) H 1 ) ∼ = π 0 S H 1 , B n S e U(1) ∼ = π 0 S H 1 , B n U(1) .

Lemma A. 7
Let K be a simplicial set, let C be an ∞-category, and let C ∈ C be an object.Let c : C → Fun(K, C) denote the constant-diagram functor.(1) If K is contractible, i.e.K ≃ * in Set ∆ with the Kan-Quillen model structure, and colim C K (cC) exists in C, then the canonical morphism colim C K (cC) → C in C is an equivalence.(2) Dually, if K is contractible and lim C K (cC) exists in C, then the canonical morphism C → lim C K (cC) in C is an equivalence.Proof.By the definition of c, there is a commutative diagram K C * coll cC C in S. By [Lur09, Cor.4.1.2.6, Thm.4.1.3.1], the morphism coll is cofinal if and only if the simplicial set K × * ∼ = K is contractible, i.e. precisely if coll : K → * is an equivalence in Set ∆ (in the Kan-Quillen model structure).The first claim then follows from the fact that cofinal morphisms preserve colimits [Lur09, Prop.4.1.1.8].The second statement follows by duality.

local objects. It follows that there are canonical equivalences
.1] there is a canonical equivalence of spaces M ≃ S e M for any M ∈ Mfd, which is natural in M .⊳ Proposition 2.4 The localisation functor Loc : H ∞ → L I H ∞ preserves finite products.The class W I of I-local equivalences in H ∞ is closed under finite products.Proof.By [Buna, Prop.2.13], the localisation L I H ∞ agrees with the localisation L W H ∞ of H ∞ at the set W of all collapse morphisms c → * , for c ∈ Cart.The set W is stable under finite products in H ∞ , since Cart has finite products.Therefore, the first claim follows from [Cis19, Cor.7.1.16].The second claim then follows since a morphism in H ∞ is in W I precisely if its image under Loc is an equivalence [Lur09, Prop.5.5.4.15].For B, C ∈ H ∞ , let C B ∈ H ∞ denote their internal hom object in H ∞ .The localisation functor Loc : H ∞ → L I H ∞ is given (up to equivalence) by Consider the adjunction c : S ⇄ H ∞ : ev * , where c assigns to a space K the constant presheaf with value K, and where ev * evaluates a presheaf on the final object * ∈ Cart.These functors induce an equivalence c : S ⇄ L I H ∞ : ev * [Buna, Thm.2.17], and there is a canonical equivalence ev * ≃ S I [−1] adjoined.For n ∈ N 0 , let ∆ +,≤n ⊂ ∆ + be the full subcategory on the objects [−1], . . ., [n].In particular, ∆ op +,≤0 is the category with two objects and one non-trivial morphism [0] → [−1].Therefore, any morphism p : P → B in an ∞-category C defines an object {p} ∈ Fun(∆ op +,≤0 , C). Definition 2.7 Given a morphism p : P → B in an ∞-category C, its Čech nerve Čp (if it exists) is the augmented simplicial object obtained as the right Kan extension Colimits in H are universal: for any diagram D : K → H, any cocone D : K ⊲ → H under D with apex C ∈ H, and for any morphism f : B → C in H, the induced morphism then lim Having established several properties of the ∞-category of group objects in H, we now define extensions of group objects: Definition 3.10 [NSS15, Def.4.26] Let A and H be group objects in an ∞-topos H.An extension of group objects of H by A is a sequence A → G → H in the ∞-category Grp(H) such that the sequence BA → BG → BH is a fibre sequence in H. Remark 3.11 This definition of a group extension has advantages from a theoretical perspective.
[NSS15] any group object G ∈ Grp(C) there is a canonical trivial action * //G on the final object * ∈ C, coming from the canonical equivalence * × G ≃ G of simplicial objects.That is, there is a canonical equivalence G ≃ * //G in Fun(∆ op , C).Theorem 3.19 Let C be an ∞-category with finite limits, let G ∈ Grp(C) be a group object in C, and let P //G ∈ Fun(∆ op , C) be an action of G on an object P ∈ C.Then, P //G is a groupoid object in C. Remark 3.20 Let L : H → H ′ be a functor between ∞-topoi which preserves finite products and geometric realisations.The relevance of Theorem 3.19 is that it will allow us to show that functors of this type map group actions in H to group actions in H ′ (see Theorem 3.32).In[NSS15], group actions are defined to be groupoid objects, but functors L : H → H ′ as above do not preserve groupoid objects in general.However, Theorem 3.19 shows that-as in the classical case of (set-theoretic) group actions-actions of group objects in ∞-topoi automatically form groupoid objects.Consequently, we do not need to require L to preserve groupoid objects.⊳WeproveTheorem 3.19 in Appendix A. For the remainder of this section, let H be an ∞-topos.Definition 3.21 Let G ∈ Grp(H) be a group object.A G-action over an object B ∈ H is an augmented simplicial object X ∈ Fun(∆ op + , H) whose underlying simplicial object is a G-action P //G on some object P ∈ H, and whose augmenting object is B, i.e.X −1 = B. Writing p : P → B for the morphism X |∆ op [Lur17,e 3.18We can now give a precise meaning to the last sentence of Example 3.13: the object G 1 //G ∈ Fun(∆ op , C) is an action of G on its underlying object G 1 ∈ H via right multiplication; see also[Lur17, Def.4.2.2.2, Example 4.2.2.4] for more background.⊳Givenan action of a group object G on an object P in C, we would like to think of the simplicial object P //G as the action groupoid associated with this action.This is indeed justified: ∆ (M • ) [Lur09, Prop.A.3.7.6].Under this equivalence, colimits in H over diagrams indexed by ordinary categories correspond to homotopy colimits in M [Lur09, Cor.4.2.4.8].It now suffices to observe that any simplicial object in M obtained as the decalage of another simplicial object has an augmentation and extra degeneracies [Rie14, Ste12].
is a group object in H, then the canonical morphism|G 1 //G| ≃ −→ * is an equivalence.Proof.Since H is presentable, there exists a combinatorial simplicial model category M and an equiva-lence of ∞-categories H ≃ N of Čech nerves over P .The level-zero component of ψ ′ is precisely the equivalence P × G 1 → P × B P which establishes that P //G → B is principal (cf.Proposition 3.29).Since ψ ′ is the image of ψ under the right Kan extension Ran ι (compare Definition 2.7), it follows that ψ ′ , and hence ψ, is an equivalence.
op , H), where a × 1 acts on P with the first copy of G and as the identity on the remaining copies of G.The induced morphism ψ is a morphism of G-principal ∞-bundles (since the triangles in the diagram commute and since a × 1 is a morphism of G-actions).It is thus equivalent to a morphism ψ ′ : Č(P × G → P ) −→ Č (P × B (P //G)) −→ P op , H): there is an equivalence Corollary 3.45 Let P //G → B be a G-principal ∞-bundle in H.For any morphism x : * → B, we have a pullback diagram BG) [NSS15, Thm.3.17].This implies that every morphism of principal G-bundles on B is an equivalence.Proposition 3.41 feeds into the proof of this equivalence by showing that the functor H(B, BG) → Bun G (B), sending a morphism B → BG to the principal ∞-bundle B × BG ( * //G), is essentially surjective.The morphism q is the canonical morphism induced from the collapse morphism LP → * .By Proposition 3.41, the front face of this diagram is a pullback in H ′ , witnessing |q| as the classifying morphism LX → BLG of the bundle LP //LG −→ LX.Since all diagonal morphisms are equivalences, the rear face of the diagram is a pullback as well, showing that L|p| is a classifying morphism for the LGprincipal ∞-bundle L(P //G) −→ LB, which is equivalent to the bundle LP //LG −→ LB.Finally, since the diagonal morphisms arise from the natural equivalences L • |−| ≃ |−| • L, it follows that |q| ≃ |Lp|.
x in Fun(∆ op , H).In particular, any fibre of P → B is canonically equivalent to G 1 in H. Remark 3.46 For any group object G in H and any object B ∈ H, there is an equivalence Bun G (B) ≃ H(B, BG) between the ∞-category of G-principal ∞-bundles on B and the mapping space H(B, ⊳ In particular, under the equivalence of Remark 3.46, the morphism |p| : B → BG in diagram (3.42) is a classifying morphism for the bundle P //G → B. Proposition 3.47 Let L : H → H ′ be a functor of ∞-topoi which preserves finite products and geometric realisations.If P //G → B is a G-principal ∞-bundle in H, classified (up to canonical equivalence) by a morphism |p| : B → BG, then the LG-principal ∞-bundle LP //LG −→ LB (compare Theorem 3.32) in H ′ is classified by the morphism |Lp| ≃ L|p|.
[FRS16] smooth singular complex functor S e : H ∞ → S. Also recall the fully faithful embedding (−) : Mfd ֒→ H ∞ , with M (c) = Mfd(c, M ); under this embedding, any Lie group (H 1 , •, e H ) canonically gives rise to a group object H in H ∞ .We can now use our results from Section 3 to transfer the definition of a string group extension to the ∞-topos H of group objects in H ∞ whose its image under S e is a string group extension in S.Note that by Theorem 3.52 the functor S e maps group extensions in H ∞ to group extensions in S. Further, even though S e induces an equivalence between S and the localisation L I H ∞ rather than the full ∞-category H ∞ , we do not need to demand that A, String(H) and H are local objects, because S e sends all I-local equivalences in H ∞ to equivalences in S (Theorem 2.2(1)).Definition 4.2 is a generalisation as well as a weakening of the following approach to smooth string group extensions (see, for instance,[FRS16]): there, one works in the localisationL τ H ∞ of H ∞ atthe Čech nerves of differentiably good open coverings {c a → c} a∈Λ of cartesian spaces c ∈ Cart.Recall that a differentiably good open covering of c ∈ Cart is an open covering {c a ֒→ c} a∈Λ such that every finite non-empty intersection of the images of the patches c a is again a cartesian space.The differentiably good open coverings endow Cart with a Grothendieck coverage τ [FSS12, Sch].
∞ : Definition 4.2 Let (H 1 , •, e H ) be a compact, simple, and simply connected Lie group.A smooth string group extension of (H 1 , •, e H ) is an extension A String(H) H ι p More concretely, for smooth string group extensions A H in the sense of Definition 4.2 it is enough if there is an I-local equivalence A 1 ≃ BU(1) in H ∞ .Therefore, this setup is considerably more general than working with the pullback (4.4).In particular, two different smooth string group extensions of a Lie group H need not be equivalent in H ∞ , but only in L I H ∞ .In Section 4.3 we present a smooth string group extension which satisfies the criteria from Definition 4.8, but not the stricter version (4.4): its fibreA is not equivalent to BU(1) in L τ H ∞ ,but only in L I H ∞ ≃ S. ⊳ Remark 4.5 Not allowing for this flexibility would lead to simply missing or being unable to recognise smooth group extensions found in nature whose underlying spaces form a string group extension in the classical sense.The smooth ∞-group A we find in the string group model in Section 4.3 is much larger than the simple delooping of U(1), but this can have advantages: for example, our string group model should act extremely naturally on the K-theory of the underlying Lie group twisted by its basic gerbe.⊳ Remark 4.6 It will be very interesting to see a Lie-algebra version of Definition 4.2.The ∞-groups A ∈ Grp(H ∞ ) that we allow to appear in string group extensions can have much larger Lie algebras than those which appear in the stricter definition via (4.4).This is true, in particular, for the smooth string group extension we present in Section 4.3 below.There might hence be a Lie-algebra version of I-local equivalences of group objects in H ∞ .⊳ p → pullback in H ≤1 , where e H is the functor that sends c ∈ Cart to the constant map c → H 1 with value the unit element of H 1 .Since p 1 is a Grothendieck fibration in groupoids, this pullback is even a homotopy pullback [BMS, App.A.1]. ⊳ Theorem 4.12 [BMS, Thms.5.23, 5.27] Let Φ : H 1 × M → M be a smooth action of a connected Lie group (H 1 , •, e H ) on a manifold M .Let G ∈ Grb(M ) be a bundle gerbe on M .
[BMS,ow state the main theorem of this section.It provides a new smooth model for smooth string group extensions which fits Definition 4.2, but which lies outside the scope of the stricter definition via the pullback (4.4).Note that applying the nerve functor N to H 1 ∈ H ≤1 yields the familiar presheaf of spaces H 1 ∈ H ∞ , defined via H 1 (c) = Mfd(c, H) for cartesian spaces c ∈ Cart.Further, N maps the smooth 2-group H in H ≤1 to the group object H in H ∞ .Theorem 4.14 Let (H 1 , •, e H ) be a compact, simple, simply connected Lie group.We consider left-action of H 1 on itself via left multiplication.Let G ∈ Grb(H) be a gerbe on H 1 whose class in H 3 (H; Z) ∼ = Z is a generator (see (4.7)).Then, the sequence of group object in H ∞ is a smooth string group extension of (H 1 , •, e H ). proof of Theorem 4.14 will occupy the remainder of this section.By Definition 4.2 we have to show that the sequence (4.15) is an extension of group objects in H ∞ and that its image under the functor S e : H ∞ → S is a string group extension in S in the sense of Definition 4.1.Proposition 4.16 The sequence (4.15) is an extension of group objects in the ∞-topos H ∞ .Proof.The nerve functor N : Cat → Cat ∞ is a right adjoint and hence maps products in H ≤1 to products in H ∞ , and final objects to the final objects.Consequently, it preserves group objects and group actions.We will now use the characterisation of group extensions from Theorem 3.48(4) to show that the sequence (4.15) of group objects in H ∞ is an extension of group objects.That is, we have to show that N Sym(G) 1 with the N HLB H -action induced by the morphism N ι (cf.Proposition 3.24) is an N HLB H -principal ∞-bundle over H 1 .According to the characterisation of principal ∞-bundles in Proposition 3.31, it suffices to prove that the morphism N p 1 is an effective epimorphism and that the action of N HLB H on N Sym(G) 1 is principal.We start by showing that the morphism N p 1 is an effective epimorphism: by[BMS, Sec.5.1] the restriction p 1|c of p 1 to any fibre is essentially surjective, hence N p 1|c is surjective on connected components.Since H ∞ is a presheaf ∞-topos (in which limits and colimits are computed objectwise), a morphism in H ∞ is an effective epimorphism if and only if it is objectwise an effective epimorphism in S. The effective epimorphisms in S, however, are exactly those morphisms which are surjective on connected components [Lur09, Cor.7.2.1.15].Therefore, N p 1 is an effective epimorphism in H ∞ .
[BMS,ction of N HLB H on N Sym(G) 1 is principal with respect to N p 1 as was shown in[BMS, Thm.5.27] (there, the principality condition was shown on the level of the sequence (4.13) of smooth 2-groupsthis suffices for the ∞-categorical context used here because of Lemma 3.29 and because the nerve functor is a right adjoint).Therefore, the sequence (4.15) is a group extension in H ∞ .It thus remains to show that the image of the sequence (4.15) under S e is a string group extension in S.