Integration of Lie 2-algebras and their morphisms

Given a strict Lie 2-algebra, we can integrate it to a strict Lie 2-group by integrating the corresponding Lie algebra crossed module. On the other hand, the integration procedure of Getzler and Henriques will also produce a 2-group. In this paper, we show that these two integration results are Morita equivalent. As an application, we integrate a non-strict morphism between Lie algebra crossed modules to a generalized morphism between their corresponding Lie group crossed modules.


Introduction
Recently people have paid much attention to the integration of Lie-algebra-like structures, such as that of Lie algebroids [7,9,27], of L ∞ -algebras [13,14] and of Courant algebroids [17,19,24]. Here "integration" is meant in the same sense in which a Lie algebra is integrated to a corresponding Lie group.
For an L ∞ -algebra h, there is an infinite dimensional Kan simplicial space h constructed in [13,14], whose k-cells are given by L ∞ -algebroid morphisms T ∆ k → h. Applying to a strict Lie 2-algebra h, the 2-truncation τ 2 ( h) is a 2-group which is believed to be the universal integration of h.
On the other hand, a strict Lie 2-algebra (resp. strict Lie 2-group) one-to-one corresponds to a Lie algebra (resp. Lie group) crossed module. Thus, a strict Lie 2-algebra can be easily integrated to a strict Lie 2-group by integrating its corresponding crossed module. In this article, we prove that these two integration results are Morita equivalent. Already noticed in [4], the classifying Postnikov data of Lie 2-algebras is the quotient Lie algebra in degree 0, a choice of Lie module and of a corresponding 3-cocycle. When the Lie module structure is trivial, as in the case of string Lie 2-algebra, the above Morita equivalence may be implied via a homotopy theoretical method (see 0 Keyword: L∞-algebras, L∞-morphisms, crossed modules, Lie 2-groups, integration 0 MSC: Primary 17B55. Secondary 18B40, 18D10. * The first author is supported by NSFC (11026046,11101179), SRFDP (20100061120096) and "the Fundamental Research Funds for the Central Universities" (200903294). The second author is supported by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Göttingen. [11,Sect. 4.1.3]). Our article further provides an explicit Morita morphism generally regardless the triviality of the Lie module. We must warn the readers that we treat finite dimensional case only because we need to use the fact that the second homotopy group of a Lie group is trivial, and this is true only in the finite dimensional case.
As L ∞ -algebras and their integration play an important role in higher gauge theory [5] and higher Chern-Weil theory [11], we believe our explicit construction will have potential application in mathematical physics.
As an application, we use the above result to integrate a nonstrict morphism between strict Lie 2-algebras to a generalized morphism between their strict Lie 2-groups. We must mention that an integration of such morphisms is also provided via the technique of butterflies in [20]. Here we also provide some mathematical physics oriented examples of such morphisms: they can encode 2-term L ∞ -modules of g in the sense of [16], or equivalently, 2-term representation up to homotopy in the sense of [2], non-abelian extensions of g, and up to homotopy Poisson actions of g in the sense of [21]. Further application of the integration is provided in [23]. Acknowledgement: We thank very much the referee for very helpful comments. Both authors give their warmest thanks to Courant Research Centre "Higher Order Structures", Göttingen University, and Jilin University, where this work was done during their visits.

Equivalence of Integrations
For an L ∞ -algebra h, there is an infinite dimensional Kan simplicial space h constructed in [13,14], Here we remind the readers that ∧ • (h) has a natural differential graded commutative algebra (d.g.c.a.) structure which generalizes the Chevalley-Eilenberg complex for a Lie algebra. It is shown in this paper, that when h is a Lie algebra, the one-truncation τ 1 [23,Remark 3.7]). In this section, we show the isomorphisms between these two 2-groups.

Background on L ∞ algebras
In this section, we briefly review the notions of L ∞ -algebras and crossed modules of Lie algebras. They both provide models for strict Lie 2-algebras.
L ∞ -algebras, sometimes called strongly homotopy Lie algebras, were introduced by Stasheff [25] as a model for "Lie algebras that satisfy Jacobi identity up to all higher homotopies". The following convention of L ∞ -algebras is the same as Lada and Markl in [16].
where the exterior powers are interpreted in the graded sense and the following relation with Ksozul sign "Ksgn" is satisfied for all n ≥ 0: where the summation is taken over all (i, n − i)-unshuffles with i ≥ 1.
If L is concentrated in degrees < n, we obtain the notion of n-term L ∞ -algebras. A semistrict Lie 2-algebra can be understood as a 2-term L ∞ -algebra. a strict Lie 2-algebra is a 2-term L ∞ -algebra, in which l 3 is zero (see [4] Here Der(h 1 ) is the derivation Lie algebra of h 1 with the commutation Lie bracket [·, ·] C .
The following result is well known.

Theorem 2.3. There is a one-to-one correspondence between strict Lie 2-algebras and crossed modules of Lie algebras.
For the precise relation between the operation l 2 and the Lie brackets [·, ·] h0 and [·, ·] h1 , please see [23]. The key difference is that l 2 (m, n) = 0, for any m, n ∈ L 1 = h 1 , and [m, n] h1 = l 2 (dm, n) = 0. On the direct sum h 0 ⊕ h 1 , there is also a Lie bracket [·, ·] h0⊕h1 , which is the semidirect product of the Lie algebra h 0 and the Lie algebra h 1 :

Background on 2-groups
A group is a monoid where every element has an inverse. A 2-group is a monoidal category where every object has a weak inverse and every morphism has an inverse. Denote the category of smooth Banach manifolds and smooth maps by Diff, a semistrict Lie 2-group is a 2-group in DiffCat, where DiffCat is the 2-category consisting of categories, functors, and natural transformations in Diff. In the sequel, all the Lie 2-groups are semistrict.

Definition 2.5. A semistrict Lie 2-group consists of an object C in DiffCat together with
• a multiplication morphism (horizontal multiplication) · h : C × C −→ C, • identity object 1, • an inverse map inv : C −→ C together with the following natural isomorphisms:: • the associator • the left and right unit • the unit and counit such that the pentagon identity for the associator, the triangle identity for the left and right unit, the first and second zig-zag identities are satisfied. We refer to [3,Definition 7.1].
As pointed out in [3,Sect. 7], if the category C carries a semistrict Lie 2-group structure, then C must be a Lie groupoid. We denote the groupoid multiplication in C by · v (vertical multiplication).
In the special case when a x,y,z , l x , r x , i x , e x are all identity isomorphisms, we obtain the concept of a strict Lie 2-group. It is well-known that strict Lie 2-groups can be described by crossed modules of Lie groups.
and t satisfies the so called Peiffer identity: The following result is well-known, see [3,12] for more details.

Theorem 2.7.
There is a one-to-one correspondence between crossed modules of Lie groups and strict Lie 2-groups.
Roughly speaking, given a crossed module (H 1 , H 0 , t, Φ) of Lie groups, the corresponding strict Lie 2-group has C 0 = H 0 and C 1 = H 0 ⋉ H 1 , the semidirect product of H 0 and H 1 . In this strict Lie 2-group, the source and target maps s, t : C 1 −→ C 0 are given by the vertical multiplication · v is given by: the horizontal multiplication · h is given by Definition 2.8. Given two Lie 2-groups C and C ′ , a unital morphism F : such that F 0 (1 C ) = 1 C ′ and the following diagrams commute: • the compatibility condition of F 2 with the associator: • the compatibility condition of F 2 with the left and right unit:

Equivalence of two 2-groups
Given a Lie algebra g, denote by P g the usual C 2 path space in g and by P 0 g the C 2 path space in g with a convenient boundary condition, Then both P g and P 0 g naturally have a smooth structure of Banach manifold because we choose C 2 paths (see for example [27,Sect. 2]). From now on, when not specially mentioned, all morphisms are of C 2 -classes. The paths a 0 and a 1 are said to be g-homotopic and we write a 0 ∼ a 1 , if there exist C 2morphisms a, b : [0, 1] ×2 → g satisfy the following differential equation with boundary value b(0, s) = 0, b(1, s) = 0, a(t, 0) = a 0 (t) and a(t, 1) = a 1 (t). This is equivalent [8] to the fact that, a(t, s)dt + b(t, s)ds : T I × T I −→ g is a Lie algebroid morphism and b(0, s) = b(1, s) = 0. The g-homotopy also restricts to P 0 g (see [27]). Then the simply connected Lie group G of g is the quotient (see [10, Sect. 1.13] for more details). Next, we recall the construction (in [14]) of the 2-group structure of τ 2 ( h) for a strict Lie 2-algebra h. Let There is an equivalence relation ∼ defined on P 2 h: with boundary conditions: Then P 2 h/ ∼⇒ P 1 h is a groupoid with the source and target evaluation of a on s = 0 and s = 1 respectively. Moreover the 2-group of τ 2 ( h) is exactly the 2-group structure on P 2 h/ ∼⇒ P 1 h with vertical multiplication the concatenation with respect to the parameter s and horizontal multiplication the concatenation with respect to the parameter t. Later on we will give a reparametrized horizontal multiplication (21) (22) (23) for convenience. However, we notice that reparametrization will not change the class in P 2 h: Lemma 2.9. Given an element (a, b, z) ∈ P 2 h and reparametrizations τ i : Proof. In general, elements in P n h are d.g.c.a. morphisms ∧ • h → Ω • ([0, 1] ×n ) with certain boundary conditions, and the homotopies ∼ are d.g.c.a. morphisms We first construct our equivalence with the help of a couple of lemmas.
Proof. The conclusion follows from s) is not necessarily zero and this is exactly the obstruction of a(−, 0) and a(−, 1) being homotopic.
Proposition 2.11. With the above notations, the concatenation of d∆b(1, −) and where the concatenation ⊙ of two paths, a(t) and b(t) is defined as follows: with a cut-off function τ : Since b(0, s) = 0, we know that g(0, s) is fixed. Since b(1, s) = 0, g(1, s) is not a constant path in H 0 . So g(t, 0) and g(t, 1) are not homotopic. However, it is obvious that the concatenation of g(1, s) and g(t, 0) is homotopic to g(t, 1) in the Lie group H 0 . Therefore, the corresponding h 0 -paths are h 0 -homotopic.
is the solution of the following ODE: with the initial value w(0, s) = v(s). (1). This completes the proof of item (1). Item (2) can be proved similarly.

Consequently, the corresponding group action of
spectively, assume that ∆b and ∆b † are the corresponding solutions of (15) respectively. We reparametrized the concatenation with respect to t, namely the bigon of horizontal multiplication and denote by ∆b ‡ the corresponding solution of (15).
When 1 ≤ t ≤ 2, by straightforward computations, we have The last equality holds because when in which ∆b is the unique solution of (15) with the initial value ∆b(0, s) = 0. To see that Ψ 1 is well defined, for two elements (a 0 , b 0 , z 0 ) and (a 1 , b 1 , z 1 ) in P 2 h equivalent through (a, b, c, x, y, z), we need to prove that ∆b(1, s, 0) and ∆b(1, s, 1) are homotopic in the Lie algebra h 1 . This follows from the next lemma.
Morita equivalence are defined for n-groupoids in an arbitrary category with a certain Grothendieck pretopology in [30]. We adapt this notation to our situation: a morphism F : C → C ′ of Lie 2group is a hypercover and denoted by F : 0 is a surjective submersion, and the natural map With the above preparations, we have Theorem 2.15. There is a Lie 2-group Morita equivalence given by a morphism (Ψ 0 , Ψ 1 , Ψ 2 = id): where Ψ 0 (a(t)) = [a(t)], which is the equivalence class of the path a(t) and Ψ 1 is given by (26).

Remark 2.16.
There is an integration obstruction proved in [14], that is, the quotient P 2 h/ ∼ might not be representable as a Banach manifold unless a certain obstruction class vanishes. In this theorem, we show directly (Prop. 2

.18) that τ 2 ( h) is always representable.
We prove it by several steps.
Proof. Obviously, (Ψ 1 , Ψ 0 ) respects the source and target maps. It is not hard to see that (Ψ 1 , Ψ 0 ) is a morphism with respect to the vertical multiplication. In fact, for h, assume that ∆b, ∆b ♯ are the corresponding solutions of (15) respectively.
By definition, we have On the other hand, it is straightforward to see that ∆b ♯ ⊙ ∆b is the solution of (15) for the bigon Next we prove that (Ψ 1 , Ψ 0 ) is also a morphism with respect to the horizontal multiplication. By (6), we have where Φ is given by (20) which integrates the action of h 0 on h 1 .
On the other hand, by Lemma 2.13 and Lemma 2.12, we have which implies that i.e. Ψ 1 is a morphism with respect to the horizontal multiplication. Finally, since the right hand side of (40) is a strict 2-group and (Ψ 1 , Ψ 0 ) preserves the horizontal multiplication strictly, condition (7) reduces to This holds obviously because a a3,a2,a1 being a reparametrization between (a 3 ⊙ a 2 ) ⊙ a 1 and a 3 ⊙ (a 2 ⊙ a 1 ) must be a homotopy by [9,Lemma 1.5]. Similarly, condition (8) holds.
It is clear that Ψ 0 sends any smooth path in P h to a smooth path in H 0 . Moreover, for a smooth family of homotopies (parametrized by u) A u = a u (t, s)dt + b u (t, s)ds, the solution ∆b u (t, s) of (15) depends smoothly on u. Thus, both Φ 0 and Φ 1 are smooth. a(−, 0), a(−, 1) is an isomorphism.

Proposition 2.18. The natural map
We first remark that P 1 h = P h 0 and H 0 is a quotient of P h, thus Ψ 0 : P 1 h → H 0 is a surjective submersion of Banach manifolds. Thus this proposition will automatically imply that P 2 h/ ∼ is representible and hence τ 2 ( h) is a Lie 2-group. The morphism we demonstrate in last lemma will further be a Lie 2-group morphism. Now we prove this lemma by constructing an inverse morphism. We notice that the Lie group H 1 = P 0 h 1 / ∼. Given an element h 0 , h 1 , a 0 , a 1 on the left hand side, we take a representative ∆b(s) ∈ P 0 h 1 of h 1 , then there are a(t, s), b(t, s) satisfying h 0 -homotopy equation (16) and the boundary conditions as in Lemma 2.10. We extend ∆b(s) to a morphism ∆b(t, s) a(1, s), ∆(s)). (41) Such extension always exists. For example, we take Proof. If we take another representative ∆b 1 ∈ P 0 h 1 which is equivalent to ∆b in P 0 h 1 via ∆b(s, u) and ∆c(s, u), that is and a 0 via a(t, s), b(t, s) and a 1 ∼ d(∆b 1 ) ⊙ a 0 via a 1 (t, s), b 1 (t, s). Since π 2 (H 0 ) = 0, there is no higher obstruction between h 0 -homotopies from being homotopic, so (a(t, s), b(t, s)) and (a 1 (t, s), b 1 (t, s)) must be homotopic via a certain homotopy with boundary conditions a (t, s, 0) = a(t, s), a(t, s, 1) = a 1 (t, s), (t, s), b(t, s, 1) = b 1 (t, s), b(0, s, u) = 0, b(1, s, u) = d(∆b(s, u)), c(0, s, u) = 0, c(1, s, u) = d (∆c(s, u)), c(t, 0, u) = 0, c(t, 1, u) = 0. Now we repeat the construction of z(t, s) for each u, and we obtain z(t, s, u) and b(t, s, u) with correct boundary conditions satisfying (11). We need to show that (a, b, z)| u=0 ∼ (a, b, z) Firstly, by a similar method, we construct y(t, s, u) and c(t, s, u) with correct boundary conditions 2 and satisfying (12).
It is obvious that ̟ • ζ = id. To finish the proof of Theorem 2.15, we still need to show that ζ • ̟ = id. Given an element (a, b, z) ∈ P 2 h, since ̟ does not depend on the choice of representative, we choose a convenient reparametrization such that z(t, s)| s=0,1 = 0. Thus the solution ∆b in Lemma 2.10 also has ∆b(t, s)| s=0,1 = 0. Following ̟ then ζ, we first restrict ∆b on t = 1, then extend it again to all t by (42), thus we might end up with another ∆b 1 , with the same boundary value, that is when either t or s is 0 or 1. Thus Theorem 2.15 follows immediately from the following lemma: Proof. We suppose that there are two such extensions ∆b(t, s) and ∆b 1 (t, s). We connect them by ∆b u := u∆b + (1 − u)∆b 1 . Then the corresponding b := b − d∆b and b 1 := b − d∆b 1 are connected by b u := b − ∆b u ; the corresponding z and z 1 are connected by z u := l 2 (a, ∆b u ) − ∂ t ∆b u . Now we take a(t, s, u) = a(t, s), b(t, s, u) = b u , c = 0, y = 0, x = ∂ u ∆b u , then it is obviously to see that (11), (12) hold. Equation (13) is implied by the fact that b does not depend on u. Equation (14) is implied by the fact that a does not depend on u. The boundary condition of x is implied by that of ∆b and ∆b 1 . Thus (a, b, z)| u=0 is homotopic to (a, b, z)| u=1 .

Application on Integration of (non-strict) Lie 2-algebra morphisms
Lie's theorem II tells us that Lie algebra morphisms can integrate to Lie group morphisms. As pointed out in [11,Def. 4.2.8] (and also easy to see), an L ∞ -morphism between L ∞ -algebras f : g → h induces a natural map f : g → h of Kan complex. Thus applying in the case of Lie 2-algebras, an L ∞ -morphism between Lie 2-algebras (also called non-strict Lie 2-algebra morphisms) f : g → h can integrate to a 2-group morphism τ 2 ( f ) : τ 2 ( g) → τ 2 ( h). Combining with our result, we have between the corresponding (simply-connected) Lie group crossed modules.

Remark 3.2.
It is fairly easy to integrate a strict morphism which consists of Lie algebra morphisms f i : g i → h i preserving all crossed module structures. One only needs to integrate f i individually as a Lie algebra morphism. The integration of nonstrict morphism is also addressed in the context of butterflies [20]. Butterflies between crossed modules are believed 3 to be equivalent to generalized morphisms between strict Lie 2-groups.
Finally, we call the generalized morphism above an integration of f based on the fact that τ 2 ( f ) should be considered as a canonical integration. However we do not justify the concept of integration by the inverse procedure, namely differentiation. Now we concentrate on Lie 2-algebra morphisms from a Lie algebra to a strict Lie 2-algebra. We will see that several interesting objects can be described by such a morphism, including 2-term representations up to homotopy of Lie algebras, non-abelian extensions of Lie algebras and up to homotopy Poisson actions.
We first recall an explicit formulation of L ∞ -morphism that we will mention in the examples: 3. An L ∞ -morphism from a Lie algebra g to a strict Lie 2-algebra L 1 d −→ L 0 consists of linear maps µ : g −→ L 0 and ν : g ∧ g −→ L 1 such that the obstruction of µ being a Lie algebra morphism is given by and ν satisfies the following condition: where c.p. means cyclic permutations.

• 2-term representations up to homotopy of Lie algebras
Associated to any k-term complex of vector spaces V, there is a natural DGLA (differential graded Lie algebra) gl(V) [16,23], which plays the same role as gl(V ) for a vector space V in the classical case. An L ∞ -module [16] of an L ∞ -algebra L is given by an L ∞ -morphism from L to gl(V). Associated to any 2-term complex of vector spaces V, by truncation of gl(V), we obtain a strict Lie 2-algebra, which we denote by End(V). The degree 0 part End 0 (V) is given by and the degree 1 part End 1 (V) is Hom(V 0 , V 1 ). The Lie bracket of End(V) is given by the commutator and the differential is induced by d. It turns out that for 2-term L ∞ -modules of a Lie algebra g, it is enough to look at morphisms to the strict Lie 2-algebra End(V): Proposition 3.4. [16]A 2-term L ∞ -module of a Lie algebra g is given by an L ∞ -morphism from g to End(V).
A 2-term L ∞ -module of a Lie algebra g is the same as a representation up to homotopy of the Lie algebra g on a 2-term complex of vector spaces, see [2,22] for more details. Thus Corollary 3.1 can be applied to integrate L ∞ -modules V of a Lie algebra g to that of a Lie group G. This is studied further in [23], where the semidirect product g⋉V is also integrated. It then has application in integrating omni-Lie algebras and Courant algebroids [24].

• Non-abelian extensions of Lie algebras
It is well known that abelian extensions of a Lie algebra g give rise to a representation of g and the equivalence classes of extensions are in one-to-one correspondence with the second cohomology. In the following, we will see that a non-abelian extension of a Lie algebra g, given by a short exact sequence of Lie algebras can be realized as an L ∞ -morphism from Lie algebra g to the strict Lie 2-algebra k ad −→ Der(k) (see Example 2.4).
By choosing a splitting of p, we can always assume thatĝ = g ⊕ k as vector spaces. Then the Lie bracket [·, ·]ĝ decomposes as below, Since p is a morphism of Lie algebras, there is a linear map ν : g ∧ g −→ k such that [X 1 , X 2 ]ĝ = [X 1 , X 2 ] g + ν(X 1 , X 2 ).
As stated in [21], another equivalent formulation of up to homotopy Poisson action is an L ∞ morphism from g to the DGLA L π M . We further simplify this statement. Denote by X(M ) π the set of vector fields preserving the Poisson structure π, i.e. X(M ) π = {X ∈ X(M ), [X, π] = 0}.
By truncation, we obtain a strict Lie 2-algebra C ∞ (M ) [π,·] −→ X(M ) π , of which the degree 1 part is C ∞ (M ), the degree 0 part is X(M ) π and the differential is [π, ·]. The extension g M of g by C ∞ (M ) is totally determined by a linear map µ : g −→ X(M ) π and a linear map ν : g ∧ g −→ C ∞ (M ), which satisfy the following equation −→ X(M ) π .
Remark 3.8. We only need to use the fact π 2 (H 0 ) = 0 in the construction of g in the last section. Without this condition, we will still have a morphism even though not a Morita morphism. The space of X(M ) π is infinite dimensional and does not admit a Banach structure. However, there is also certain infinite-dimensional calculus available in this case (see for example [28,App.A]). Thus our result can not be applied directly, however certain modification may be applied.