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Stacky Lie Algebroids Stacky Lie algebroids Miquel Cueca Ten (joint with Daniel Alvarez)´ Georg-August-Universit¨at G¨ottingen,Germany October 9, 2020 Motivation LG = fLie groupoids categoryg; LA = fLie algebroids categoryg and we have the Lie functor between them Lie : LG ! LA; but Lie is not essentially surjective Molino, Weinstein... (Explicit obstructions computed by Crainic-Fernandes). What is the integration of a general Lie algebroid? Answer Tseng-Zhu: an ´etale stacky Lie groupoid. What is the infinitesimal counterpart of a stacky Lie groupoid? A stacky Lie algebroid. UFO!! Moreover we want the following diagram to commute Lie LG −−! LA \\ SLie SLG −−! SLA Structure of the talk Goal Introduce and study stacky Lie algebroids (A) Two related problems: 1. Stacky Lie algebroids. 2. Lie algebroids over a differentiable stack. (B) Defining stacky Lie algebroids. (C) Main properties. (D) Work in progress: 1. SLie functor. 2. Integration of Courant algebroids. Part A Two related problems: Stacky Lie algebroids and Lie algebroids over a differentiable stack. Differentiable stacks Grothendieck, Behrend-Xu Idea: Differentiable stacks are good models for singular spaces. Man = fSmooth manifolds categoryg with the Grothendieck topology given by open covers. Differentiable stack: A category X with a functor FX : X ! Man satisfying: 1. Pullbacks exists and are \universal". 2. Gluing properties (sheaves like conditions). 3. It has a presentation p : X ! X. Differentiable stacks form a 2-category: I Objects: Differentiable stacks X; J... I 1-morphisms: Functors F : X ! J s.t. FJF = FX. 0 I 2-morphisms: Natural trans η : F ) F s.t. FJη(x) = idFX(x). The dictionary Given G ⇒ M a Lie groupoid define the category [M=G]: I Objects: Principal right G-bundles. I Morphism: morphisms of principal G-bundles. Theorem Behrend-Xu: 1. For any lie groupoid G ⇒ M,[M=G] is a differentiable stack. 2. Given a differentiable stack with a presentation p : X ! X; ∼ X ×X X ⇒ X is a Lie groupoid and X = [X =X ×X X ]. ∼ 3. G ⇒ M and H ⇒ N Morita equivalent iff [M=G] = [N=H]. Remark:The dictionary can be extended to 1- and 2-morphisms. Conclusion Differentiable stacks can be thought as the Morita class of a Lie groupoid. Stacky Lie groupoids Tseng-Zhu A Stacky Lie groupoid G ⇒ M consist of a differentiable stack G and a manifold M together with I 1-morphisms: s; t; m; 1; i. I 2-isomorphisms: α : m(id ×m) ) m(m × id), ul : mh1t; idi ) id, ur : mhid; 1si ) id, ιl : mhi; idi ) 1s, ιr : mhid; ii ) 1t. satisfying 1. s1 = idM , t1 = idM , si = t, ti = s, sm = sns , tm = tnt . 2. Higher coherence conditions for the 2-isomorphism. Why? Integrate Lie algebroids Tseng-Zhu, symmetries of stacks Blohmann, actions on differentiable stacks Bursztyn-Noseda-Zhu, hamiltonian actions on symplectic stacks Hoffman-Sjamaar-Zhu... Examples I Any Lie groupoid G ⇒ M. I If H ⇒ G is a strict 2-groups then [G=H] ⇒ ∗ is a stacky Lie group. More generally, given D / K M / M a double Lie groupoid covering the identity groupoid then [K=D] ⇒ M is a stacky Lie groupoid. I Tseng-Zhu: For any Lie algebroid A ! M the Weinstein groupoid W(A) ⇒ M is a stacky Lie groupoid. Guiding Theorem (Zhu) There is a 1-1 correspondence between stacky Lie groupoids and Lie 2-groupoids modulo 1-Morita equivalence. Problem 1: Stacky Lie algebroids 1. Strategy 1 : Define vector bundle E ! M and a Lie algebroid structure there. Waldron, still work in progress!! 2. Strategy 2: Using the dictionary re-interpret stacky Lie groupoids as a double structure and differenciate. G ⇒ X ! G ?? ## M ⇒ M ! M Then a stacky Lie algebroid will be the \Morita" class of one of such structures. Problem 2: Lie algebroids over a differentiable stack Including double groupoids and applying the Lie functor −−!Lie D / K / [K=D] AD / AK / [AK =AD ] G / M / [M=G] G // M / [M=G] If G = M \Stacky side", strict arrows \Double side". Part B Defining stacky Lie algebroids via quasi LA-groupoids Semistric Lie 2-algebras Baez-Crans If G = M = {∗} qLA-groupoids already appear in the literature. A semistrict Lie 2-algebra L ≡ L1 ⇒ L0 is a 2 vector space endowed with 1. A skew-symmetric bilinear functor [·; ·]: L × L ! L 2. An antisymmetric trilinear natural isomorphism αx;y;z : [[x; y]; z] ) [x; [y; z]] + [[x; z]; y]. satisfying α[w;x];y;z [αw;x;z ; y] + 1 αw;[x;z];y + α[w;z];x;y + αw;x;[y;z] = [αw;x;y ; z] α[w;y];x;z + αw;[x;y];z [αw;y;z ; x] + 1 [w; αx;y;z ] + 1 Why it works? Theorem Baez-Crans: There is a 1-1 correspondence between semistrict Lie 2-algebras and 2-term L1-algebras. Graded manifolds perspective Replacing a 2 vector space by 2 vector bundle VB-groupoid. Problem: for Lie algebroids, [·; ·] is not a functor!! Theorem Vaintrob: There is a 1-1 correspondence between Lie algebroids and degree 1 Q-manifolds. Theorem Mehta: There is a 1-1 correspondece between VB-groupoids and degree 1 Lie groupoids. Moreover, LA-groupoids are in correspondence with degree 1 Q-groupoids. Definition Mehta A degree n Q-groupoid (G ⇒ M; Q) is a Lie groupoid object between degree n manifolds endowed with Q 2 X1(G) such that Q2 = 0 and Q : G! T [1]G groupoid morphism. Quasi Q-groupoids Q2;0 G / T [2]G ; α M / T [2]M Definition 1 A degree n quasi Q-groupoid (G ⇒ M; Q; α) is a Lie groupoid object in the category of degree n manifolds, Q 2 X1(G) and α 2 Γ2AG such that Q : G! T [1]G groupoid morphism and Q2 = αl − αr [Q; αl ] = 0: 0 0 0 0 (F ; R):(G ⇒ M; Q; α) ! (G ⇒ M ; Q ; α ) is a morphism if 1. F : G!G0 groupoid morphism. 2. R : T [1]F ◦ Q ) Q0 ◦ F natural transformation such that 0 T [1]F ◦ αb = (αb ◦ F )R(R ◦ Q) Classical definition Given H // E G / M a VB-groupoid then H[1] ⇒ E[1] defines a degree 1 groupoid. Definition 2 A quasi LA-groupoid structure on (H ⇒ E; G ⇒ M) is a quasi Q-groupoid structure on H[1] ⇒ E[1]. That means that H ! G carries an anchor and a bracket but also I A new 3-bracket that controls Jacobi. I A new 2-anchor that controls the anchor being bracket preserving. many equations!! Particular case If the base groupoid is M ⇒ M (this is the case that we need) we have: Proposition 1 A qLA-groupoid on (H ⇒ E; M ⇒ M) is equivalent to an almost Lie algebroid (H ! M; [·; ·]; a) and φ : ^3E ! C satisfying: 1. graph(mH ) ! graph(mM ) is an almost Lie subalgebroid of H × H × H ! M × M × M. l r 2. Jac[·;·](h0; h1; h2) = (φ − φ )(h0; h1; h2). l 3. a(φ (h0; h1; h2)) = 0. l l l l 4. [φ (h0;h1;h2);h3]+[φ (h2;h3;h0);h1]+φ ([h0;h2];h1;h3)+φ ([h1;h3];h0;h2)= l l l l φ ([h0;h1];h2;h3)+φ ([h0;h3];h1;h2)+φ ([h1;h2];h0;h3)+φ ([h2;h3];h0;h1) l l +[φ (h1;h2;h3);h0]+[φ (h3;h0;h1);h2]. l where φ (h0; h1; h2) = mH 0; φ(tH (h0); tH (h1); tH (h2)) Morphisms can also be re-interpret in classical terms. Stacky Lie algebroids A morphism (F ; R):(H; Q; α) ! (H0; Q; α0) between qLA-groupoids is a Morita map iff F : H ! H0 is VB-Morita as defined by Del Hoyo-Ortiz. Two qLA-groupoids over M ⇒ M,(H1; [·; ·]1; a1; φ1) and (H2; [·; ·]2; a2; φ2) are Morita equivalent if there exists (H3; [·; ·]3; a3; φ3) and Morita maps F1;R1 F2;R2 H1 −−− H3 −−−! H2 such that Fi on the base are the identity on M. Definition 3 A stacky Lie algebroid over M is the Morita class of a qLA-groupoid over M ⇒ M. Part C Main properties of Stacky Lie algebroids. L1-algebroids Sheng-Zhu A n-term L1-algebroid is a non-positively graded vector bundle n−1 A = ⊕i=0 A−i ! M together with an anchor ρ : A0 ! TM and graded antisymmetric brackets fli g of degree 2 − i such that X X Ksgn(σ)lj (li (xσ(1); ··· ; xσ(i)); ··· ; xσ(r)) = 0 i+j=r+1 σ2Sh(i;j−1) I For n = 1 (A0; ρ, [·; ·]) Lie algebroid, @ I For n = 2 (A−1 −! A0; ρ, [·; ·]; r; [·; ·; ·]). Theorem Bonavolonta-Poncin There is an equivalence of categories between n-term L1-algebroids and degree n Q-manifolds. The infinitesimal counterpart of Lie 2-groupoids are 2-term L1-algebroids Severa, Zhu, Li-Zhu, Severa-Siran... Dold-Kan Correspondence Theorem Gracia-Saz−Mehta: VB-groupoids (H ⇒ E; G ⇒ M) 2-term representations up to homotopy of G ⇒ M. Observe that a 2-term representation up to homotopy of M ⇒ M is the same as a 2-term complex C −!@ E. Theorem 1 There is a 1-1 correspondence between I quasi LA-groupoids covering the unit groupoid. I 2-term L1-algebroids. Moreover, the quasi LA-groupoids are Morita equivalent iff the 2-term L1-algebroids are quasi-isomorphic. Infinitesimal counterpart of Zhu's Guiding Theorem. More equivalences Corollary 1 There is a 1-1 correspondence between the following sets: 1. Stacky Lie algebroids. 2. Morita classes of quasi LA-groupoids over the unit groupoid. 3. Quasi-isomorphism classes of 2-term L1-algebroids.
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