Formality for algebroid stacks The 2-groupoid of Hochschild - PowerPoint PPT Presentation

Formality for algebroid stacks Ryszard Nest 2-groupoids 2-groupoid associated to a DGLA Σ construction Formality for algebroid stacks The 2-groupoid of Hochschild cochains Star products and the Deligne 2-groupoid Deformations Ryszard Nest of smooth manifolds Stacks and cocycles University of Copenhagen Jets Formality for stacks 22nd August 2014 Twisting jets Formality theorem

Formality for algebroid stacks Ryszard Nest 2-groupoids 2-groupoid associated to a DGLA Σ construction The 2-groupoid of Hochschild cochains Joint work with Star products and the Deligne 2-groupoid Paul Bressler, Alexander Gorokhovsky, Boris Tsygan Deformations of smooth manifolds Stacks and cocycles Jets Formality for stacks Twisting jets Formality theorem

� � Formality for algebroid stacks 2-groupoids Ryszard Nest Data: 2-groupoids 2-groupoid 1 Units G 0 • x associated to a DGLA Σ construction γ The 2-groupoid of � • y 2 Arrows G 1 • x Hochschild cochains Star products and the Deligne composable when range of one coincides with the source 2-groupoid Deformations of the next one. of smooth manifolds 3 Two-morphisms G 2 Stacks and cocycles • x Jets Formality for stacks Twisting jets Formality theorem θ γ 1 = γ 2 ⇒ • y

� � � � � � � � Formality for Two-morphisms have a "natural" composition structure algebroid stacks satisfying natural associativity conditions. Ryszard Nest Horizontal 2-groupoids 2-groupoid associated to a DGLA γ 2 Σ construction θ τ The 2-groupoid of Hochschild cochains γ 1 γ 3 Star products and the Deligne 2-groupoid Deformations τ ◦ θ of smooth manifolds and Stacks and cocycles Jets Vertical Formality for stacks Twisting jets Formality theorem θ γ 1 = γ 2 ⇒ µ 1 ◦ γ 1 µ 2 ◦ γ 2

Formality for algebroid stacks Ryszard Nest 2-groupoids 2-groupoid associated to a DGLA Σ construction Canonical example - 2-groupoid of categories The 2-groupoid of Hochschild cochains Star products and the Deligne 1 Objects: Categories 2-groupoid Deformations 2 Arrows: Functors of smooth manifolds 3 2-morphisms: Natural transformations of functors Stacks and cocycles Jets Formality for stacks Twisting jets Formality theorem

Simplicial nerve of a 2-groupoid Formality for algebroid stacks 1 N 0 G is the set of objects of G . Ryszard Nest 2 For n ≥ 1, N n G is the set of data: 2-groupoids 2-groupoid (( µ i ) 0 ≤ i ≤ n , ( g ij ) 0 ≤ i < j ≤ n , ( c ijk ) 0 ≤ i < j < k ≤ n ) , associated to a DGLA Σ construction where The 2-groupoid of Hochschild cochains Star products and 1 ( µ i ) is an n -tuple of objects of G , the Deligne 2-groupoid 2 ( g ij ) is a collection of 1-morphisms with g ij : µ j → µ i and Deformations 3 ( c ijk ) is a collection of 2-morphisms with c ijk : g ij g jk → g ik of smooth manifolds which satisfies c ijl c jkl = c ikl c ijk (in the set of 1-morphisms Stacks and cocycles g ij g jk g kl → g il ). Jets Formality for stacks Twisting jets Formality theorem Nerve N ∗ G of a 2-groupoid is a Kan simplicial set with π n ( N ∗ G ) = 0 for n > 2. Definition Two 2-groupoids are equivalent, if their nerves are weakly homotopy equivalent.

Formality for algebroid stacks A bigroupoid is a similar data as a 2-groupoid, except that the associativity Ryszard Nest of the composition G 1 ( x , y ) × G 1 ( y , z ) → G 1 ( x , z ) is satisfied up to a natural transformation which satisfies the pentagonal identity. 2-groupoids 2-groupoid Duskin associated to a DGLA Σ construction There exist indempotent endofunctors The 2-groupoid of Hochschild cochains Π n : { simplicial Kan sets } → { simplicial Kan sets } , π k (Π n ( · )) = 0 for k > n , Star products and the Deligne 2-groupoid and the range of Π 2 consist of nerves of bigroupoids (up to weak homotopy Deformations of smooth equivalence). manifolds Stacks and cocycles As above, two bigroupoids are called equivalent, if their nerves are weakly Jets homotopy equivalent. Formality for stacks Twisting jets Formality theorem Suppose that g is a DGLA. A Maurer-Cartan element of g is an element γ ∈ g 1 satisfying d γ + 1 2 [ γ, γ ] = 0 . (1)

Formality for algebroid 2-groupoid of a nilpotent DGLA with g i = 0 for i < − 1 stacks Ryszard Nest 1 MC 2 ( g ) 0 is the set of Maurer-Cartan elements of g . 2-groupoids 2 MC 2 ( g ) 1 ( γ 1 , γ 2 ) = exp g 0 . 2-groupoid associated to a Here the product in the unipotent group exp g 0 is defined by the DGLA Σ construction Hausdorff-Cambell formula and exp g 0 acts on the set of The 2-groupoid of Hochschild cochains Maurer-Cartan elements of g by d + ad γ 2 = Ad exp X ( d + ad γ 1 ) . Star products and the Deligne 3 Given γ ∈ MC 2 ( g ) 0 , MC 2 ( g ) 2 ( exp X , exp Y ) = exp γ g − 1 . 2-groupoid Deformations Here g − 1 has the Lie bracket given by [ a , b ] γ = [ a , db + [ γ, b ]] and of smooth manifolds exp γ t acts by ( exp γ t ) · ( exp X ) = exp ( dt + [ γ, t ]) exp X . Stacks and cocycles Jets Formality for A morphism of nilpotent DGLA φ : g → h induces a functor stacks φ : MC 2 ( g ) → MC 2 ( h ) . The following holds: Twisting jets Formality theorem Suppose that φ : g → h is a quasi-isomorphism (isomorphism on cohomology) of DGLA’s and let m be a nilpotent commutative ring. Then the induced map φ : MC 2 ( g ⊗ m ) → MC 2 ( h ⊗ m ) is an equivalence of 2-groupoids.

Let g be a L ∞ -algebra. Recall that an L ∞ -algebra is a graded vector space Formality for g equipped with operations algebroid stacks � k g → g [ 2 − k ]: x 1 ∧ . . . ∧ x k �→ [ x 1 , . . . , x k ] Ryszard Nest defined for k = 1 , 2 , . . . . which satisfy a sequence of Jacobi identities. 2-groupoids It follows from the Jacobi identities that the unary operation [ . ]: g → g [ 1 ] 2-groupoid associated to a is a differential, which we will usually denote by d . DGLA Σ construction An easiest way to visualize an L ∞ -algebra is to observe that, if g ∗ is a The 2-groupoid of Hochschild cochains Star products and DGLA, then d + [ , ] is a transpose of an odd coderivation δ on the tensor the Deligne coalgebra T k [ 1 ] satisfying δ 2 = 0. Now allow δ to have higher terms: 2-groupoid Deformations 1 d t : k → k of smooth manifolds 2 [ , ] t : k → k ⊗ 2 Stacks and cocycles Jets 3 m t 3 : k → k ⊗ 3 Formality for stacks 4 and so on. Twisting jets Formality theorem The lower central series of an L ∞ -algebra g is the canonical decreasing filtration F • g with F i g = g for i ≤ 1 and defined recursively for i ≥ 1 by � ∞ � F i + 1 g = [ F i 1 g , . . . , F i k g ] . k = 2 i = i 1 + ··· + i k i k � i An L ∞ -algebra is nilpotent if there exists an i such that F i g = 0.

Formality for Suppose that g is a nilpotent L ∞ -algebra. An element µ ∈ g 1 is called a algebroid stacks Maurer-Cartan element (of g ) if it satisfies the condition Ryszard Nest � ∞ 1 k ![ µ ∧ k ] = 0 ( ∈ g 2 ) . F ( µ ) := δµ + 2-groupoids 2-groupoid k = 2 associated to a DGLA Σ construction We will denote by MC ( g ) the set of Maurer Cartan elements of g . The 2-groupoid of Hochschild cochains Hinich, Getzler Star products and the Deligne 2-groupoid Σ( g ) is the simplicial set Deformations of smooth [ n ] → MC ( g ) ⊗ Ω(∆ n )) , manifolds Stacks and cocycles Jets where Ω(∆ n ) is the differential graded commutative algebra of differential Formality for forms on the standard n-simplex. Σ( g ) is a Kan simplicial set and, if g i = 0 stacks for i < − n + 1, π k (Σ( g )) = 0 for k > n . Twisting jets Formality theorem Theorem Let g → h be a L ∞ quasiisomorphism of nilpotent L ∞ algebras. The induced map of simplicial sets Σ( g ) → Σ( h ) is a weak homotopy equivalence.

Formality for algebroid stacks Ryszard Nest Let g be a nilpotent DGLA satisfying g i = 0 for i ≤ − 2. By now we have two simplicial Kan sets associated to g , the nerve of Deligne 2-groupoid 2-groupoids 2-groupoid N ( MC 2 ( g )) and Σ( g ) . associated to a DGLA Σ construction Theorem The 2-groupoid of Hochschild cochains Let g be a nilpotent DGLA satisfying g i = 0 for i ≤ − 2. Then N ( MC 2 ( g )) Star products and the Deligne and Σ( g ) are weakly homotopy equivalent. 2-groupoid Deformations of smooth In particular, for nilpotent DGLA’s g such that g i = 0 for i ≤ − 2 the two manifolds notions of equivalence coincide, and the following definition is Stacks and cocycles unambiguous. Jets Formality for Definition stacks Twisting jets The bigroupoid associated to a nilpotent DGLA g such that g i = 0 for Formality theorem i ≤ − 2 is Π 2 (Σ( g ))