Unital associahedra and homotopy unital homotopy associative algebras Andrew TONKS London Metropolitan University with Fernando MURO Universidad de Sevilla
Outline Classical algebraic definition arises from a topological operad Unital Associahedra: spaces of trees-with-corks D.g. operad ua governing homotopy unital A algebras [Fukaya-Ono-Oh-Ohta, Hirsh-Millès, Lyubashenko] Operad of Associahedra: spaces of trees [Boardman-Vogt, Stasheff] Cellular chain complex D.g. operad A governing homotopy associative d.g. algebras
Outline Algebra Differential graded operads A algebras ua algebras Combinatorics Rooted planar trees with leaves and corks Operations: contracting and grafting Pruning binary trees Cell complexes Classical associahedra Cubical construction Polytope construction Relation to A algebras Unital associahedra Cubical construction Cellular construction Relation to ua algebras
Prototypical d.g. operad: multilinear endo morphisms If (M, d) is a differential graded k-module, consider E M (n) =hom k (M n, M), (n 0) with the identity and substitution operations u =(M id M) E M (1), E M (m) k E M (n) i EM (n+m 1) These satisfy the relations u 1 a = a = a i u and { ( 1) b c (a j c) i+n 1 b ( 1 j < i ) (a i b) j c = a i (b j i+1 c) ( i j < m + i ) for M l a M, M m b M, M n c M in E M.
Definition A d.g. operad O is a sequence of d.g. k-modules (O(n)) n 0, with u O(1), O(m) k O(n) i O(n + m 1) for each 1 i m, satisfying u 1 a = a = a i u and { ( 1) b c (a j c) i+n 1 b ( 1 j < i ) (a i b) j c = a i (b j i+1 c) ( i j < m + i ) for a O(l), b O(m), c O(n). E M is termed the endomorphism operad of M. All operads here will be non-symmetric. [Later: all trees are planar and rooted]
Algebras over d.g. operads An O-algebra is an object M together with an operad morphism O E M (or O(n) k M n M, n 0) Examples Consider the non-symmetric operad A with A (0) =0, A (n) =k (n 1) Then an A -algebra is just an associative algebra. Consider the non-symmetric operad ua with ua (0) =k, ua (n) =k (n 1) Then a ua -algebra is just an associative algebra with unit.
Homotopy algebra Problem: Given two homotopy equivalent complexes A B, A is an O-algebra B is an O-algebra Solution: Replace O by a minimal resolution O O Then A is an O -algebra B is an O -algebra
Theorem (Stasheff 1963) A CW complex is a loop space it is an A -space. The A structure is a lifting of the obvious associative algebra structure on the homology of the a loop space. The following more recent result has the same form: Deligne Conjecture Let A be an associative algebra. The Hochschild cochain complex C (A, A) has a G -algebra structure inducing the known Gerstenhaber algebra structure on Hochschild cohomology.
Koszul Duality for Operads [GK,... ] Consider the adjoint functors Bar and Cobar and the Koszul dual dg AlgebrasOperads 2 B( ) dg CoalgebrasCooperads 2 Ω (C, C, d C ) = Ω ( ) T (s 1 C), = s 1 d C s +(s 1 ) 2 C s ΩBO
A quadratic operad O is Koszul iff the cobar construction of the Koszul dual cooperad is a resolution: quadratic: Ω(O ) O 0 (R) Free(X) O 0 Generators X concentrated in valence 2 Relations R in valence 3 (that is, weight 2) Example: associative operad A Koszul dual cooperad: 0 O Cofree(sX) (s 2 R) 0 Example: (O = A ) A structure on M coalgebra structure on s 1 M
The A operad From the purely algebraic formalism of Koszul duality: A is the d.g. operad such that As a graded operad, it is freely generated by µ n A (n) of degree n 2 (n 2) It has a differential defined on the generators by d(µ n ) = ( 1) qp+(q 1)i µ p i µ q. p+q 1=n 1 i p Thus an A -algebra is a d.g. k-module M with a binary operation m 2 and (higher) homotopies M n mn M, dm n +m n d = ( 1) qp+(q 1)i m p (1 M (i 1) m q 1 M (p i)). p+q 1=n 1 i p
Resolutions of Quadratic + Linear + Constant Operads Let O be an operad admitting a presentation of the form 0 (R) Free(X) O 0 where the ideal is generated by its part of weight 2, R = (R) (k X Free(X) (2) ) R X = 0 Let qr be the quadratic image of the relations, where q : Free(X) Free(X) (2) and define a quadratic operad 0 (qr) Free(X) qo 0 with its classical Koszul dual cooperad 0 qo Cofree(sX) (s 2 qr) 0 Consider the morphism ϕ : qr X k whose graph is R, R = { a ϕ(a) : a qr }
The above data induce a curved cooperad structure (qo, d ϕ, θ φ ) termed the Koszul dual dg cooperad O of O. The operad O is termed Koszul if the associated quadratic operad qo is Koszul in the classical sense. Theorem (Gálvez-Tonks-Vallette, Hirsch-Millès) The cobar construction of the Koszul dual dg cooperad O is a resolution of a Koszul (linear + quadratic + constant) operad O, Examples O := Ω(qO, d ϕ, θ ϕ ) O Batalin-Vilkovisky algebras are quadratic+linear, (a b) ( (a) b) (a (b)) [ a, b ] = 0 Unital associative algebras are quadratic+constant, a 1 a = 1 a a = 0
Appplication: definition of homotopy unital algebras Fukaya Oh Ohta Ono introduced homotopy units into the picture in their work in symplectic geometry. Lyubashenko, Millès Hirsch gave equivalent definitions. Definition ua has underlying graded operad freely generated by µ S n+m O(n), S [n + m], S = m, in degree µ S n+m = 2m + n 2, (n 2ifm = 0), with d(µ {i} 1+1 )=µ 2+0 i µ {1} 0+1 u, i {1, 2}. d(µ S n+m) = ±µ S 1 p+s i µ S 2 q+t, p+q 1=n s+t=m 1 i p S 1 i S 2 =S
Some easy combinatorial structures
Planar rooted trees with corks and leaves v 5 v 6 v 7 v 8 v 9 v 4 v3 v 1 v 2 v 0 v 8 v 4 v3 v 1 Vertices may not have degree two. The vertex order determines the planar structure. Usually we only draw the internal vertices and the corks. In the picture: v 1, v 3, v 4 are internal, v 8 is a cork.
Contraction of inner edges, and grafting at leaf edges v 8 v 4 e v3 v 1 v 8 [e] v 1 2 =
Binary trees with one cork and 2leaves
Pruning of one edge in a binary tree T = e1 e 2 e 3, T \ e 1 =, T \ e 2 =, T \ e 3 =.
Classical associahedra
Boardman Vogt constructed the classical associahedra as / K n = H T, H T =[0, 1] n 2 binary trees T with n leaves and no corks (x 1,...,x n 2 ) H T is a length labelling of the inner edges of T. x 2 x 1 Whenever T /e = T /e then identifies the corresponding negative faces of H T and H T. 0. 0
The associahedra K 3 and K 4 as cubical complexes
The associahedra form a topological operad Operad structure i on the family (K n ) is defined by maps H T H T H T i T given by an insertion of coordinates in a grafting x 1 x 2 2 y 2 y1 = y 2 y1 1 x 1 x 2 = (x 1, 1, y 1, y 2, x 2 ). with an inclusion as a positive face.
Associahedra as polytopes One may forget the cubical subdivision: the associahedron K n can be represented as an (n 2)-dimensional polytope. Its face poset is the category of planar rooted trees with n leaves, no corks and no degree 2 vertices. Inclusion of faces corresponds to collapses T T /e. The unique (n 2)-cell of K n is the corolla C n. We may write K T for the codimension r face of K n corresponding to a tree T with n leaves and r inner edges. Recall: codimension 1 faces of K n are given by K p i K q. These correspond to trees with a unique internal edge e = {v 1, v i+1 }, obtained by grafting corollas C p i C q. More generally K T i K T = K T i T.
K 3. K 4
Associahedra and A algebras Associahedra admit orientations such that the d.g. operad A can be recovered by taking the cellular chains: A = (C (K n, k)) n 2 It is freely generated as a graded operad by the fundamental classes µ n =[K n ] C n 2 (K n, k), n 2, The differential is given by the polytope boundary d(µ n ) = ( 1) qp+(q 1)i µ p i µ q. p+q 1=n 1 i p The exact signs depend on the orientations chosen. That is, the cellular chains on the (polytopal) associahedra gives the operad encoding A -algebras.
Unital associahedra
We define unital associahedra using quotients of cubes, / Kn u = H T, H T =[0, 1] 2m+n 2 binary trees T with n leaves and 0 m < corks (x 1,...,x 2m+n 2 ) H T is a length labelling of the inner edges of T. x 3 x 2 =(x 1, x 2, x 3 ) H T. x 1 If T /e = T /e, identifies negative faces of H T and H T. Of course, both of these faces have the same dimension. However, if T contains a corked edge e, then will identify a negative face of H T with the lower-dimensional cube H T \e indexed by the pruned tree.
0 b a max(a,b) b 0 a 0 a 0 a 0 a a 0 0 0
Example: 2-cells H T of the unital associahedron K u 2 These cells correspond to binary trees with 2 leaves and 1 cork: 1 1 1 1 Edges with white corks have length 1 (as do others so labelled)
The unital associahedra form a topological operad Theorem The cubical maps H T H T H T i T given by grafting and inclusion of positive faces x 1 x 3 1 x 2 x 4 y 3 y 2 y1 = y 3 y 2 y1 1 =(x 1, x 2,1, y 1, y 2, y 3, x 3, x 4 ) x 1 x 3 x 2 x 4 are compatible with the identifications. The collection (K u n ) with the induced operations j forms a cofibrant object in the model category of topological operads.
A better cellular structure: cork filtration Theorem There is a sequence of trivial cofibrations K u n,0 K u n,1 K u n,2 K u n where: The first stage Kn,0 u is the classical associahedron Each further stage Kn,m u is the quotient of binary trees T with n leaves and mcorks The cofibrations K u n,m 1 K u n,m are formed by attaching copies of (2m + n 2)-dimensional cells K n+m [0, 1] m. The attachment is along K n+m ([0, 1] m ). H T
The first stage of the cork filtration After the classical associahedron K u n,0 = K n, the next stage in the cork filtration also has a straightforward description: Theorem For n 1 the space Kn,1 u is the mapping cylinder of (σ i ) i : n+1 K n+1 K n, i=1 where σ i : K n+1 K n are the associahedral degeneracy maps.
Stage Kn,1 u of the first four unital associahedra K u 0,1 :, K u 1,1 :, K u 2,1 :, K u 3,1 :
The second stage of the cork filtration: The smallest example is given by trees with no leaves and at most two corks, K 0,2
Remarks Each unital associahedron K u n is a contractible CW complex (but not a poytope). The cells K u T of the unital associahedron K u n are in bijection with the set of all planar rooted trees T with n leaves, m b black corks and m w white corks, in which no vertex has degree two, and the case (n, m b )=(0, 1) is also excluded. The operad composition is cellular: K u T i K u T = K u T i T
Theorem The cells of the unital associahedra admit orientations (inherited by those on K n+m [0, 1] m ) such that the d.g. operad (C (K u n, k)) obtained by taking cellular chains is naturally isomorphic to the operad ua for the homotopy unital associative algebras of Fukaya Oh Ohta Ono.
Final Example: attempting to visualize K u 1,2 This stage of the cork filtration of K1 u indexed by consists of three 3-cells, The first 3-cell (and, symmetrically, the last) has faces.
The second 3-cell has faces of the form
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