Higher categorified algebras versus bounded homotopy algebras
aa r X i v : . [ m a t h . QA ] J un Higher categorified algebras versus boundedhomotopy algebras
David Khudaverdyan, Ashis Mandal, and Norbert Poncin ∗ Abstract
We define Lie 3-algebras and prove that these are in 1-to-1 correspondence with the3-term Lie infinity algebras whose bilinear and trilinear maps vanish in degree ( , ) andin total degree 1, respectively. Further, we give an answer to a question of [Roy07] per-taining to the use of the nerve and normalization functors in the study of the relationshipbetween categorified algebras and truncated sh algebras. Mathematics Subject Classification (2000) :
Key words : Higher category, homotopy algebra, monoidal category, Eilenberg-Zilber map.
Higher structures – infinity algebras and other objects up to homotopy, higher categories,“oidified” concepts, higher Lie theory, higher gauge theory... – are currently intensively in-vestigated. In particular, higher generalizations of Lie algebras have been conceived undervarious names, e.g. Lie infinity algebras, Lie n -algebras, quasi-free differential graded com-mutative associative algebras ( qfDGCAs for short), n -ary Lie algebras, see e.g. [Dzh05],crossed modules [MP09] ...See also [AP10], [GKP11].More precisely, there are essentially two ways to increase the flexibility of an algebraicstructure: homotopification and categorification.Homotopy, sh or infinity algebras [Sta63] are homotopy invariant extensions of differ-ential graded algebras. This property explains their origin in BRST of closed string fieldtheory. One of the prominent applications of Lie infinity algebras [LS93] is their appearancein Deformation Quantization of Poisson manifolds. The deformation map can be extendedfrom differential graded Lie algebras ( DGLA s) to L ¥ -algebras and more precisely to a functorfrom the category L ¥ to the category Set . This functor transforms a weak equivalence intoa bijection. When applied to the
DGLA s of polyvector fields and polydifferential operators,the latter result, combined with the formality theorem, provides the 1-to-1 correspondencebetween Poisson tensors and star products. ∗ University of Luxembourg, Campus Kirchberg, Mathematics Research Unit, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg City, Grand-Duchy of Luxembourg. The research of D. Khudaverdyan and N.Poncin was supported by UL-grant SGQnp2008. A. Mandal thanks the Luxembourgian NRF for support viaAFR grant PDR-09-062. n -algebras are often defined as sh Lie algebras concentratedin the first n degrees [Hen08]. However, this ‘definition’ is merely a terminological conven-tion, see e.g. Definition 4 in [SS07b]. On the other hand, Lie infinity algebra structures on an N -graded vector space V are in 1-to-1 correspondence with square 0 degree -1 (with respectto the grading induced by V ) coderivations of the free reduced graded commutative associa-tive coalgebra S c ( sV ) , where s denotes the suspension operator, see e.g. [SS07b] or [GK94].In finite dimension, the latter result admits a variant based on qfDGCAs instead of coalge-bras. Higher morphisms of free DGCAs have been investigated under the name of derivationhomotopies in [SS07b]. Quite a number of examples can be found in [SS07a].Besides the proof of the mentioned correspondence between Lie 2-algebras and 2-termLie infinity algebras, the seminal work [BC04] provides a classification of all Lie infinityalgebras, whose only nontrivial terms are those of degree 0 and n −
1, by means of a Liealgebra, a representation and an ( n + ) -cohomology class; for a possible extension of thisclassification, see [Bae07].In this paper, we give an explicit categorical definition of Lie 3-algebras and prove thatthese are in 1-to-1 correspondence with the 3-term Lie infinity algebras, whose bilinear andtrilinear maps vanish in degree ( , ) and in total degree 1, respectively. Note that a ‘3-term’Lie infinity algebra implemented by a 4-cocycle [BC04] is an example of a Lie 3-algebra inthe sense of the present work.The correspondence between categorified and bounded homotopy algebras is expected toinvolve classical functors and chain maps, like e.g. the normalization and Dold-Kan functors,the (lax and oplax monoidal) Eilenberg-Zilber and Alexander-Whitney chain maps, the nervefunctor... We show that the challenge ultimately resides in an incompatibility of the cartesianproduct of linear n -categories with the monoidal structure of this category, thus answering aquestion of [Roy07].The paper is organized as follows. Section 2 contains all relevant higher categorical def-initions. In Section 3, we define Lie 3-algebras. The fourth section contains the proof of thementioned 1-to-1 correspondence between categorified algebras and truncated sh algebras –the main result of this paper. A specific aspect of the monoidal structure of the category oflinear n -categories is highlighted in Section 5. In the last section, we show that this featureis an obstruction to the use of the Eilenberg-Zilber map in the proof of the correspondence“bracket functor – chain map”.ategorificationand homotopification 3 Let us emphasize that notation and terminology used in the present work originate in[BC04], [Roy07], as well as in [Lei04]. For instance, a linear n -category will be an (a strict) n -category [Lei04] in Vect . Categories in
Vect have been considered in [BC04] and also calledinternal categories or 2-vector spaces. In [BC04], see Sections 2 and 3, the correspondingmorphisms (resp. 2-morphisms) are termed as linear functors (resp. linear natural transfor-mations), and the resulting 2-category is denoted by
VectCat and also by . Therefore,the ( n + ) -category made up by linear n -categories ( n -categories in Vect or ( n + ) -vectorspaces), linear n -functors... will be denoted by Vect n - Cat or ( n + ) Vect .The following result is known. We briefly explain it here as its proof and the involvedconcepts are important for an easy reading of this paper.
Proposition 1.
The categories
Vect n- Cat of linear n-categories and linear n-functors and C n + ( Vect ) of ( n + ) -term chain complexes of vector spaces and linear chain maps areequivalent. We first recall some definitions.
Definition 1. An n-globular vector space L, n ∈ N , is a sequenceL n s , t ⇒ L n − s , t ⇒ . . . s , t ⇒ L ⇒ , (1) of vector spaces L m and linear maps s , t such thats ( s ( a )) = s ( t ( a )) and t ( s ( a )) = t ( t ( a )) , (2) for any a ∈ L m , m ∈ { , . . . , n } . The maps s , t are called source map and target map , respec-tively, and any element of L m is an m-cell . By higher category we mean in this text a strict higher category. Roughly, a linear n -category, n ∈ N , is an n -globular vector space endowed with compositions of m -cells, 0 < m ≤ n , along a p -cell, 0 ≤ p < m , and an identity associated to any m -cell, 0 ≤ m < n . Two m -cells ( a , b ) ∈ L m × L m are composable along a p -cell, if t m − p ( a ) = s m − p ( b ) . The composite m -cell will be denoted by a ◦ p b (the cell that ‘acts’ first is written on the left) and the vectorsubspace of L m × L m made up by the pairs of m -cells that can be composed along a p -cell willbe denoted by L m × L p L m . The following figure schematizes the composition of two 3-cellsalong a 0-, a 1-, and a 2-cell. ~ ~ ~ ~ • (cid:28) (cid:28) B B / / • (cid:28) (cid:28) B B / / • x x & & / / • / / (cid:25) (cid:25) E E x x & & • / / (cid:15) (cid:15) ~ ~ • (cid:28) (cid:28) B B / / / / • ategorificationand homotopification 4 Definition 2. A linear n-category , n ∈ N , is an n-globular vector space L (with source andtarget maps s , t) together with, for any m ∈ { , . . . , n } and any p ∈ { , . . . , m − } , a linearcomposition map ◦ p : L m × L p L m → L m and, for any m ∈ { , . . . , n − } , a linear identity map L m → L m + , such that the properties • for ( a , b ) ∈ L m × L p L m ,if p = m − , then s ( a ◦ p b ) = s ( a ) and t ( a ◦ p b ) = t ( b ) , if p ≤ m − , then s ( a ◦ p b ) = s ( a ) ◦ p s ( b ) and t ( a ◦ p b ) = t ( a ) ◦ p t ( b ) , • s ( a ) = t ( a ) = a , • for any ( a , b ) , ( b , c ) ∈ L m × L p L m , ( a ◦ p b ) ◦ p c = a ◦ p ( b ◦ p c ) , • m − ps m − p a ◦ p a = a ◦ p m − pt m − p a = aare verified, as well as the compatibility conditions • for q < p, ( a , b ) , ( c , d ) ∈ L m × L p L m and ( a , c ) , ( b , d ) ∈ L m × L q L m , ( a ◦ p b ) ◦ q ( c ◦ p d ) = ( a ◦ q c ) ◦ p ( b ◦ q d ) , • for m < n and ( a , b ) ∈ L m × L p L m , a ◦ p b = a ◦ p b . The morphisms between two linear n -categories are the linear n -functors. Definition 3. A linear n-functor F : L → L ′ between two linear n-categories is made up bylinear maps F : L m → L ′ m , m ∈ { , . . . , n } , such that the categorical structure – source andtarget maps, composition maps, identity maps – is respected. Linear n -categories and linear n -functors form a category Vect n - Cat , see Proposition1. To disambiguate this proposition, let us specify that the objects of C n + ( Vect ) are thecomplexes whose underlying vector space V = ⊕ ni = V i is made up by n + V i .The proof of Proposition 1 is based upon the following result. Proposition 2.
Let L be any n-globular vector space with linear identity maps. If s m denotesthe restriction of the source map to L m , the vector spaces L m and L ′ m : = ⊕ mi = V i , V i : = ker s i ,m ∈ { , . . . , n } , are isomorphic. Further, the n-globular vector space with identities can becompleted in a unique way by linear composition maps so to form a linear n-category. If weidentify L m with L ′ m , this unique linear n-categorical structure readss ( v , . . . , v m ) = ( v , . . . , v m − ) , (3)ategorificationand homotopification 5 t ( v , . . . , v m ) = ( v , . . . , v m − + tv m ) , (4)1 ( v ,..., v m ) = ( v , . . . , v m , ) , (5) ( v , . . . , v m ) ◦ p ( v ′ , . . . , v ′ m ) = ( v , . . . , v p , v p + + v ′ p + , . . . , v m + v ′ m ) , (6) where the two m-cells in Equation (6) are assumed to be composable along a p-cell.Proof. As for the first part of this proposition, if m = a L : L ′ = V ⊕ V ⊕ V ∋ ( v , v , v ) v + v + v ∈ L and b L : L ∋ a ( s a , s ( a − s a ) , a − s ( a − s a ) − s a ) ∈ V ⊕ V ⊕ V = L ′ are inverses of each other. For arbitrary m ∈ { , . . . , n } and a ∈ L m , we set b L a = s m a , . . . , s m − i ( a − i − (cid:229) j = m − jp j b L a ) , . . . , a − m − (cid:229) j = m − jp j b L a ! ∈ V ⊕ . . . ⊕ V i ⊕ . . . ⊕ V m = L ′ m , where p j denotes the projection p j : L ′ m → V j and where the components must be computedfrom left to right.For the second claim, note that when reading the source, target and identity mapsthrough the detailed isomorphism, we get s ( v , . . . , v m ) = ( v , . . . , v m − ) , t ( v , . . . , v m ) =( v , . . . , v m − + tv m ) , and 1 ( v ,..., v m ) = ( v , . . . , v m , ) . Eventually, set v = ( v , . . . , v m ) and let ( v , w ) and ( v ′ , w ′ ) be two pairs of m -cells that are composable along a p -cell. The compos-ability condition, say for ( v , w ) , reads ( w , . . . , w p ) = ( v , . . . , v p − , v p + tv p + ) . It follows from the linearity of ◦ p : L m × L p L m → L m that ( v + v ′ ) ◦ p ( w + w ′ ) = ( v ◦ p w ) +( v ′ ◦ p w ′ ) . When taking w = m − pt m − p v and v ′ = m − ps m − p w ′ , we find ( v + w ′ , . . . , v p + w ′ p , v p + , . . . , v m ) ◦ p ( v + w ′ , . . . , v p + w ′ p + tv p + , w ′ p + , . . . , w ′ m )= ( v + w ′ , . . . , v m + w ′ m ) , so that ◦ p is necessarily the composition given by Equation (6). It is easily seen that, con-versely, Equations (3) – (6) define a linear n -categorical structure. Proof of Proposition 1.
We define functors N : Vect n - Cat → C n + ( Vect ) and G : C n + ( Vect ) → Vect n - Cat that are inverses up to natural isomorphisms.If we start from a linear n -category L , so in particular from an n -globular vector space L ,we define an ( n + ) -term chain complex N ( L ) by setting V m = ker s m ⊂ L m and d m = t m | V m : V m → V m − . In view of the globular space conditions (2), the target space of d m is actually V m − and we have d m − d m v m = . Moreover, if F : L → L ′ denotes a linear n -functor, the value N ( F ) : V → V ′ is defined on V m ⊂ L m by N ( F ) m = F m | V m : V m → V ′ m . It is obvious that N ( F ) is a linear chain map.ategorificationand homotopification 6It is obvious that N respects the categorical structures of Vect n - Cat and C n + ( Vect ).As for the second functor G , if ( V , d ) , V = ⊕ ni = V i , is an ( n + ) -term chain complex ofvector spaces, we define a linear n -category G ( V ) = L , L m = ⊕ mi = V i , as in Proposition 2: thesource, target, identity and composition maps are defined by Equations (3) – (6), except that tv m in the RHS of Equation (4) is replaced by dv m .The definition of G on a linear chain map f : V → V ′ leads to a linear n -functor G ( f ) : L → L ′ , which is defined on L m = ⊕ mi = V i by G ( f ) m = ⊕ mi = f i . Indeed, it is readily checkedthat G ( f ) respects the linear n -categorical structures of L and L ′ .Furthermore, G respects the categorical structures of C n + ( Vect ) and
Vect n - Cat .Eventually, there exist natural isomorphisms a : NG ⇒ id and g : GN ⇒ id.To define a natural transformation a : NG ⇒ id, note that L ′ = ( NG )( L ) is the linear n -category made up by the vector spaces L ′ m = ⊕ mi = V i , V i = ker s i , as well as by the source,target, identities and compositions defined from V = N ( L ) as in the above definition of G ( V ) ,i.e. as in Proposition 2. It follows that a L : L ′ → L , defined by a L : L ′ m ∋ ( v , . . . , v m ) mv + . . . + v m − + v m ∈ L m , m ∈ { , . . . , n } , which pulls the linear n -categorical structureback from L to L ′ , see Proposition 2, is an invertible linear n -functor. Moreover a is naturalin L .It suffices now to observe that the composite GN is the identity functor.Next we further investigate the category Vect n - Cat . Proposition 3.
The category
Vect n- Cat admits finite products.
Let L and L ′ be two linear n -categories. The product linear n -category L × L ′ is definedby ( L × L ′ ) m = L m × L ′ m , S m = s m × s ′ m , T m = t m × t ′ m , I m = m × ′ m , and (cid:13) p = ◦ p × ◦ ′ p . Thecompositions (cid:13) p coincide with the unique compositions that complete the n -globular vectorspace with identities, thus providing a linear n -category. It is straightforwardly checked thatthe product of linear n -categories verifies the universal property for binary products. Proposition 4.
The category
Vect - Cat admits a 3-categorical structure. More precisely, its2-cells are the linear natural 2-transformations and its 3-cells are the linear 2-modifications.
This proposition is the linear version (with similar proof) of the well-known result thatthe category 2-
Cat is a 3-category with 2-categories as 0-cells, 2-functors as 1-cells, natural2-transformations as 2-cells, and 2-modifications as 3-cells. The definitions of n -categoriesand 2-functors are similar to those given above in the linear context (but they are formulatedwithout the use of set theoretical concepts). As for (linear) natural 2-transformations and(linear) 2-modifications, let us recall their definition in the linear setting: Definition 4. A linear natural 2-transformation q : F ⇒ G between two linear 2-functorsF , G : C → D , between the same two linear 2-categories, assigns to any a ∈ C a unique q a : F ( a ) → G ( a ) in D , linear with respect to a and such that for any a : f ⇒ g in C ,f , g : a → b in C , we have F ( a ) ◦ q b = q a ◦ G ( a ) . (7)ategorificationand homotopification 7If C = L × L is a product linear 2-category, the last condition reads F ( a , b ) ◦ q t a , t b = q s a , s b ◦ G ( a , b ) , for all ( a , b ) ∈ L × L . As functors respect composition, i.e. as F ( a , b ) = F ( a ◦ t a , s b ◦ b ) = F ( a , s b ) ◦ F ( t a , b ) , this naturality condition is verified if and only if it holds true in case all but one of the 2-cells are identities 1 − , i.e. if and only if the transformation is natural with respect to all itsarguments separately. Definition 5.
Let C , D be two linear 2-categories. A linear 2-modification m : h ⇛ e be-tween two linear natural 2-transformations h , e : F ⇒ G, between the same two linear 2-functors F , G : C → D , assigns to any object a ∈ C a unique m a : h a ⇒ e a in D , which islinear with respect to a and such that, for any a : f ⇒ g in C , f , g : a → b in C , we haveF ( a ) ◦ m b = m a ◦ G ( a ) . (8)If C = L × L is a product linear 2-category, it suffices again that the preceding modificationproperty be satisfied for tuples ( a , b ) , in which all but one 2-cells are identities 1 − . Theexplanation is the same as for natural transformations.Beyond linear 2-functors, linear natural 2-transformations, and linear 2-modifications, weuse below multilinear cells. Bilinear cells e.g., are cells on a product linear 2-category, withlinearity replaced by bilinearity. For instance, Definition 6.
Let L, L ′ , and L ′′ be linear -categories. A bilinear 2-functor F : L × L ′ → L ′′ is a -functor such that F : L m × L ′ m → L ′′ m is bilinear for all m ∈ { , , } . Similarly,
Definition 7.
Let L, L ′ , and L ′′ be linear -categories. A bilinear natural 2-transformation q : F ⇒ G between two bilinear 2-functors F , G : L × L ′ → L ′′ , assigns to any ( a , b ) ∈ L × L ′ a unique q ( a , b ) : F ( a , b ) → G ( a , b ) in L ′′ , which is bilinear with respect to ( a , b ) and such thatfor any ( a , b ) : ( f , h ) ⇒ ( g , k ) in L × L ′ , ( f , h ) , ( g , k ) : ( a , b ) → ( c , d ) in L × L ′ , we haveF ( a , b ) ◦ q ( c , d ) = q ( a , b ) ◦ G ( a , b ) . (9) We now recall the definition of a Lie infinity (strongly homotopy Lie, sh Lie, L ¥ − ) alge-bra and specify it in the case of a 3-term Lie infinity algebra. Definition 8. A Lie infinity algebra is an N -graded vector space V = ⊕ i ∈ N V i together witha family ( ℓ i ) i ∈ N ∗ of graded antisymmetric i-linear weight i − maps on V , which verify thesequence of conditions (cid:229) i + j = n + (cid:229) ( i , n − i ) – shuffles s c ( s )( − ) i ( j − ) ℓ j ( ℓ i ( a s , . . . , a s i ) , a s i + , . . . , a s n ) = , (10) where n ∈ { , , . . . } , where c ( s ) is the product of the signature of s and the Koszul signdefined by s and the homogeneous arguments a , . . . , a n ∈ V . ategorificationand homotopification 8For n =
1, the L ¥ -condition (10) reads ℓ = n =
2, it means that ℓ is a gradedderivation of ℓ , or, equivalently, that ℓ is a chain map from ( V ⊗ V , ℓ ⊗ id + id ⊗ ℓ ) to ( V , ℓ ) . In particular,
Definition 9. A is a 3-term graded vector space V = V ⊕ V ⊕ V endowed with graded antisymmetric p-linear maps ℓ p of weight p − , ℓ : V i → V i − ( ≤ i ≤ ) ,ℓ : V i × V j → V i + j ( ≤ i + j ≤ ) ,ℓ : V i × V j × V k → V i + j + k + ( ≤ i + j + k ≤ ) ,ℓ : V × V × V × V → V (11) (all structure maps ℓ p , p > , necessarily vanish), which satisfy L ¥ -condition (10) (that istrivial for all n > ). In this 3-term situation, each L ¥ -condition splits into a finite number of equations deter-mined by the various combinations of argument degrees, see below.On the other hand, we have the Definition 10. A Lie 3-algebra is a linear 2-category
L endowed with a bracket , i.e. anantisymmetric bilinear 2-functor [ − , − ] : L × L → L, which verifies the Jacobi identity up toa
Jacobiator , i.e. a skew-symmetric trilinear natural 2-transformationJ xyz : [[ x , y ] , z ] → [[ x , z ] , y ] + [ x , [ y , z ]] , (12) x , y , z ∈ L , which in turn satisfies the Baez-Crans Jacobiator identity up to an Identiator , i.e.a skew-symmetric quadrilinear 2-modification m xyzu : [ J x , y , z , u ] ◦ ( J [ x , z ] , y , u + J x , [ y , z ] , u ) ◦ ([ J xzu , y ] + ) ◦ ([ x , J yzu ] + ) ⇒ J [ x , y ] , z , u ◦ ([ J xyu , z ] + ) ◦ ( J x , [ y , u ] , z + J [ x , u ] , y , z + J x , y , [ z , u ] ) , (13) x , y , z , u ∈ L , required to verify the coherence law a + a − = a + a − , (14) where a – a are explicitly given in Definitions 12 – 15 and where superscript − denotesthe inverse for composition along a 1-cell. Just as the
Jacobiator is a natural transformation between the two sides of the
Jacobi identity, the
Identiator is a modification between the two sides of the Baez-Crans Jacobiator identity .In this definition “skew-symmetric 2-transformation” (resp. “skew-symmetric 2-modification”)means that, if we identify L m with ⊕ mi = V i , V i = ker s i , as in Proposition 2, the V -componentof J xyz ∈ L (resp. the V -component of m xyzu ∈ L ) is antisymmetric. Moreover, the def-inition makes sense, as the source and target in Equation (13) are quadrilinear natural 2-transformations between quadrilinear 2-functors from L × to L . These 2-functors are simplestobtained from the
RHS of Equation (13). Further, the mentioned source and target actuallyare natural 2-transformations, since a 2-functor composed (on the left or on the right) with anatural 2-transformation is again a 2-transformation.ategorificationand homotopification 9
Remark 1.
In the following, we systematically identify the vector spaces L m , m ∈ { , . . . , n } ,of a linear n-category with the spaces L ′ m = ⊕ mi = V i , V i = ker s i , so that the categorical struc-ture is given by Equations (3) – (6). In addition, we often substitute common, index-freenotations ( e.g. a = ( x , f , a )) for our notations ( e.g. v = ( v , v , v ) ∈ L ) . The next theorem is the main result of this paper.
Theorem 1.
There exists a 1-to-1 correspondence between Lie 3-algebras and 3-term Lieinfinity algebras ( V , ℓ p ) , whose structure maps ℓ and ℓ vanish on V × V and on triplets oftotal degree 1, respectively. Example 1.
There exists a 1-to-1 correspondence between ( n + ) -term Lie infinity algebrasV = V ⊕ V n (whose intermediate terms vanish), n ≥ , and ( n + ) -cocycles of Lie algebrasendowed with a linear representation, see [BC04], Theorem 6.7. A 3-term Lie infinity algebraimplemented by a 4-cocycle can therefore be viewed as a special case of a Lie 3-algebra. The proof of Theorem 1 consists of five lemmas.
First, we recall the correspondence between the underlying structures of a Lie 3-algebraand a 3-term Lie infinity algebra.
Lemma 1.
There is a bijective correspondence between linear 2-categories L and 3-termchain complexes of vector spaces ( V , ℓ ) .Proof. In the proof of Proposition 1, we associated to any linear 2-category L a unique 3-term chain complex of vector spaces N ( L ) = V , whose spaces are given by V m = ker s m , m ∈ { , , } , and whose differential ℓ coincides on V m with the restriction t m | V m . Conversely,we assigned to any such chain complex V a unique linear 2-category G ( V ) = L , with spaces L m = ⊕ mi = V i , m ∈ { , , } and target t ( x ) = , t ( x , f ) = x + ℓ f , t ( x , f , a ) = ( x , f + ℓ a ) . Inview of Remark 1, the maps N and G are inverses of each other. Remark 2.
The globular space condition is the categorical counterpart of L ¥ -condition n = . We assume that we already built ( V , ℓ ) from L or L from ( V , ℓ ) . Lemma 2.
There is a bijective correspondence between antisymmetric bilinear 2-functors [ − , − ] on L and graded antisymmetric chain maps ℓ : ( V ⊗ V , ℓ ⊗ id + id ⊗ ℓ ) → ( V , ℓ ) thatvanish on V × V . ategorificationand homotopification 10 Proof.
Consider first an antisymmetric bilinear “2-map” [ − , − ] : L × L → L that verifies allfunctorial requirements except as concerns composition. This bracket then respects the com-positions, i.e., for each pairs ( v , w ) , ( v ′ , w ′ ) ∈ L m × L m , m ∈ { , } , that are composable alonga p -cell, 0 ≤ p < m , we have [ v ◦ p v ′ , w ◦ p w ′ ] = [ v , w ] ◦ p [ v ′ , w ′ ] , (15)if and only if the following conditions hold true, for any f , g ∈ V and any a , b ∈ V : [ f , g ] = [ t f , g ] = [ f , t g ] , (16) [ a , b ] = [ t a , b ] = [ a , t b ] = , (17) [ f , b ] = [ t f , b ] = . (18)To prove the first two conditions, it suffices to compute [ f ◦ t f , ◦ g ] , for the next threeconditions, we consider [ a ◦ t a , ◦ b ] and [ a ◦ , ◦ b ] , and for the last two, we focus on [ f ◦ t f , ◦ b ] and [ f ◦ ( t f + f ′ ) , b ◦ b ′ ] . Conversely, it can be straightforwardly checkedthat Equations (16) – (18) entail the general requirement (15).On the other hand, a graded antisymmetric bilinear weight 0 map ℓ : V × V → V com-mutes with the differentials ℓ and ℓ ⊗ id + id ⊗ ℓ , i.e., for all v , w ∈ V , we have ℓ ( ℓ ( v , w )) = ℓ ( ℓ v , w ) + ( − ) v ℓ ( v , ℓ w ) (19)(we assumed that v is homogeneous and denoted its degree by v as well), if and only if, forany y ∈ V , f , g ∈ V , and a ∈ V , ℓ ( ℓ ( f , y )) = ℓ ( ℓ f , y ) , (20) ℓ ( ℓ ( f , g )) = ℓ ( ℓ f , g ) − ℓ ( f , ℓ g ) , (21) ℓ ( ℓ ( a , y )) = ℓ ( ℓ a , y ) , (22)0 = ℓ ( ℓ f , b ) − ℓ ( f , ℓ b ) . (23) Remark 3.
Note that, in the correspondence ℓ ↔ t and ℓ ↔ [ − , − ] , Equations (20) and(22) read as compatibility requirements of the bracket with the target and that Equations (21)and (23) correspond to the second conditions of Equations (16) and (18), respectively.Proof of Lemma 2 (continuation) . To prove the announced 1-to-1 correspondence, wefirst define a graded antisymmetric chain map N ([ − , − ]) = ℓ , ℓ : V ⊗ V → V from anyantisymmetric bilinear 2-functor [ − , − ] : L × L → L .Let x , y ∈ V , f , g ∈ V , and a , b ∈ V . Set ℓ ( x , y ) = [ x , y ] ∈ V and ℓ ( x , g ) = [ x , g ] ∈ V . However, we must define ℓ ( f , g ) ∈ V , whereas [ f , g ] ∈ V . Moreover, in this case, theantisymmetry properties do not match. The observation [ f , g ] = [ t f , g ] = [ f , t g ] = ℓ ( ℓ f , g ) = ℓ ( f , ℓ g ) and Condition (21) force us to define ℓ on V × V as a symmetric bilinear map valued inV ∩ ker ℓ . We further set ℓ ( x , b ) = [ x , b ] ∈ V , and, as ℓ is required to have weight 0, weategorificationand homotopification 11must set ℓ ( f , b ) = ℓ ( a , b ) = . It then follows from the functorial properties of [ − , − ] that the conditions (20) – (22) are verified. In view of Equation (18), Property (23) reads0 = [ t f , b ] − ℓ ( f , ℓ b ) = − ℓ ( f , ℓ b ) . In other words, in addition to the preceding requirement, we must choose ℓ in a way thatit vanishes on V × V if evaluated on a 1-coboundary. These conditions are satisfied if wechoose ℓ = V × V .Conversely, from any graded antisymmetric chain map ℓ that vanishes on V × V , wecan construct an antisymmetric bilinear 2-functor G ( ℓ ) = [ − , − ] . Indeed, using obviousnotations, we set [ x , y ] = ℓ ( x , y ) ∈ L , [ x , y ] = [ x , y ] ∈ L , [ x , g ] = ℓ ( x , g ) ∈ V ⊂ L . Again [ f , g ] ∈ L cannot be defined as ℓ ( f , g ) ∈ V . Instead, if we wish to build a 2-functor,we must set [ f , g ] = [ t f , g ] = [ f , t g ] = ℓ ( ℓ f , g ) = ℓ ( f , ℓ g ) ∈ V ⊂ L , which is possible in view of Equation (21), if ℓ is on V × V valued in 2-cocycles (and inparticular if it vanishes on this subspace). Further, we define [ x , y ] = [ x , y ] ∈ L , [ x , g ] = [ x , g ] ∈ L , [ x , b ] = ℓ ( x , b ) ∈ V ⊂ L , [ f , g ] = [ f , g ] ∈ L . Finally, we must set [ f , b ] = [ t f , b ] = ℓ ( ℓ f , b ) = , which is possible in view of Equation (23), if ℓ vanishes on V × V when evaluated on a1-coboundary (and especially if it vanishes on the whole subspace V × V ), and [ a , b ] = [ t a , b ] = [ a , t b ] = , which is possible.It follows from these definitions that the bracket of a = ( x , f , a ) = x + f + a ∈ L and b = ( y , g , b ) = y + g + b ∈ L is given by [ a , b ] = ( ℓ ( x , y ) , ℓ ( x , g ) + ℓ ( f , tg ) , ℓ ( x , b ) + ℓ ( a , y )) ∈ L , (24)where g = ( y , g ) . The brackets of two elements of L or L are obtained as special cases ofthe latter result.We thus defined an antisymmetric bilinear map [ − , − ] that assigns an i -cell to any pairof i -cells, i ∈ { , , } , and that respects identities and sources. Moreover, since Equations(16) – (18) are satisfied, the map [ − , − ] respects compositions provided it respects targets.For the last of the first three defined brackets, the target condition is verified due to Equation(20). For the fourth bracket, the target must coincide with [ t f , t g ] = ℓ ( ℓ f , ℓ g ) and it actu-ally coincides with t [ f , g ] = ℓ ℓ ( ℓ f , g ) = ℓ ( ℓ f , ℓ g ) , again in view of (20). As regards theseventh bracket, the target t [ x , b ] = ℓ ℓ ( x , b ) = ℓ ( x , ℓ b ) , due to (22), must coincide with [ x , t b ] = ℓ ( x , ℓ b ) . The targets of the two last brackets vanish and [ f , t b ] = ℓ ( f , ℓ ℓ b ) = [ t a , t b ] = ℓ ( ℓ a , ℓ ℓ b ) = . It is straightforwardly checked that the maps N and G are inverses.ategorificationand homotopification 12Note that N actually assigns to any antisymmetric bilinear 2-functor a class of gradedantisymmetric chain maps that coincide outside V × V and whose restrictions to V × V arevalued in 2-cocycles and vanish when evaluated on a 1-coboundary. The map N , with valuesin chain maps, is well-defined thanks to a canonical choice of a representative of this class.Conversely, the values on V × V of the considered chain map cannot be encrypted intothe associated 2-functor, only the mentioned cohomological conditions are of importance.Without the canonical choice, the map G would not be injective. Remark 4.
The categorical counterpart of L ¥ -condition n = is the functor condition oncompositions. Remark 5.
A 2-term Lie infinity algebra (resp. a Lie 2-algebra) can be viewed as a 3-termLie infinity algebra (resp. a Lie 3-algebra). The preceding correspondence then of coursereduces to the correspondence of [BC04].
We suppose that we already constructed ( V , ℓ , ℓ ) from ( L , [ − , − ]) or ( L , [ − , − ]) from ( V , ℓ , ℓ ) . Lemma 3.
There exists a bijective correspondence between skew-symmetric trilinear natural2-transformations J : [[ − , − ] , • ] ⇒ [[ − , • ] , − ] + [ − , [ − , • ]] and graded antisymmetric trilinearweight 1 maps ℓ : V × → V that verify L ¥ -condition n = and vanish in total degree 1.Proof. A skew-symmetric trilinear natural 2-transformation J : [[ − , − ] , • ] ⇒ [[ − , • ] , − ] +[ − , [ − , • ]] is a map that assigns to any ( x , y , z ) ∈ L × a unique J xyz : [[ x , y ] , z ] → [[ x , z ] , y ] +[ x , [ y , z ]] in L , such that for any a = ( z , f , a ) ∈ L , we have [[ x , y ] , a ] ◦ J x , y , t a = J x , y , s a ◦ (cid:0) [[ x , a ] , y ] + [ x , [ y , a ]] (cid:1) (as well as similar equations pertaining to naturality with respect to the other two variables).A short computation shows that the last condition decomposes into the following two require-ments on the V - and the V -component: J x , y , t f + [ [ x , y ] , f ] = [[ x , f ] , y ] + [ x , [ y , f ]] , (25) [ [ x , y ] , a ] = [[ x , a ] , y ] + [ x , [ y , a ]] . (26)A graded antisymmetric trilinear weight 1 map ℓ : V × → V verifies L ¥ -condition n = ℓ ( ℓ ( u , v , w )) + ℓ ( ℓ ( u , v ) , w ) − ( − ) vw ℓ ( ℓ ( u , w ) , v ) + ( − ) u ( v + w ) ℓ ( ℓ ( v , w ) , u )+ ℓ ( ℓ ( u ) , v , w ) − ( − ) uv ℓ ( ℓ ( v ) , u , w ) + ( − ) w ( u + v ) ℓ ( ℓ ( w ) , u , v ) = , (27)for any homogeneous u , v , w ∈ V . This condition is trivial for any arguments of total degree d = u + v + w >
2. For d =
0, we write ( u , v , w ) = ( x , y , z ) ∈ V × , for d =
1, we considerategorificationand homotopification 13 ( u , v ) = ( x , y ) ∈ V × and w = f ∈ V , for d =
2, either ( u , v ) = ( x , y ) ∈ V × and w = a ∈ V ,or u = x ∈ V and ( v , w ) = ( f , g ) ∈ V × , so that Equation (4.3) reads ℓ ( ℓ ( x , y , z )) + ℓ ( ℓ ( x , y ) , z ) − ℓ ( ℓ ( x , z ) , y ) + ℓ ( ℓ ( y , z ) , x ) = , (28) ℓ ( ℓ ( x , y , f )) + ℓ ( ℓ ( x , y ) , f ) − ℓ ( ℓ ( x , f ) , y ) + ℓ ( ℓ ( y , f ) , x ) + ℓ ( ℓ ( f ) , x , y ) = , (29) ℓ ( ℓ ( x , y ) , a ) − ℓ ( ℓ ( x , a ) , y ) + ℓ ( ℓ ( y , a ) , x ) + ℓ ( ℓ ( a ) , x , y ) = , (30) ℓ ( ℓ ( x , f ) , g ) + ℓ ( ℓ ( x , g ) , f ) + ℓ ( ℓ ( f , g ) , x ) − ℓ ( ℓ ( f ) , x , g ) − ℓ ( ℓ ( g ) , x , f ) = . (31)It is easy to associate to any such map ℓ a unique Jacobiator G ( ℓ ) = J : it suffices toset J xyz : = ([[ x , y ] , z ] , ℓ ( x , y , z )) ∈ L , for any x , y , z ∈ L . Equation (28) means that J xyz hasthe correct target. Equations (25) and (26) exactly correspond to Equations (29) and (30),respectively, if we assume that in total degree d = , ℓ is valued in 2-cocycles and vanisheswhen evaluated on a 1-coboundary . These conditions are verified if we start from a structuremap ℓ that vanishes on any arguments of total degree 1. Remark 6.
Remark that the values ℓ ( x , y , f ) ∈ V cannot be encoded in a natural 2-transformation J : L × ∋ ( x , y , z ) → J xyz ∈ L (and that the same holds true for Equation (31),whose first three terms are zero, since we started from a map ℓ that vanishes on V × V ).Proof of Lemma 3 (continuation) . Conversely, to any Jacobiator J corresponds a uniquemap N ( J ) = ℓ . Just set ℓ ( x , y , z ) : = J xyz ∈ V and ℓ ( x , y , f ) =
0, for all x , y , z ∈ V and f ∈ V (as ℓ is required to have weight 1, it must vanish if evaluated on elements of degree d ≥ NG and GN are identity maps. Remark 7.
The naturality condition is, roughly speaking, the categorical analogue of theL ¥ -condition n = . For x , y , z , u ∈ L , we set h xyzu : = [ J x , y , z , u ] ◦ ( J [ x , z ] , y , u + J x , [ y , z ] , u ) ◦ ([ J xzu , y ] + ) ◦ ([ x , J yzu ] + ) ∈ L (32)and e xyzu : = J [ x , y ] , z , u ◦ ([ J xyu , z ] + ) ◦ ( J x , [ y , u ] , z + J [ x , u ] , y , z + J x , y , [ z , u ] ) ∈ L , (33)see Definition 10. The identities 1 are uniquely determined by the sources of the involvedfactors. The quadrilinear natural 2-transformations h and e are actually the left and right handcomposites of the Baez-Crans octagon that pictures the coherence law of a Lie 2-algebra, see[BC04], Definition 4.1.3. They connect the quadrilinear 2-functors F , G : L × L × L × L → L ,whose values at ( x , y , z , u ) are given by the source and the target of the 1-cells h xyzu and e xyzu ,as well as by the top and bottom sums of triple brackets of the mentioned octagon. Lemma 4.
The skew-symmetric quadrilinear 2-modifications m : h ⇛ e are in 1-to-1 cor-respondence with the graded antisymmetric quadrilinear weight 2 maps ℓ : V × → V thatverify the L ¥ -condition n = . ategorificationand homotopification 14 Proof.
A skew-symmetric quadrilinear 2-modification m : h ⇛ e maps every tuple ( x , y , z , u ) ∈ L × to a unique m xyzu : h xyzu ⇒ e xyzu in L , such that, for any a = ( u , f , a ) ∈ L , we have F ( x , y , z , a ) ◦ m x , y , z , u + t f = m xyzu ◦ G ( x , y , z , a ) (34)(as well as similar results concerning naturality with respect to the three other variables). Ifwe decompose m xyzu ∈ L = V ⊕ V ⊕ V , m xyzu = ( F ( x , y , z , u ) , h xyzu , m xyzu ) = h xyzu + m xyzu , Condition (34) reads F ( x , y , z , f ) + h x , y , z , u + t f = h xyzu + G ( x , y , z , f ) , (35) F ( x , y , z , a ) + m x , y , z , u + t f = m xyzu + G ( x , y , z , a ) . (36)On the other hand, a graded antisymmetric quadrilinear weight 2 map ℓ : V × → V , andmore precisely ℓ : V × → V , verifies L ¥ -condition n =
4, if ℓ ( ℓ ( a , b , c , d )) − ℓ ( ℓ ( a , b , c ) , d ) + ( − ) cd ℓ ( ℓ ( a , b , d ) , c ) − ( − ) b ( c + d ) ℓ ( ℓ ( a , c , d ) , b )+( − ) a ( b + c + d ) ℓ ( ℓ ( b , c , d ) , a ) + ℓ ( ℓ ( a , b ) , c , d ) − ( − ) bc ℓ ( ℓ ( a , c ) , b , d )+( − ) d ( b + c ) ℓ ( ℓ ( a , d ) , b , c ) + ( − ) a ( b + c ) ℓ ( ℓ ( b , c ) , a , d ) − ( − ) ab + ad + cd ℓ ( ℓ ( b , d ) , a , c ) + ( − ) ( a + b )( c + d ) ℓ ( ℓ ( c , d ) , a , b ) − ℓ ( ℓ ( a ) , b , c , d ) + ( − ) ab ℓ ( ℓ ( b ) , a , c , d ) − ( − ) c ( a + b ) ℓ ( ℓ ( c ) , a , b , d ) + ( − ) d ( a + b + c ) ℓ ( ℓ ( d ) , a , b , c ) = , (37)for all homogeneous a , b , c , d ∈ V . The condition is trivial for d ≥
2. For d =
0, we write ( a , b , c , d ) = ( x , y , z , u ) ∈ V × , and, for d =
1, we take ( a , b , c , d ) = ( x , y , z , f ) ∈ V × × V , sothat – since ℓ and ℓ vanish on V × V and for d =
1, respectively – Condition (37) reads ℓ ( ℓ ( x , y , z , u )) − h xyzu + e xyzu = , (38) ℓ ( ℓ ( f ) , x , y , z ) = , (39)where h xyzu and e xyzu are the V -components of h xyzu and e xyzu , see Equations (32) and (33).We can associate to any such map ℓ a unique 2-modification G ( ℓ ) = m , m : h ⇛ e . Itsuffices to set, for x , y , z , u ∈ L , m xyzu = ( F ( x , y , z , u ) , h xyzu , − ℓ ( x , y , z , u )) ∈ L . In view of Equation (38), the target of this 2-cell is t m xyzu = ( F ( x , y , z , u ) , h xyzu − ℓ ( ℓ ( x , y , z , u ))) = e xyzu ∈ L . ategorificationand homotopification 15Note now that the 2-naturality equations (25) and (26) show that 2-naturality of h : F ⇒ G means that F ( x , y , z , f ) + h x , y , z , u + t f = h xyzu + G ( x , y , z , f ) , F ( x , y , z , a ) = G ( x , y , z , a ) . When comparing with Equations (35) and (36), we conclude that m is a 2-modification if andonly if ℓ ( ℓ ( f ) , x , y , z ) = , which is exactly Equation (39).Conversely, if we are given a skew-symmetric quadrilinear 2-modification m : h ⇛ e ,we define a map N ( m ) = ℓ by setting ℓ ( x , y , z , u ) = − m xyzu , with self-explaining notations. L ¥ -condition n = m xyzu must have the target e xyzu and the second requires that m t f , x , y , z vanish – a consequence of the2-naturality of h and of Equation (36).The maps N and G are again inverses. L ¥ -condition n = Lemma 5.
Coherence law (14) is equivalent to L ¥ -condition n = .Proof. The sh Lie condition n = ℓ ( ℓ ( x , y , z , u ) , v ) − ℓ ( ℓ ( x , y , z , v ) , u ) + ℓ ( ℓ ( x , y , u , v ) , z ) − ℓ ( ℓ ( x , z , u , v ) , y ) + ℓ ( ℓ ( y , z , u , v ) , x )+ ℓ ( ℓ ( x , y ) , z , u , v ) − ℓ ( ℓ ( x , z ) , y , u , v ) + ℓ ( ℓ ( x , u ) , y , z , v ) − ℓ ( ℓ ( x , v ) y , z , u ) + ℓ ( ℓ ( y , z ) , x , u , v ) − ℓ ( ℓ ( y , u ) , x , z , v ) + ℓ ( ℓ ( y , v ) , x , z , u ) + ℓ ( ℓ ( z , u ) , x , y , v ) − ℓ ( ℓ ( z , v ) , x , y , u ) + ℓ ( ℓ ( u , v ) , x , y , z )= , (40)for any x , y , z , u , v ∈ V . It is trivial in degree d ≥
1. Let us mention that it follows fromEquation (28) that ( V , ℓ ) is a Lie algebra up to homotopy, and from Equation (30) that ℓ isa representation of V on V . Condition (40) then requires that ℓ be a Lie algebra 4-cocycleof V represented upon V .The coherence law for the 2-modification m corresponds to four different ways to re-bracket the expression F ([ x , y ] , z , u , v ) = [[[[ x , y ] , z ] , u ] , v ] by means of m , J , and [ − , − ] . Moreprecisely, we define, for any tuple ( x , y , z , u , v ) ∈ L × , four 2-cells a i : s i ⇒ t i , i ∈ { , , , } , in L , where s i , t i : A i → B i . Dependence on the considered tuple is under-stood. We omit temporarily also index i . Of course, s and t read s = ( A , s ) ∈ L and t = ( A , t ) ∈ L . If a = ( A , s , a ) ∈ L , we set a − = ( A , t , − a ) ∈ L , which is, as easily seen, the inverse of a for composition along 1-cells.ategorificationand homotopification 16 Definition 11.
The coherence law for the 2-modification m of a Lie 3-algebra ( L , [ − , − ] , J , m ) reads a + a − = a + a − , (41) where a – a are detailed in the next definitions. Definition 12.
The first 2-cell a is given by a = ◦ (cid:0) m x , y , z , [ u , v ] + [ m xyzv , u ] (cid:1) ◦ ◦ (cid:0) m [ x , v ] , y , z , u + m x , [ y , v ] , z , u + m x , y , [ z , v ] , u + (cid:1) , (42) where = J [[ x , y ] , z ] , u , v , = [ J x , [ z , v ] , y , u ]+[ J [ x , v ] , z , y , u ]+[ J x , z , [ y , v ] , u ] + , (43) and where the are the identity 2-cells associated with the elements of L provided by thecomposability condition. For instance, the squared target of the second factor of a is G ( x , y , z , [ u , v ]) +[ G ( x , y , z , v ) , u ] ,whereas the squared source of the third factor is [[[ x , [ z , v ]] , y ] , u ] + [[[[ x , v ] , z ] , y ] , u ] + [[[ x , z ] , [ y , v ]] , u ] + . . .. As the three first terms of this sum are three of the six terms of [ G ( x , y , z , v ) , u ] , the object“ . . . ”, at which 1 in 1 is evaluated, is the sum of the remaining terms and G ( x , y , z , [ u , v ]) . Definition 13.
The fourth 2-cell a is equal to a = [ m xyzu , v ] ◦ ◦ (cid:0) m [ x , u ] , y , z , v + m x , [ y , u ] , z , v + m x , y , [ z , u ] , v (cid:1) ◦ , (44) where = [ J [ x , u ] , z , y , v ]+[ J x , z , [ y , u ] , v ]+[ J x , [ z , u ] , y , v ] + , = [[ J xuv , z ] , y ]+[ J xuv , [ y , z ] ]+[ x , [ J yuv , z ]]+[[ x , J zuv ] , y ]+[ x , [ y , J zuv ]]+[ [ x , z ] , J yuv ] + . (45) Definition 14.
The third 2-cell a reads a = m [ x , y ] , z , u , v ◦ ◦ (cid:0) [ m xyuv , z ] + (cid:1) ◦ ◦ , (46) where = [ J [ x , y ] , v , u , z ] + , = [ J xyv , [ z , u ] ]+ J x , y , [[ z , v ] , u ] + J x , y , [ z , [ u , v ]] + J [[ x , v ] , u ] , y , z + J [ x , v ] , [ y , u ] , z + J [ x , u ] , [ y , v ] , z + J x , [[ y , v ] , u ] , z + J [ x , [ u , v ]] , y , z + J x , [ y , [ u , v ]] , z +[ J xyu , [ z , v ] ] , = J x , [ y , v ] , [ z , u ] + J [ x , v ] , y , [ z , u ] + J x , [ y , u ] , [ z , v ] + J [ x , u ] , y , [ z , v ] + . (47) Definition 15.
The second 2-cell a is defined as a = ◦ (cid:0) m [ x , z ] , y , u , v + m x , [ y , z ] , u , v (cid:1) ◦ ◦ (cid:0) [ x , m yzuv ] + [ m xzuv , y ] + (cid:1) ◦ , (48) where = [[ J xyz , u ] , v ] , = [ x , J [ y , z ] , v , u ]+[ J [ x , z ] , v , u , y ] + , = [ J xzv , [ y , u ] ]+[ J xzu , [ y , v ] ]+[ [ x , v ] , J yzu ]+[ [ x , u ] , J yzv ] + . (49)ategorificationand homotopification 17To get the component expression ( A + A , s + t , a − a ) = ( A + A , s + t , a − a ) (50)of the coherence law (41), we now comment on the computation of the components ( A i , s i , a i ) (resp. ( A i , t i , − a i ) ) of a i (resp. a − i ).As concerns a , it is straightforwardly seen that all compositions make sense, that its V -component is A = F ([ x , y ] , z , u , v ) , and that the V -component is a = − ℓ ( x , y , z , ℓ ( u , v )) − ℓ ( ℓ ( x , y , z , v ) , u ) − ℓ ( ℓ ( x , v ) , y , z , u ) − ℓ ( x , ℓ ( y , v ) , z , u ) − ℓ ( x , y , ℓ ( z , v ) , u ) . When actually examining the composability conditions, we find that 1 in the fourth factor of a is 1 G ( x , y , z , [ u , v ]) and thus that the target t a is made up by the 24 terms G ([ x , v ] , y , z , u ) + G ( x , [ y , v ] , z , u ) + G ( x , y , [ z , v ] , u ) + G ( x , y , z , [ u , v ]) . The computation of the V -component s is tedious but simple – it leads to a sum of 29 termsof the type “ ℓ ℓ ℓ , ℓ ℓ ℓ , or ℓ ℓ ℓ ”. We will comment on it in the case of a − , which isslightly more interesting.The V -component of a − is A = [ F xyzu , v ] = F ([ x , y ] , z , u , v ) and its V -component is equal to − a = ℓ ( ℓ ( x , y , z , u ) , v ) + ℓ ( ℓ ( x , u ) , y , z , v ) + ℓ ( x , ℓ ( y , u ) , z , v ) + ℓ ( x , y , ℓ ( z , u ) , v ) . The V -component t of a − is the V -component of the target of a . This target is thecomposition of the targets of the four factors of a and its V -component is given by t = [ e xyzu , v ] + [ J [ x , u ] , z , y , v ] + [ J x , z , [ y , u ] , v ] + [ J x , [ z , u ] , y , v ] + e [ x , u ] , y , z , v + e x , [ y , u ] , z , v + e x , y , [ z , u ] , v +[[ J xuv , z ] , y ] + [ J xuv , [ y , z ] ] + [ x , [ J yuv , z ]] + [[ x , J zuv ] , y ] + [ x , [ y , J zuv ]] + [ [ x , z ] , J yuv ] . The definition (33) of e immediately provides its V -component e as a sum of 5 terms of thetype “ ℓ ℓ or ℓ ℓ ”. The preceding V -component t of a − can thus be explicitly written asa sum of 29 terms of the type “ ℓ ℓ ℓ , ℓ ℓ ℓ , or ℓ ℓ ℓ ”. It can moreover be checked that thetarget t a − is again a sum of 24 terms – the same as for t a .The V -component of a is A = F ([ x , y ] , z , u , v ) , the V -component s can be computed as before and is a sum of 25 terms of the usual type“ ℓ ℓ ℓ , ℓ ℓ ℓ , or ℓ ℓ ℓ ”, whereas the V -component is equal to a = − ℓ ( ℓ ( x , y ) , z , u , v ) − ℓ ( ℓ ( x , y , u , v ) , z ) . ategorificationand homotopification 18Again t a is made up by the same 24 terms as t a and t a − .Eventually, the V -component of a − is A = F ([ x , y ] , z , u , v ) , the V -component t is straightforwardly obtained as a sum of 27 terms of the form “ ℓ ℓ ℓ ,ℓ ℓ ℓ , or ℓ ℓ ℓ ”, and the V -component reads − a = ℓ ( ℓ ( x , z ) , y , u , v ) + ℓ ( x , ℓ ( y , z ) , u , v ) + ℓ ( x , ℓ ( y , z , u , v )) + ℓ ( ℓ ( x , z , u , v ) , y ) . The target t a − is the same as in the preceding cases.Coherence condition (41) and its component expression (50) can now be understood. Thecondition on the V -components is obviously trivial. The condition on the V -componentsis nothing but L ¥ -condition n =
5, see Equation (40). The verification of triviality of thecondition on the V -components is lengthy: 6 pairs (resp. 3 pairs) of terms of the LHS s + t (resp. RHS s + t ) are opposite and cancel out, 25 terms of the LHS coincide with terms ofthe
RHS , and, finally, 7 triplets of
LHS -terms combine with triplets of
RHS -terms and provide7 sums of 6 terms, e.g. ℓ ( ℓ ( ℓ ( x , y ) , z ) , u , v ) + ℓ ( ℓ ( x , y , z ) , ℓ ( u , v )) + ℓ ( ℓ ( ℓ ( x , y , z ) , v ) , u ) − ℓ ( ℓ ( ℓ ( x , y , z ) , u ) , v ) − ℓ ( ℓ ( ℓ ( x , z ) , y ) , u , v ) − ℓ ( ℓ ( x , ℓ ( y , z )) , u , v ) . Since, for f = ℓ ( x , y , z ) ∈ V , we have ℓ ( f ) = t J xyz = ℓ ( ℓ ( x , z ) , y ) + ℓ ( x , ℓ ( y , z )) − ℓ ( ℓ ( x , y ) , z ) , the preceding sum vanishes in view of Equation (29). Indeed, if we associate a Lie 3-algebrato a 3-term Lie infinity algebra, we started from a homotopy algebra whose term ℓ van-ishes in total degree 1, and if we build an sh algebra from a categorified algebra, we alreadyconstructed an ℓ -map with that property. Finally, the condition on V -components is reallytrivial and the coherence law (41) is actually equivalent to L ¥ -condition n = Vect n - Cat
In this section we exhibit a specific aspect of the natural monoidal structure of the categoryof linear n -categories. Proposition 5.
If L and L ′ are linear n-categories, a family F m : L m → L ′ m of linear maps thatrespects sources, targets, and identities, commutes automatically with compositions and thusdefines a linear n-functor F : L → L ′ .Proof. If v = ( v , . . . , v m ) , w = ( w , . . . , w m ) ∈ L m are composable along a p -cell, then F m v =( F v , . . . , F m v m ) and F m w = ( F w , . . . , F m w m ) are composable as well, and F m ( v ◦ p w ) =( F m v ) ◦ p ( F m w ) in view of Equation (6). Proposition 6.
The category
Vect n- Cat admits a canonical symmetric monoidal structure ⊠ . ategorificationand homotopification 19 Proof.
We first define the product ⊠ of two linear n -categories L and L ′ . The n -globularvector space that underlies the linear n -category L ⊠ L ′ is defined in the obvious way, ( L ⊠ L ′ ) m = L m ⊗ L ′ m , S m = s m ⊗ s ′ m , T m = t m ⊗ t ′ m . Identities are clear as well, I m = m ⊗ ′ m . Thesedata can be completed by the unique possible compositions (cid:3) p that then provide a linear n -categorical structure.If F : L → M and F ′ : L ′ → M ′ are two linear n -functors, we set ( F ⊠ F ′ ) m = F m ⊗ F ′ m ∈ Hom K ( L m ⊗ L ′ m , M m ⊗ M ′ m ) , where K denotes the ground field. Due to Proposition 5, the family ( F ⊠ F ′ ) m defines a linear n -functor F ⊠ F ′ : L ⊠ L ′ → M ⊠ M ′ .It is immediately checked that ⊠ respects composition and is therefore a functor from theproduct category ( Vect n - Cat ) × to Vect n - Cat . Further, the linear n -category K , defined by K m = K , s m = t m = id K ( m > m = id K ( m < n ), acts as identity object for ⊠ . Its isnow clear that ⊠ endows Vect n - Cat with a symmetric monoidal structure.
Proposition 7.
Let L, L ′ , and L ′′ be linear n-categories. For any bilinear n-functor F : L × L ′ → L ′′ , there exists a unique linear n-functor ˜ F : L ⊠ L ′ → L ′′ , such that ⊠ ˜ F = F . Here ⊠ : L × L ′ → L ⊠ L ′ denotes the family of bilinear maps ⊠ m : L m × L ′ m ∋ ( v , v ′ ) v ⊗ v ′ ∈ L m ⊗ L ′ m , and juxtaposition denotes the obvious composition of the first with the second factor.Proof. The result is a straightforward consequence of the universal property of the tensorproduct of vector spaces.The next remark is essential.
Remark 8.
Proposition 7 is not a Universal Property for the tensor product ⊠ of Vect n- Cat ,since ⊠ : L × L ′ → L ⊠ L ′ is not a bilinear n-functor. It follows that bilinear n-functors ona product category L × L ′ cannot be identified with linear n-functors on the correspondingtensor product category L ⊠ L ′ . The point is that the family ⊠ m of bilinear maps respects sources, targets, and identities,but not compositions (in contrast with a similar family of linear maps, see Proposition 5).Indeed, if ( v , v ′ ) , ( w , w ′ ) ∈ L m × L ′ m are two p -composable pairs (note that this condition isequivalent with the requirement that v , w ∈ L m and v ′ , w ′ ∈ L ′ m be p -composable), we have ⊠ m (( v , v ′ ) ◦ p ( w , w ′ )) = ( v ◦ p w ) ⊗ ( v ′ ◦ p w ′ ) ∈ L m ⊗ L ′ m , (51)and ⊠ m ( v , v ′ ) ◦ p ⊠ m ( w , w ′ ) = ( v ⊗ v ′ ) ◦ p ( w ⊗ w ′ ) ∈ L m ⊗ L ′ m . (52)As the elements (51) and (52) arise from the compositions in L m × L ′ m and L m ⊗ L ′ m , re-spectively, – which are forced by linearity and thus involve the completely different linearstructures of these spaces – it can be expected that the two elements do not coincide.Indeed, when confining ourselves, to simplify, to the case n = ( v ◦ w ) ⊗ ( v ′ ◦ w ′ ) = ( v ⊗ v ′ ) ◦ ( w ⊗ w ′ ) + ( v − tv ) ⊗ w ′ + w ⊗ ( v ′ − tv ′ ) . (53)ategorificationand homotopification 20Observe also that the source spaces of the linear maps ◦ L ⊗ ◦ ′ L ′ : ( L × L L ) ⊗ ( L ′ × L ′ L ′ ) ∋ ( v , w ) ⊗ ( v ′ , w ′ ) ( v ◦ w ) ⊗ ( v ′ ◦ w ′ ) ∈ L ⊗ L ′ and ◦ L ⊠ L ′ : ( L ⊗ L ′ ) × L ⊗ L ′ ( L ⊗ L ′ ) ∋ (( v ⊗ v ′ ) , ( w ⊗ w ′ )) ( v ⊗ v ′ ) ◦ ( w ⊗ w ′ ) ∈ L ⊗ L ′ are connected by ℓ : ( L × L L ) ⊗ ( L ′ × L ′ L ′ ) ∋ ( v , w ) ⊗ ( v ′ , w ′ ) ( v ⊗ v ′ , w ⊗ w ′ ) ∈ ( L ⊗ L ′ ) × L ⊗ L ′ ( L ⊗ L ′ ) (54)– a linear map with nontrivial kernel. We continue working in the case n = ℓ : N ( L ) ⊗ N ( L ) → N ( L ) from a bilinear functor [ − , − ] : L × L → L .When denoting by [ − , − ] : L ⊠ L → L the induced linear functor, we get a chain map N ([ − , − ]) : N ( L ⊠ L ) → N ( L ) , so that it is natural to look for a second chain map f : N ( L ) ⊗ N ( L ) → N ( L ⊠ L ) . The informed reader may skip the following subsection.
The objects of the simplicial category D are the finite ordinals n = { , . . . , n − } , n ≥ f : m → n are the order respecting functions between the sets m and n . Let d i : n n + i , i ∈ { , . . . , n } , and let s i : n + ։ n be thesurjection that assigns the same image to i and i + i ∈ { , . . . , n − } . Any order respectingfunction f : m → n reads uniquely as f = s j . . . s j h d i . . . d i k , where the j r are decreasing andthe i s increasing. The application of this epi-monic decomposition to binary composites d i d j , s i s j , and d i s j yields three basic commutation relations.A simplicial object in the category Vect is a functor S ∈ [ D + op , Vect ] , where D + denotesthe full subcategory of D made up by the nonzero finite ordinals. We write this functor n + S ( n + ) = : S n , n ≥
0, ( S n is the vector space of n -simplices), d i S ( d i ) = : d i : S n → S n − , i ∈ { , . . . , n } ( d i is a face operator), s i S ( s i ) = : s i : S n → S n + , i ∈ { , . . . , n } ( s i is adegeneracy operator). The d i and s j verify the duals of the mentioned commutation rules.The simplicial data ( S n , d ni , s ni ) (we added superscript n ) of course completely determine thefunctor S . Simplicial objects in Vect form themselves a category, namely the functor category s ( Vect ) : = [ D + op , Vect ] , for which the morphisms, called simplicial morphisms, are thenatural transformations between such functors. In view of the epi-monic factorization, asimplicial map a : S → T is exactly a family of linear maps a n : S n → T n that commute withthe face and degeneracy operators.ategorificationand homotopification 21The nerve functor N : VectCat → s ( Vect ) is defined on a linear category L as the sequence L , L , L : = L × L L , L : = L × L L × L L . . . of vector spaces of 0 , , , . . . simplices, together with the face operators“composition” and the degeneracy operators “insertion of identity”, which verify the sim-plicial commutation rules. Moreover, any linear functor F : L → L ′ defines linear maps F n : L n ∋ ( v , . . . , v n ) → ( F ( v ) , . . . , F ( v n )) ∈ L ′ n that implement a simplicial map.The normalized or Moore chain complex of a simplicial vector space S = ( S n , d ni , s ni ) isgiven by N ( S ) n = ∩ ni = ker d ni ⊂ S n and ¶ n = d n . Normalization actually provides a functor N : s ( Vect ) ↔ C + ( Vect ) : G valued in the category of nonnegatively graded chain complexes of vector spaces. Indeed, if a : S → T is a simplicial map, then a n − d ni = d ni a n . Thus, N ( a ) : N ( S ) → N ( T ), defined on c n ∈ N ( S ) n by N ( a ) n ( c n ) = a n ( c n ) , is valued in N ( T ) n and is further a chain map. Moreover,the Dold-Kan correspondence claims that the normalization functor N admits a right adjoint G and that these functors combine into an equivalence of categories.It is straightforwardly seen that, for any linear category L , we have N ( N ( L )) = N ( L ) . (55)The categories s ( Vect ) and C + ( Vect ) have well-known monoidal structures (we de-note the unit objects by I s and I C , respectively). The normalization functor N : s ( Vect ) → C + ( Vect ) is lax monoidal, i.e. it respects the tensor products and unit objects up to coherentchain maps e : I C → N ( I s ) and EZ S , T : N ( S ) ⊗ N ( T ) → N ( S ⊗ T ) (functorial in S , T ∈ s ( Vect ) ), where EZ S , T is the Eilenberg-Zilber map. Functor N islax comonoidal or oplax monoidal as well, the chain morphism being here the Alexander-Whitney map AW S , T . These chain maps are inverses of each other up to chain homotopy,
EZ AW = , AW EZ ∼ . The Eilenberg-Zilber map is defined as follows. Let a ⊗ b ∈ N ( S ) p ⊗ N ( T ) q ⊂ S p ⊗ T q bean element of degree p + q . The chain map EZ S , T sends a ⊗ b to an element of N ( S ⊗ T ) p + q ⊂ ( S ⊗ T ) p + q = S p + q ⊗ T p + q . We have EZ S , T ( a ⊗ b ) = (cid:229) ( p , q ) − shuffles ( m , n ) sign ( m , n ) s n q ( . . . ( s n a )) ⊗ s m p ( . . . ( s m b )) ∈ S p + q ⊗ T p + q , where the shuffles are permutations of ( , . . . , p + q − ) and where the s i are the degeneracyoperators. We now come back to the construction of a chain map f : N ( L ) ⊗ N ( L ) → N ( L ⊠ L ) .ategorificationand homotopification 22For L ′ = L , the linear map (54) reads ℓ : ( N ( L ) ⊗ N ( L )) ∋ ( v , w ) ⊗ ( v ′ , w ′ ) ( v ⊗ v ′ , w ⊗ w ′ ) ∈ N ( L ⊠ L ) . If its obvious extensions ℓ n to all other spaces ( N ( L ) ⊗ N ( L )) n define a simplicial map ℓ : N ( L ) ⊗ N ( L ) → N ( L ⊠ L ) , then N ( ℓ ) : N ( N ( L ) ⊗ N ( L )) → N ( N ( L ⊠ L )) is a chain map. Its composition with the Eilenberg-Zilber chain map EZ N ( L ) , N ( L ) : N ( N ( L )) ⊗ N ( N ( L )) → N ( N ( L ) ⊗ N ( L )) finally provides the searched chain map f , see Equation (55).However, the ℓ n do not commute with all degeneracy and face operators. Indeed, we havefor instance ℓ (( d ⊗ d )(( u , v , w ) ⊗ ( u ′ , v ′ , w ′ ))) = ( u ⊗ u ′ , ( v ◦ w ) ⊗ ( v ′ ◦ w ′ )) , whereas d ( ℓ (( u , v , w ) ⊗ ( u ′ , v ′ , w ′ ))) = ( u ⊗ u ′ , ( v ⊗ v ′ ) ◦ ( w ⊗ w ′ )) . Equation (53), which means that ⊠ : L × L ′ → L ⊠ L ′ is not a functor, shows that these resultsdo not coincide.A natural idea would be to change the involved monoidal structures ⊠ of VectCat or ⊗ of C + ( Vect ). However, even if we substitute the Loday-Pirashvili tensor product ⊗ LP of 2-termchain complexes of vector spaces, i.e. of linear maps [LP98], for the usual tensor product ⊗ ,we do not get N ( L ) ⊗ LP N ( L ) = N ( L ⊠ L ) . Acknowledgements.
The authors thank the referee for having pointed out to them additionalrelevant literature.
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