Higher algebra of A_\infty and ΩB As-algebras in Morse theory II
HHIGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II
THIBAUT MAZUIR
Abstract.
This paper introduces the notion of n -morphisms between two A ∞ -algebras, suchthat 0-morphisms correspond to standard A ∞ -morphisms and 1-morphisms correspond to A ∞ -homotopies between A ∞ -morphisms. The set of higher morphisms between two A ∞ -algebras thendefines a simplicial set which has the property of being an algebraic ∞ -category. The operadicstructure of n − A ∞ -morphisms is also encoded by new families of polytopes, which we call the n -multiplihedra and which generalize the standard multiplihedra. These are constructed from thestandard simplices and multiplihedra by lifting the Alexander-Whitney map to the level of simplices.Rich combinatorics arise in this context, as conveniently described in terms of overlapping parti-tions. Shifting from the A ∞ to the Ω BAs framework, we define the analogous notion of n -morphismsbetween Ω BAs -algebras, which are again encoded by the n -multiplihedra, endowed with a thinnercell decomposition by stable gauged ribbon tree type. We then realize this higher algebra of A ∞ and Ω BAs -algebras in Morse theory. Given two Morse functions f and g , we construct n − Ω BAs -morphisms between their respective Morse cochain complexes endowed with their Ω BAs -algebrastructures, by counting perturbed Morse gradient trees associated to an admissible simplex of per-turbation data. We moreover show that any inner horn of higher morphisms arising from a count ofperturbed Morse gradient trees can always be filled, not only algebraically but also geometrically.
The -multiplihedron ∆ × J ... a r X i v : . [ m a t h . S G ] F e b THIBAUT MAZUIR
Contents n − A ∞ -morphisms 82. The HOM -simplicial sets
HOM A ∞ − alg ( A, B ) • n -multiplihedra 214. n − Ω BAs -morphisms 305. Signs for n -morphisms 34 n -morphisms in Morse theory 472. Transversality, signs and orientations 563. Towards the problem of the composition 61References 63 ... and its Ω BAs -cell decomposition
IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 3
Introduction
Summary and results of article I . This article is the direct sequel to [Maz21]. We thus beginby summarizing our first article, after which we outline the main results and constructions carriedout in the present paper.The structure of strong homotopy associative algebra, or equivalently A ∞ -algebra, was introducedin the seminal paper of Stasheff [Sta63]. It provides an operadic model for the notion of differentialgraded algebra whose product is associative up to homotopy. It is defined as the datum of a setof operations { m m : A ⊗ m → A } m (cid:62) of degree − m on a dg- Z -module ( A, ∂ ) , which satisfy thesequence of equations [ ∂, m m ] = (cid:88) i + i + i = m (cid:54) i (cid:54) m − ± m i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) . The first two equations respectively ensure that m is compatible with ∂ and that it is associa-tive up to the homotopy m . This algebraic structure is encoded by an operad in dg- Z -modules,called the operad A ∞ . As shown in [MTTV19], this operad stems in fact from an operad in thecategory of polytopes, whose arity m space of operations is defined to be the ( m − -dimensionalassociahedron K m .Similarly, the notion of A ∞ -morphism between two A ∞ -algebras A and B offers an operadic modelfor the notion of morphism of strong homotopy associative algebras which preserves the product upto homotopy. It is defined as the datum of a set of operations { f m : A ⊗ m → B } m (cid:62) of degree − m which satisfy the sequence of equations [ ∂, f m ] = (cid:88) i + i + i = mi (cid:62) ± f i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) + (cid:88) i + ··· + i s = ms (cid:62) ± m s ( f i ⊗ · · · ⊗ f i s ) . The first two equations show this time that f commutes with the differentials and that it preservesthe product up to the homotopy f . From the point of view of operadic algebra, A ∞ -morphisms areencoded by an operadic bimodule in dg- Z -modules : the operadic bimodule A ∞ − Morph . It occursfrom an operadic bimodule in polytopes, whose arity m space of operations is the ( m − -dimensionalmultiplihedron J m as shown in [MV]. A ∞ -algebras and A ∞ -morphisms between them provide a satisfactory framework for homotopytheory. The most famous instance of this statement is the homotopy transfer theorem : given ( A, ∂ A ) and ( H, ∂ H ) two cochain complexes and a homotopy retract diagram ( A, d A ) ( H, d H ) , h pi if ( A, ∂ A ) is endowed with an associative dg-algebra structure, then H can be made into an A ∞ -algebra such that i and p extend to A ∞ -morphisms. A more general statement of this theorem canbe found in [Mar06]. See also [Val20] and [LH02] for an extensive study on the homotopy theory of A ∞ -algebras. THIBAUT MAZUIR
The associahedra and multiplihedra, respectively encoding the operad A ∞ and the operadic bi-module A ∞ − Morph , can in fact be both realized as moduli spaces of metric trees. The associahedron K m is isomorphic as a CW-complex to the compactified moduli space of stable metric ribbon trees T m as first pointed out in [BV73]. The multiplihedron J m is isomorphic as a CW-complex to thecompactified moduli space of stable gauged metric ribbon trees CT m as shown in [For08] and [MW10].These moduli spaces come in fact with thinner cell decompositions, called their Ω BAs -cell decom-positions : the cell decomposition by stable ribbon tree type for T m , and the cell decomposition bystable gauged ribbon tree type for CT m . These thinner decompositions provide another operadicmodel for strong homotopy associative algebras with morphisms preserving the product up to ho-motopy between them : the standard operad Ω BAs and the operadic bimodule Ω BAs − Morph introduced in [Maz21]. We show moreover in [Maz21] that one can naturally shift from the Ω BAs tothe A ∞ framework via a morphism of operads A ∞ → Ω BAs and a morphism of operadic bimodules A ∞ − Morph → Ω BAs − Morph .Consider now a Morse function f on a closed oriented Riemannian manifold M together with aMorse-Smale metric. Following [Hut08], the Morse cochain complex C ∗ ( f ) is a homotopy retractof the singular cochain complex C ∗ sing ( M ) which is a dg-algebra with respect to the standard cupproduct. The dg-algebra structure on C ∗ sing ( M ) can thus be transferred to an A ∞ -algebra structureon C ∗ ( f ) using the homotopy transfer theorem. In fact, one can directly define an Ω BAs -algebrastructure on the Morse cochains C ∗ ( f ) by realizing the moduli spaces of stable metric ribbon trees T m in Morse theory. Given a choice of perturbation data { X m } m (cid:62) on the moduli spaces T m asintroduced by Abouzaid in [Abo11] and further studied by Mescher in [Mes18], we define the modulispaces of perturbed Morse gradient trees modeled on a stable ribbon tree type t and connecting thecritical points x , . . . , x m ∈ Crit( f ) to the critical point y ∈ Crit( f ) , denoted T X t ( y ; x , . . . , x m ) . Weprove in [Maz21] that under generic assumptions on the choice of perturbation data, these modulispaces are in fact orientable manifolds of finite dimension. If they have dimension 1, they canmoreover be compactified to 1-dimensional manifolds with boundary, whose boundary is modeledon the top dimensional strata in the boudary of the compactified moduli space T m . The Ω BAs -algebra structure on the Morse cochains C ∗ ( f ) is finally defined by counting the points of the0-dimensional moduli spaces T X t ( y ; x , . . . , x m ) .Consider now two Morse functions f and g on M together with generic choices of perturbationdata X f and X g . Endow the Morse cochains C ∗ ( f ) and C ∗ ( g ) with their associated Ω BAs -algebrastructures. We prove in [Maz21] that one can adapt the construction of the previous paragraph, todefine an Ω BAs -morphism from the Ω BAs -algebra C ∗ ( f ) to the Ω BAs -algebra C ∗ ( g ) . We countthis time 0-dimensional moduli spaces of perturbed Morse stable gauged trees modeled on a stablegauged ribbon tree type t g and connecting the critical points x , . . . , x m ∈ Crit( f ) to the criticalpoint y ∈ Crit( g ) , denoted CT Y t g ( y ; x , . . . , x m ) , after making a generic choice of perturbation data Y on the moduli spaces CT m . Motivational question . Let Y and Y (cid:48) be two choices of perturbations data on the moduli spaces CT m . Writing µ Y resp. µ Y (cid:48) for the Ω BAs -morphisms they define, the question which motivates thispaper is to know whether µ Y and µ Y (cid:48) are always homotopic or not C ∗ ( f ) C ∗ ( g ) µ Y µ Y (cid:48) . IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 5
In particular, one needs to determine what the correct notion of an homotopy between two Ω BAs -morphisms is.
Outline of the present paper and main results . The first step towards answering this problemis carried on the algebraic side in part 1, where we define the notion of n -morphisms between A ∞ -algebras and n -morphisms between Ω BAs -algebras. In section 1, we recall at first the suspendedbar construction point of view on A ∞ -algebras and the definition of an A ∞ -homotopy between A ∞ -morphisms from [LH02]. After introducing the cosimplicial dg-coalgebra ∆ n together with thelanguage of overlapping partitions, we can finally define a n -morphism between A ∞ -algebras A and B : Definition 7.
Let A and B be two A ∞ -algebras. A n -morphism from A to B is defined to be acollection of maps f ( m ) I : A ⊗ m −→ B of degree − m − dim( I ) for I ⊂ ∆ n and m (cid:62) , that satisfy (cid:104) ∂, f ( m ) I (cid:105) = dim( I ) (cid:88) j =0 ( − j f ( m ) ∂ j I + ( − | I | (cid:88) i + i + i = mi (cid:62) ± f ( i +1+ i ) I (id ⊗ i ⊗ m i ⊗ id ⊗ i ) + (cid:88) i + ··· + i s = mI ∪···∪ I s = Is (cid:62) ± m s ( f ( i ) I ⊗ · · · ⊗ f ( i s ) I s ) . The set of higher morphisms between A and B form a simplicial set HOM A ∞ − alg ( A, B ) • . Afterrecalling the basic definitions on ∞ -categories, which are simplicial sets satisfying the inner hornfilling property, we prove the following theorem in section 2 : Theorem 1.
For A and B two A ∞ -algebras, the simplicial set HOM A ∞ ( A, B ) • is an ∞ -category. This ∞ -category is in fact an algebraic ∞ -category as explained in proposition 2. However, theHOM-simplicial sets HOM A ∞ − alg ( A, B ) • fall short of defining a natural simplicial enrichment ofthe category A ∞ − alg : the composition of A ∞ -morphisms cannot be naturally lifted to define acomposition between n − A ∞ -morphisms. While the operad A ∞ stems from the associahedra K m and the operadic bimodule A ∞ − Morph stems from the multiplihedra J m , we introduce in section 3a family of polytopes encoding the A ∞ -equations for n -morphisms : the n -multiplihedra n − J m . Inthis regard, we begin by introducing a lifting of the Alexander-Whitney coproduct AW at the levelof the polytopes ∆ n , following [MTTV19]. The map AW ◦ s := (id × ( s − × AW) ◦ · · · ◦ (id × AW) ◦ AW then induces a thinner polytopal subdivision of ∆ n , whose higher dimensional cells can be labeledby all overlapping ( s + 1) -partitions of ∆ n . Using these thinner subdivisions of the ∆ n , we canconstruct a thinner polytopal subdivision of the polytopes ∆ n × J m : Definition 12.
The polytopes ∆ n × J m endowed with the polytopal subdivisions induced by themaps AW ◦ s will be called the n -multiplihedra and denoted n − J m .The boundary of the n -multiplihedra n − J m yield the n − A ∞ -equations : Proposition 7.
The boundary of the top dimensional cell [ n − J m ] of the n -multiplihedron n − J m is given by ∂ sing [ n − J m ] ∪ (cid:91) h + k = m +11 (cid:54) i (cid:54) kh (cid:62) [ n − J k ] × i [ K h ] ∪ (cid:91) i + ··· + i s = mI ∪···∪ I s =∆ n s (cid:62) [ K s ] × [dim( I ) − J i ] × · · · × [dim( I s ) − J i s ] , where I ∪ · · · ∪ I s = ∆ n is an overlapping partition of ∆ n . THIBAUT MAZUIR
We then show in section 4 that these constructions can be transported from the A ∞ to the Ω BAs realm. We define n -morphisms between Ω BAs -algebras as follows :
Definition 13. n − Ω BAs -morphisms are the higher morphisms between Ω BAs -algebras encodedby the quasi-free operadic bimodule generated by all pairs (face I ⊂ ∆ n , two-colored stable ribbontree), n − Ω BAs − Morph := F Ω BAs, Ω BAs ( I , I , I , I , · · · , ( I, sCRT n ) , · · · ; I ⊂ ∆ n ) . An operation t I,g := (
I, t g ) , whose underlying stable ribbon tree t has e ( t ) inner edges, and such thatits gauge crosses j vertices of t , is defined to have degree | t I,g | := j − − e ( t ) − dim( I ) = | I | + | t g | . Thedifferential of t I,g is given by the rule prescribed by the top dimensional strata in the boundary of CT m ( t g ) combined with the algebraic combinatorics of overlapping partitions, added to the simplicialdifferential of I , i.e. ∂t I,g = t ∂ sing I,g + ± ( ∂ CT m t g ) I . We show that the n − Ω BAs -equations are also encoded by the n -multiplihedra, endowed thistime with a thinner cell decomposition taking the Ω BAs -decomposition of the multiplihedra J m into account. What’s more, a n -morphism between Ω BAs -algebras naturally yields a n -morphismbetween A ∞ -algebras : Proposition 8.
There exists a morphism of ( A ∞ , A ∞ ) -operadic bimodules n − A ∞ − Morph → n − Ω BAs − Morph . Using the same tools as in [Maz21], we finally unravel all sign conventions in section 5.In part 2 we illustrate how n -morphisms naturally arise in geometry, here in the context of Morsetheory, solving our motivational question at the same time. In section 1 we detail the constructionof n -morphisms between Ω BAs -algebras in Morse theory. Given two Morse functions f and g on aclosed oriented manifold M , endow their Morse cochains with their Ω BAs -algebra structure comingfrom a choice of perturbation data on the moduli spaces T m . A n -morphism between C ∗ ( f ) and C ∗ ( g ) can be constructed by adapting the techniques of [Abo11] and [Mes18] that we used in [Maz21] formoduli spaces of perturbed Morse gradient trees. We define to this extent the notion of n -simplicesof perturbation data Y ∆ n : Definition 22. A n -simplex of perturbation data for a gauged metric stable ribbon tree T g is definedto be a choice of perturbation data Y δ,T g for T g for every δ ∈ ˚∆ n .Given a smooth n -simplex of perturbation data Y ∆ n ,t g on the moduli space CT m ( t g ) , we introducethe following moduli spaces of perturbed Morse gradient trees : Definition 24.
Let y ∈ Crit( g ) and x , . . . , x m ∈ Crit( f ) , we define the moduli spaces CT Y ∆ n,tg ∆ n ,t g ( y ; x , . . . , x m ) := (cid:91) δ ∈ ˚∆ n CT Y δ,tg t g ( y ; x , . . . , x m ) . As in [Maz21], these moduli spaces are orientable manifolds under some generic transversality as-sumptions on the perturbation data :
IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 7
Theorems 3 and 4.
Under some generic assumptions on the choice of perturbation data ( Y I,m ) m (cid:62) I ⊂ ∆ n ,the moduli spaces CT I,t g ( y ; x , . . . , x m ) are orientable manifolds. If they have dimension 0 they aremoreover compact. If they have dimension 1 they can be compactified to 1-dimensional manifoldswith boundary, whose boundary is modeled on the boundary of the n -multiplihedron n − J m endowedwith its n − Ω BAs -cell decomposition.
Perturbation data ( Y I,m ) m (cid:62) I ⊂ ∆ n satisfying the generic assumptions under which theorems 3 and 4 holdwill be called admissible. Given admissible choices of perturbation data X f and X g , we constructa n − Ω BAs -morphism between the Ω BAs -algebras C ∗ ( f ) and C ∗ ( g ) by counting 0-dimensionalmoduli spaces of Morse gradient trees : Theorem 5.
Let ( Y I,m ) m (cid:62) I ⊂ ∆ n be an admissible choice of perturbation data. For every m and t g ∈ sCRT m , and every I ⊂ ∆ n we define the operation µ I,t g as µ I,t g : C ∗ ( f ) ⊗ · · · ⊗ C ∗ ( f ) −→ C ∗ ( g ) x ⊗ · · · ⊗ x m (cid:55)−→ (cid:88) | y | = (cid:80) mi =1 | x i | + | t I,g | CT Y I,tg
I,t g ( y ; x , · · · , x m ) · y . This set of operations then defines a n − Ω BAs -morphism ( C ∗ ( f ) , m X f t ) → ( C ∗ ( g ) , m X g t ) . This n -morphism is in fact a twisted n -morphism as defined in [Maz21]. We subsequently prove afilling theorem for simplicial complexes of perturbation data : Theorem 6.
For every admissible choice of perturbation data Y S parametrized by a simplicial sub-complex S ⊂ ∆ n , there exists an admissible n -simplex of perturbation data Y ∆ n extending Y S . Consider in particular an inner horn in
HOM A ∞ − alg ( C ∗ ( f ) , C ∗ ( g )) • induced by an inner horn ofperturbation data. This inner horn can always be filled algebraically because this HOM-simplicalset is an ∞ -category. Theorem 6 tells us in particular that this inner horn can in fact always befilled geometrically, by directly filling the horn of perturbation data to a n -simplex of perturbationdata. Moreover, the motivational question to this paper is then a simple corollary of theorem 6.All transversality arguments and sign computations are performed in section 2 : they are mereadaptations of the analogous constructions in [Maz21]. We finally recall the second question statedat the end of [Maz21] in section 3, which is going to be tackled in an upcoming article. Acknowledgements . My first thanks go to my advisor Alexandru Oancea, for his continuous helpand support through the settling of this series of papers. I also express my gratitude to BrunoVallette for his constant reachability and his suggestions and ideas on the algebra underlying thiswork. I specially thank Jean-Michel Fischer and Guillaume Laplante-Anfossi who repeatedly tookthe time to offer explanations on higher algebra and ∞ -categories. I finally adress my thanks toFlorian Bertuol, Thomas Massoni, Amiel Peiffer-Smadja, Victor Roca Lucio and the members of theRoberta seminar for useful discussions. THIBAUT MAZUIR
Part 1
Algebra n − A ∞ -morphisms This section is dedicated to the study of the higher algebra of A ∞ -algebras . Our starting point isthe study of homotopy theory in the category of A ∞ -algebras. Putting it simply, considering two A ∞ -morphisms F, G between A ∞ -algebras, we would like to determine which notion would give asatisfactory meaning to the sentence " F and G are homotopic". This question is solved in section 1.2following [LH02], where we define the notion of an A ∞ -homotopy .Studying higher algebra of A ∞ -algebras means that we will be concerned with the higher homotopytheory of A ∞ -algebras. Typically, the questions arising are the following ones. Homotopies beingdefined, what is now a good notion of a homotopy between homotopies ? And of a homotopy betweentwo homotopies between homotopies ? And so on. Higher algebra is a general term standing for allproblems that involve defining coherent sets of higher homotopies (also called n -morphisms ) whenstarting from a basic homotopy setting.The sections following the definition of A ∞ -homotopies will then be concerned with defining agood notion of n -morphisms between A ∞ -algebras, i.e. such that A ∞ -morphisms correspond to 0-morphisms and A ∞ -homotopies to 1-morphisms. This will be done using the viewpoint of section 1.1,which defines the category of A ∞ -algebras as a full subcategory of the category of dg-coalgebras.Sections 1.3 and 1.4 consist in a pedestrian approach to the construction of these n -morphisms, andsection 1.5 sums it all up. We postpone moreover all sign computations to section 5.2.1.1. Recollections and definitions.
Let A be a graded Z -module. We introduce its suspension sA defined as the graded Z -module ( sA ) i := A i +1 . In other words, | sa | = | a | − . This is merelya notation that gives a convenient way to handle certain degrees. Note for instance that a degree − n map A ⊗ n → A is simply a degree +1 map ( sA ) ⊗ n → sA .Our main category of interest will be the category whose objects are A ∞ -algebras and whosemorphisms are A ∞ -morphisms. It will be written as A ∞ − alg . Recall that a structure of A ∞ -algebraon a dg- Z -module A can equivalently be defined as a collection of operations m n : A ⊗ n → A satisfyingthe A ∞ -equations, or as a codifferential D A on its shifted bar construction T ( sA ) . Similarly, an A ∞ -morphism is equivalently defined as a collection of operations f n : A ⊗ n → B satisfying the A ∞ -equations, or as a morphism of dg-coalgebras ( T ( sA ) , D A ) → ( T ( sB ) , D B ) . We refer to the firstarticle of this series [Maz21] for a detailed discussion on these results.As a consequence, the shifted bar construction functor identifies the category A ∞ − alg with afull subcategory of the category of dg-coalgebras dg − coalg , that is A ∞ − alg ⊂ dg − coalg . This basic idea is the key to our first construction of n -morphisms in this section. We will performsome natural constructions in the category dg − coalg , and then specialize them to the category IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 9 A ∞ − alg using the above inclusion. As before, these natural constructions will then admit aninterpretation in terms of operations A ⊗ n → B , using the universal property of the bar construction.1.2. A ∞ -homotopies. The material presented in this section is taken from the thesis of Lefèvre-Hasegawa [LH02].1.2.1.
Homotopies between morphisms of dg-coalgebras.
Definition 1 ([LH02]) . Let C and C (cid:48) be two dg-coalgebras. Let F and G be morphisms C → C (cid:48) of dg-coalgebras. A ( F, G ) -coderivation is defined to be a map H : C → C (cid:48) such that ∆ C (cid:48) H = ( F ⊗ H + H ⊗ G )∆ C . The morphisms F and G are then said to be homotopic if there exists a ( F, G ) -coderivation H ofdegree -1 such that [ ∂, H ] = G − F .
Introduce the dg-coalgebra ∆ := Z [0] ⊕ Z [1] ⊕ Z [0 < . Its differential is the singular differential ∂ sing ∂ sing ([0 < − [0] ∂ sing ([0]) = 0 ∂ sing ([1]) = 0 , its coproduct is the Alexander-Whitney coproduct ∆ ∆ ([0 < ⊗ [0 <
1] + [0 < ⊗ [1] ∆ ∆ ([0]) = [0] ⊗ [0] ∆ ∆ ([1]) = [1] ⊗ [1] , the elements [0] and [1] have degree , and the element [0 < has degree − . We refer to subsec-tion 1.3.1 for a broader interpretation of ∆ . Proposition 1 ([LH02]) . There is a one-to-one correspondence between ( F, G ) -coderivations andmorphisms of dg-coalgebras ∆ ⊗ C −→ C (cid:48) .Proof. One checks indeed that :(i) F and G are the restrictions to the summands Z [0] ⊗ C and Z [1] ⊗ C , H is the restrictionto the summand Z [0 < ⊗ C ;(ii) the coderivation relation is given by the compatibility with the coproduct ;(iii) the homotopy relation is given by the compatibility with the differential. (cid:3) A ∞ -homotopies. Using the inclusion A ∞ − alg ⊂ dg − coalg , this yields a notion of homo-topy between two A ∞ -morphisms, which we call a A ∞ -homotopy : Definition 2 ([LH02]) . Let ( T ( sA ) , D A ) and ( T ( sB ) , D B ) be two A ∞ -algebras. Given two A ∞ -morphisms F, G : ( T ( sA ) , D A ) → ( T ( sB ) , D B ) , an A ∞ -homotopy from F to G is defined to be amorphism of dg-coalgebras H : ∆ ⊗ T ( sA ) −→ T ( sB ) , whose restriction to the [0] summand is F and whose restriction to the [1] summand is G .An alternative and equivalent definition ensues then as follows (see subsection 1.4.2 for a moregeneral proof of the equivalence between the two definitions) : Definition 3 ([LH02]) . An A ∞ -homotopy between two A ∞ -morphisms ( f n ) n (cid:62) and ( g n ) n (cid:62) of A ∞ -algebras A and B is defined to be a collection of maps h n : A ⊗ n −→ B , of degree − n , which satisfy the equations [ ∂, h n ] = g n − f n + (cid:88) i + i + i = mi (cid:62) ± h i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i )+ (cid:88) i + ··· + i s + l + j + ··· + j t = ns +1+ t (cid:62) ± m s +1+ t ( f i ⊗ · · · ⊗ f i s ⊗ h l ⊗ g j ⊗ · · · ⊗ g j t ) . The signs will be made explicit in section 5.2. Using the same symbolic formalism as in [Maz21],this can be represented as [ ∂, [0 < ] = [1] − [0] + (cid:88) ± [0 < + (cid:88) ± [1][1][0 < [0] [1] , where we denote [0] , [0 < and [1] respectively for the f n , the h n and the g n .1.2.3. On this notion of homotopy.
The relation being A ∞ -homotopic on the class of A ∞ -morphismsis in fact an equivalence relation. It is moreover stable under composition. These results cannot beproven using naive tools, and are obtained through considerations of model categories. We refer toLefèvre-Hasegawa [LH02] for the reader interested in the proof of these two results.1.3. Some definitions.
The cosimplicial dg-coalgebra ∆ n . Definition 4.
Define ∆ n to be the graded Z -module generated by the faces of the standard n -simplex ∆ n , ∆ n = (cid:77) (cid:54) i < ···
These dg-coalgebras are to be seen as the realizations of the simplices ∆ n in the world of dg-coalgebras. The collection of dg-coalgebras ∆ • := { ∆ n } n (cid:62) is then naturally a cosimplicial dg-coalgebra. The coface map δ i : ∆ n − −→ ∆ n , (cid:54) i (cid:54) n , is obtained by seeing the simplex ∆ n − as the i -th face of the simplex ∆ n . The codegeneracy map σ i : ∆ n +1 −→ ∆ n , (cid:54) i (cid:54) n , is defined as [ j < · · · < j r < ˆ i < j r +1 < · · · < j s ] (cid:55)−→ [ j < · · · < j r < j r +1 − < · · · < j s − , [ j < · · · < j r < (cid:91) i + 1 < j r +1 < · · · < j s ] (cid:55)−→ [ j < · · · < j r < j r +1 − < · · · < j s − , [ j < · · · < j s ] (cid:55)−→ if [ i < i + 1] ⊂ [ j < · · · < j s ] . In other words, the face [0 < · · · < ˆ i < · · · < n + 1] and its subfaces are identified with ∆ n and itssubfaces. The same goes for [0 < · · · < (cid:91) i + 1 < · · · < n + 1] and its subfaces. All faces of ∆ n +1 thatcontain [ i < i + 1] are taken to 0.Heuristically, the coface and codegeneracy maps are obtained by applying the functor C sing −∗ : Spaces −→ dg − coalg to the cosimplicial space ∆ n , and then quotienting out each C sing −∗ (∆ n ) by the subcomplex generatedby all degenerate singular simplices. For instance, the codegeneracy map σ i : ∆ n +1 → ∆ n isobtained by contracting the edge [ i < i + 1] of ∆ n +1 , which yields the above codegeneracy map σ i : ∆ n +1 → ∆ n . We refer to [GJ09] for more details on the matter.1.3.2. Overlapping partitions.
Definition 5 ([MS03]) . Let I be a face of ∆ n . An overlapping partition of I is defined to be asequence of faces ( I l ) (cid:54) (cid:96) (cid:54) s of I such that(i) the union of this sequence of faces is I , i.e. ∪ (cid:54) (cid:96) (cid:54) s I l = I ;(ii) for all (cid:54) (cid:96) < s , max( I (cid:96) ) = min( I (cid:96) +1 ) .These two requirements then imply in particular that min( I ) = min( I ) and max( I s ) = max( I ) .If the overlapping partition has s components I (cid:96) , we will refer to it as an overlapping s -partition .These sequences of faces are those which naturally arise when applying several times the Alexander-Whitney coproduct to a face I . For instance, the Alexander-Whitney coproduct corresponds to thesum of all overlapping 2-partitions of I . Iterating n times the Alexander-Whitney coproduct, we getthe sum of all overlapping ( n + 1) -partitions of I . An overlapping 6-partition for [0 < < is forinstance [0 < <
2] = [0] ∪ [0] ∪ [0 < ∪ [1] ∪ [1 < ∪ [2] . n -morphisms between A ∞ -algebras. We now want to define a notion of higher homotopies ,or n -morphisms , between A ∞ -algebras, such that 0-morphisms are A ∞ -morphisms and 1-morphismsare A ∞ -homotopies. Since A ∞ -morphisms correspond to the set Hom dg − cog ( T ( sA ) , T ( sB )) and A ∞ -homotopies correspond to the set Hom dg − cog ( ∆ ⊗ T ( sA ) , T ( sB )) , a natural candidate for the set of n -morphisms is HOM A ∞ − alg ( A, B ) n := Hom dg − cog ( ∆ n ⊗ T ( sA ) , T ( sB )) . n -morphisms between dg -coalgebras. We begin by making explicit the n -simplices of the HOM -simplicial sets
HOM dg − cog ( C, C (cid:48) ) n := Hom dg − cog ( ∆ n ⊗ C, C (cid:48) ) . Take a morphism of dg-coalgebras f : ∆ n ⊗ C −→ C (cid:48) . Write f [ i < ···
Let A and B be two A ∞ -algebras. A n -morphism from A to B is defined to be amorphism of dg-coalgebras F : ∆ n ⊗ T ( sA ) −→ T ( sB ) . We will write b n for the degree +1 maps associated to the A ∞ -operations m n , which define thecodifferentials on T ( sA ) and T ( sB ) . The property of being a morphism of coalgebras is equivalentto the property of satisfying equations 1.2. Using the universal property of the bar construction,this is equivalent to saying that the n -morphism is given by a collection of maps of degree | I | , F ( m ) I : ( sA ) ⊗ m −→ sB , where I is a face of ∆ n and m (cid:62) . The restriction of the map F I : T ( sA ) → T ( sB ) to ( sA ) ⊗ m isthen given by F ( m ) I + (cid:88) i + i = mI ∪ I = I F ( i ) I ⊗ F ( i ) I + · · · + (cid:88) i + ··· + i s = mI ∪···∪ I s = I F ( i ) I ⊗ · · · ⊗ F ( i s ) I s + · · · + (cid:88) I ∪···∪ I m = I F (1) I ⊗ · · · ⊗ F (1) I m , IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 13 where I ∪ · · · ∪ I s = I stands for an overlapping partition of I . Corestricting to B ⊗ s yields themorphism (cid:88) i + ··· + i s = mI ∪···∪ I s = I F ( i ) I ⊗ · · · ⊗ F ( i s ) I s : ( sA ) ⊗ m −→ ( sB ) ⊗ s . The property of being compatible with the differentials is equivalent to the property of satisfyingequations 1.1. This is itself equivalent to the fact that the collection of morphisms F ( m ) I satisfies thefollowing family of equations involving morphisms ( sA ) ⊗ m → sB , dim( I ) (cid:88) j =0 ( − j F ( m ) ∂ j I + ( − | I | (cid:88) i + i + i = m F ( i +1+ i ) I (id ⊗ i ⊗ b i ⊗ id ⊗ i ) = (cid:88) i + ··· + i s = mI ∪···∪ I s = I b s ( F ( i ) I ⊗ · · · ⊗ F ( i s ) I s ) . We unwind the signs obtained by changing the b n into the m n and the degree | I | maps F ( m ) I :( sA ) ⊗ m −→ sB into degree − m + | I | maps f ( m ) I : A ⊗ m −→ B in subsection 5.2.3. The finalequations read as (cid:104) ∂, f ( m ) I (cid:105) = dim( I ) (cid:88) j =0 ( − j f ( m ) ∂ j I + ( − | I | (cid:88) i + i + i = mi (cid:62) ± f ( i +1+ i ) I (id ⊗ i ⊗ m i ⊗ id ⊗ i ) ( (cid:63) ) + (cid:88) i + ··· + i s = mI ∪···∪ I s = Is (cid:62) ± m s ( f ( i ) I ⊗ · · · ⊗ f ( i s ) I s ) , or equivalently and more visually, [ ∂, I ] = k (cid:88) j =1 ( − j ∂ singj I + (cid:88) I ∪···∪ I s = I ± I s I + (cid:88) ± I . Definition 7.
Let A and B be two A ∞ -algebras. A n -morphism from A to B is defined to be acollection of maps f ( m ) I : A ⊗ m −→ B of degree − m + | I | for I ⊂ ∆ n and m (cid:62) , that satisfyequations (cid:63) .1.5. Résumé.
Given A and B two A ∞ -algebras, we define a n -morphism between A and B to bean element of the simplicial set HOM A ∞ − alg ( A, B ) n := Hom dg − cog ( ∆ n ⊗ T ( sA ) , T ( sB )) , or equivalently a collection of operations I : A ⊗ m → B of degree − m − dim( I ) for all faces I of ∆ n and all m (cid:62) , satisfying the A ∞ -equations [ ∂, I ] = k (cid:88) j =1 ( − j ∂ singj I + (cid:88) I ∪···∪ I s = I ± I s I + (cid:88) ± I , where we refer to subsection 5.2.3 for signs. The
HOM -simplicial sets
HOM A ∞ − alg ( A, B ) • The
HOM -simplicial sets
HOM A ∞ − alg ( A, B ) • provide a satisfactory framework to study the higheralgebra of A ∞ -algebras thanks to the following theorem : Theorem 1.
For A and B two A ∞ -algebras, the simplicial set HOM A ∞ ( A, B ) • is an ∞ -category. This section is cut out as follows. In 2.1, we provide a brief exposition of ∞ -categories. Theorem 1is explained in 2.2 and then proven in 2.3. We finally show in section 2.4 how the natural approachto define a simplicial enrichment of the category A ∞ − alg using these HOM -simplicial sets fails.2.1. ∞ -categories. Motivation.
The operads A ∞ and Ω BAs provide two equivalent frameworks to study thenotion of "dg-algebras which are associative up to homotopy". See section III.2 of [Maz21] for adetailed account on the matter. In fact, the operad A ∞ can also be used to define the notion of"dg-categories whose composition is associative up to homotopy" : these categories are called A ∞ -categories . They are of prime interest in symplectic topology for instance, where they appear as theFukaya categories of symplectic manifolds. The notion of Ω BAs -categories can be defined similarly,but it has never appeared in the litterature to the author’s knowledge. A ∞ -categories are thus "categories" which are endowed with a collection of operations corre-sponding to all the higher coherent homotopies arising from the associativity up to homotopy oftheir composition. They are thus operadic in essence . The notion of ∞ -category that we are goingto define in the following lines, provides another framework to study "categories whose compositionis associative up to homotopy" but is, on the other hand, not operadic : it does not come with aspecific set of operations encoding rigidly all the higher coherent homotopies.2.1.2. Intuition.
A category can be seen as the data of a set of points, its objects, together with aset of arrows between them, the morphisms. The composition is then simply an operation whichproduces from two arrows A → B and B → C a new arrow A → C .Part of the data of an ∞ -category will also consist in a set of objects and arrows between them.The difference will lie in the notion of composition. Given two arrows u : A → B and v : B → C ,an ∞ -category will have the property that there always exists a new arrow A → C , which can becalled a composition of u and v . But this arrow is not necessarily unique, and above all, it resultsfrom a property of the "category" and is not produced by an operation of composition. It is in thissense that an ∞ -category is not operadic.2.1.3. Definition.
The correct framework to formulate this paradigm is the one of simplicial sets. Wewrite ∆ n for the simplicial set naturally realizing the standard n -simplex ∆ n , and Λ kn for the simplicialset realizing the simplicial subcomplex obtained from ∆ n by removing the faces [0 < · · · < n ] and [0 < · · · < (cid:98) k < · · · < n ] . The simplicial set Λ kn is called a horn , and if < k < n it is called an innerhorn .An ∞ -category is then defined to be a simplicial set X which has the left-lifting property withrespect to all inner horn inclusions Λ kn → ∆ n : for each n (cid:62) and each < k < n , every simplicial IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 15 map u : Λ kn → X extends to a simplicial map u : ∆ n → X whose restriction to Λ kn is u . This isillustrated in the diagram below. Λ kn X ∆ n u ∃ u The vertices of X are then to be seen as objects, while its edges correspond to morphisms.The left-lifting property with respect to Λ → ∆ ensures that the following diagram can alwaysbe filled by the dashed arrows . The [0 < edge will represent a composition of the morphisms associated to [0 < and [1 < .The intuition of subsection 2.1.2 is thus realized, and comes with a wide range of higher homotopiescontrolled by the combinatorics of simplicial algebra.2.1.4. Further properties.
The notion of ∞ -category contains the classical notion of category. Thereexists a functor N : Cat → sSets called the nerve functor , which associates to every small category C a simplicial set N ( C ) which is an ∞ -category. In fact, an A ∞ -category also yields an ∞ -category,through the A ∞ -nerve functor N A ∞ : A ∞ − cat → sSets constructed in [Fao17]. Moreover, an ∞ -category X determines canonically a classical category, called its homotopy category Ho( X ) , whoseobjects are the same as those of X , and whose morphisms are the equivalence classes of morphismsof X for the homotopy equivalence relation. For more details we refer to [Lur09].As a matter of fact, most categorical constructions that hold for classical categories can be carriedover to the framework of ∞ -categories : limits, fiber sequences, cones, terminal and initial objects,to only cite a few. André Joyal refers to them as quasi-categories for this reason.Finally, we mention that ∞ -categories are the fibrant-cofibrant objects of the model category ofsimplicial sets endowed with the Joyal model category structure : in this model category, everyobject is cofibrant and the fibrant objects are exactly those which admit the left-lifting propertywith respect to the cofibrations Λ kn → ∆ n , n (cid:62) and < k < n . This is thoroughly explainedin [Joy].2.2. ∞ -categories and the higher algebra of A ∞ -algebras. As stated in the introduction, the
HOM -simplicial sets
HOM A ∞ − alg ( A, B ) • provide a satisfactory framework to study higher algebraof A ∞ -algebras : Theorem 1.
For A and B two A ∞ -algebras, the simplicial set HOM A ∞ ( A, B ) • is an ∞ -category. Our proof will even show that they are algebraic ∞ -categories as explained in subsection 2.3.2.One aspect of this construction needs moreover to be clarified. The points of these ∞ -categories arethe A ∞ -morphisms, and the arrows between them are the A ∞ -homotopies. This can be misleading atfirst sight, but the points are the morphisms and NOT the algebras and the arrows are the homotopiesand NOT the morphisms . Proof that the
HOM A ∞ ( A, B ) • are ∞ -categories. Let A and B be two A ∞ -algebras. Wenow prove that the HOM -simplicial set
HOM A ∞ ( A, B ) • is an ∞ -category, using the shifted barconstruction framework, that is by defining an A ∞ -algebra to be a set of degree +1 operations b n : ( sA ) ⊗ n → sA satisfying equations (cid:88) i + i + i = n b i +1+ i (id ⊗ i ⊗ b i ⊗ id ⊗ i ) = 0 . The proof will mainly consist of easy but tedious combinatorics. We recommend reading it in twosteps : first ignoring the signs ; then adding them at the second reading stage and referring tosection 5.2 for the sign conventions on the shifted A ∞ -equations.2.3.1. Proof.
Consider an inner horn Λ kn → HOM A ∞ ( A, B ) • , where < k < n . We want to provethat the following diagram can be completed Λ kn HOM A ∞ ( A, B ) • ∆ n . This inner horn corresponds to a collection of degree − dim( I ) morphisms F ( m ) I : ( sA ) ⊗ m −→ sB for I ⊂ Λ kn , which satisfy the A ∞ -equations dim( I ) (cid:88) j =0 ( − j F ( m ) ∂ j I +( − | I | (cid:88) i + i + i = m F ( i +1+ i ) I (id ⊗ i ⊗ b i ⊗ id ⊗ i ) = (cid:88) i + ··· + i s = mI ∪···∪ I s = I b s ( F ( i ) I ⊗· · ·⊗ F ( i s ) I s ) . Filling this horn amounts then to defining a collection of operations F ( m )[0 < ··· < (cid:98) k< ··· We now prove that the following choice of morphisms fills the inner horn : F ( m )∆ n := 0 ,F ( m )[0 < ··· < (cid:98) k< ··· HOM A ∞ − alg ( A, B ) • are algebraic ∞ -categories. We constructed in the previous subsec-tion an explicit filler of any inner horn Λ kn −→ HOM A ∞ − alg ( A, B ) • which was defined as F ( m )∆ n := 0 ,F ( m )[0 < ··· < (cid:98) k< ··· Proposition 2. There is a natural one-to-one correspondence fillers Λ kn HOM A ∞ ( A, B ) • ∆ n ←→ (cid:26) families of maps of degree − nU ( m ) : ( sA ) ⊗ m → sB, m (cid:62) (cid:27) . We will say that HOM A ∞ − alg ( A, B ) • is an algebraic ∞ -category . This terminology is borrowedfrom [RNV20].2.3.3. Remark on the proof. We point out that this proof does not adapt to the more general caseof a HOM -simplicial set HOM dg − cog ( C, C (cid:48) ) • . Indeed, while we can always solve the equation [ ∂, f ∆ n ] = n (cid:88) j =0 ( − j f [0 < ··· < (cid:98) j< ··· Let F : ∆ n ⊗ C → C (cid:48) and G : ∆ n ⊗ C (cid:48) → C (cid:48)(cid:48) be two morphisms of dg-coalgebras. The only natural candidate to construct acomposition is the Alexander-Whitney coproduct ∆ ∆ n , i.e. we define G ◦ F to be the followingcomposite of maps ∆ n ⊗ C ∆ n ⊗ ∆ n ⊗ C ∆ n ⊗ C (cid:48) C (cid:48)(cid:48) ∆ ∆ n ⊗ id C id ∆ n ⊗ F G . Note that we use the word "map" and not "morphism" because we have yet to check that thiscomposite is indeed a morphism of dg-coalgebras.Before moving on, we point out that for the composition of continuous maps of topological spaces ∆ n × X → Y we use the diagonal map of ∆ n , ∆ n × X −→ diag ∆ n × id X ∆ n × ∆ n × X −→ id ∆ n × F ∆ n × Y −→ G Z . This construction cannot be reproduced in our case, as the diagonal map ∆ n → ∆ n ⊗ ∆ n does notrespect the gradings, nor does it respect the differentials.Set ∆ n := ∆ n , ∆ n := ∆ n and write ∆ ∆ n : ∆ n → ∆ n ⊗ ∆ n for the Alexander-Whitney mapseen as a map from the dg-coalgebra ∆ n to the product dg-coalgebra ∆ n ⊗ ∆ n . In the previouscomposition, it is sufficient to prove that ∆ ∆ n : ∆ n → ∆ n ⊗ ∆ n is a morphism of dg-coalgebrasto prove that G ◦ F is a morphism of dg-coalgebras. This map does preserve the differential, but itdoes not preserve the coproduct ! Indeed, consider the following diagram ∆ n ∆ n ⊗ ∆ n ∆ n ⊗ ∆ n ⊗ ∆ n ⊗ ∆ n ∆ n ⊗ ∆ n ( ∆ n ⊗ ∆ n ) ⊗ ( ∆ n ⊗ ∆ n ) ∆ ∆ n ∆ ∆ n ∆ ∆ n ⊗ ∆ ∆ n id ⊗ τ ⊗ id∆ ∆ n ⊗ ∆ ∆ n . Up to specifying the correct signs, the upper composite path of the square is the map I (cid:55)−→ (cid:88) I ∪ I ∪ I ∪ I = I I ⊗ I ⊗ I ⊗ I , where I ∪ I ∪ I ∪ I denotes an overlapping partition of the face I ⊂ ∆ n , while the lower compositepath of the square is the map I (cid:55)−→ (cid:88) I ∪ I ∪ I ∪ I = I I ⊗ I ⊗ I ⊗ I . These two maps are not equal, the square does not commute.The map G ◦ F is in particular not a morphism of dg-coalgebras, and as a result does not belongto HOM dg − cog ( A, C ) n . It ensues that the composition fails to be lifted to higher morphisms withthis naive approach. IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 21 An open question and a result. We proved in the previous subsection that the natural approachto lift the composition in A ∞ − alg to HOM A ∞ − alg ( A, B ) • does not work. Hence, it is still anopen question to know whether these HOM -simplicial sets could fit into a simplicial enrichment ofthe category A ∞ − alg . This would then endow A ∞ − alg with a structure of ( ∞ , -category asexplained at p.6 of [Lur09]. In fact, it is unclear to the author why such a statement should be true.Still, something more can be said about the previous non-commutative square. Again, up tocomputing the correct signs, the map ∆ n −→ ( ∆ n ⊗ ∆ n ) ⊗ ( ∆ n ⊗ ∆ n ) I (cid:55)−→ (cid:88) I ∪ I ∪ I ∪ I ∪ I = I I ⊗ I ⊗ ( I ∪ I ) ⊗ I , defines an homotopy between the upper composite path and the lower composite path of the square :it fills the square to make it homotopy-commutative. In the language introduced in [MS03], the uppercomposite path is equal to , the lower one is equal to , and the filler is equal to .Using the results of [MS03], the author proved in [Maz] that : Theorem 2. The Alexander-Whitney coproduct can be lifted to an A ∞ -morphism between the dg-coalgebras ∆ n and ∆ n ⊗ ∆ n , whose first higher homotopy is the map . The n -multiplihedra Recall from [Maz21] that, in the language of operadic algebra, A ∞ -algebras are governed by theoperad A ∞ , and A ∞ -morphisms are governed by the ( A ∞ , A ∞ ) -operadic bimodule A ∞ − Morph .These two operadic objects actually stem from collections of polytopes. Under the functor C cell −∗ the associahedra { K m } realise the operad A ∞ , while the multiplihedra { J m } form a ( { K m } , { K m } ) -operadic bimodule realising A ∞ − Morph .The first section shows that the operadic bimodule formalism for A ∞ -morphisms can be gener-alised to the setting of n − A ∞ -morphisms : for each n (cid:62) there exists an ( A ∞ , A ∞ ) -operadicbimodule n − A ∞ − Morph , which encodes n -morphisms between A ∞ -algebras. In fact, they fit intoa cosimplicial operadic bimodule { n − A ∞ − Morph } n (cid:62) . Reproducing the previous progression, wewould like to realise the combinatorics of n -morphisms at the level of polytopes. The first step inthis direction is performed in section 3.2 : we explain how to lift the Alexander-Whitney coproductto the level of the standard simplices ∆ n and study the rich combinatorics that arise in this problem.Section 3.3 subsequently introduces the n -multiplihedra n − J m , which are the polytopes ∆ n × J m en-dowed with a thinner polytopal subdivision. These polytopes do not form a ( { K m } , { K m } ) -operadicbimodule, but they suffice to recover all the combinatorics of n -morphisms.3.1. The cosimplicial ( A ∞ , A ∞ ) -operadic bimodule encoding higher morphisms. The ( A ∞ , A ∞ ) -operadic bimodules n − A ∞ − Morph . The ( A ∞ , A ∞ ) -operadic bimodule en-coding A ∞ -morphisms is the quasi-free ( A ∞ , A ∞ ) -operadic bimodule generated in arity n by oneoperation of degree − n , A ∞ − Morph = F A ∞ ,A ∞ ( , , , , · · · ) . Representing the generating operations of the operad A ∞ acting on the right in blue and theones of the operad A ∞ acting on the left in red , its differential is defined by ∂ ( m ) = (cid:88) h + k = m +11 (cid:54) i (cid:54) kh (cid:62) ± ki h + (cid:88) i + ··· + i s = ms (cid:62) ± i s i . Definition 8. The ( A ∞ , A ∞ ) -operadic bimodule encoding n − A ∞ -morphisms is the quasi-free ( A ∞ , A ∞ ) -operadic bimodule generated in arity m by the operations f ( m ) I of degree − m + | I | , forall faces I of ∆ n , and whose differential is defined by ∂ ( f ( m ) I ) = dim I (cid:88) j =0 ( − j f ( m ) ∂ singj I + (cid:88) i + i + i = mi (cid:62) ± f ( i +1+ i ) I (id ⊗ i ⊗ m i ⊗ id ⊗ i ) + (cid:88) i + ··· + i s = mI ∪··· I s = Is (cid:62) ± m s ( f ( i ) I ⊗ · · · ⊗ f ( i s ) I s ) . Representing the operations f ( m ) I as I , this can be rewritten as n − A ∞ − Morph = F A ∞ ,A ∞ ( I , I , I , I , · · · ; I ⊂ ∆ n ) . where ∂ ( I ) = dim I (cid:88) j =0 ( − j ∂ singj I + (cid:88) I ∪···∪ I s = I ± I s I + (cid:88) ± I . The collection of ( A ∞ , A ∞ ) -operadic bimodules { n − A ∞ − Morph } n (cid:62) forms a cosimplicial ( A ∞ , A ∞ ) -operadic bimodule whose coface and codegeneracy maps are built out of those of sec-tion 1.3. Given two A ∞ -algebras A ∞ → Hom( A ) and A ∞ → Hom( B ) , the set of n -morphisms isthen simply given by HOM A ∞ − alg ( A, B ) n = Hom ( A ∞ , A ∞ ) − op . bimod . ( n − A ∞ − Morph , Hom( A, B )) . The two-colored operadic viewpoint. Recall that A ∞ -algebras and A ∞ -morphisms betweenthem are naturally encoded by the quasi-free two-colored operad A ∞ := F ( , , , · · · , , , , · · · , , , , , · · · ) , with differential given by the A ∞ -algebra relations on the one-colored operations, and the A ∞ -morphism relations on the two-colored operations.Similarly, A ∞ -algebras and n − A ∞ -morphisms between them are naturally encoded by the quasi-free two-colored operad n − A ∞ := F ( , , , · · · , , , , · · · , ( I , I , I , I , · · · ; I ⊂ ∆ n )) , with differential given by the A ∞ -algebra relations on the one-colored operations, and the n − A ∞ -morphism relations on the two-colored operations. The collection of two-colored operads { n − A ∞ } n (cid:62) constitutes again a cosimplicial two-colored operad. IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 23 Polytopal subdivisions on ∆ n induced by the Alexander-Whitney coproduct. Oneway of interpreting the Alexander-Whitney coproduct ∆ ∆ n : ∆ n −→ ∆ n ⊗ ∆ n is to say that it is a diagonal on the dg- Z -module ∆ n . The following natural question then arises. Does there exist a diagonal (i.e. a polytopal map that is homotopic to the usual diagonal - the usualdiagonal map failing to be polytopal in general) on the standard n -simplex ∆ n , AW : ∆ n −→ ∆ n × ∆ n , such that its image under the functor C cell −∗ is AW −∗ = ∆ ∆ n ? The answer to this question is positive, and contains rich combinatorics that we now lay out.3.2.1. The map AW . We recall in this section the construction of a diagonal on the standard sim-plices explained in [MTTV19] (example 1 of section 2.3.). Definition 9 ([MTTV19]) . Consider the realizations of the standard n -simplices ∆ n := conv { (1 , . . . , , , . . . , ∈ R n } = { ( z , . . . , z n ) ∈ R n | (cid:62) z (cid:62) · · · (cid:62) z n (cid:62) } . We define the map AW by the formula AW( z , · · · , z n ) = ((2 z − , . . . , z i − , , . . . , , (1 , · · · , , z i +1 , . . . , z n )) , for (cid:62) z (cid:62) · · · (cid:62) z i (cid:62) / (cid:62) z i +1 (cid:62) · · · (cid:62) z n (cid:62) .In particular, the map AW comes with a thinner polytopal subdivision of ∆ n , whose n + 1 topdimensional strata are given by the subsets { ( z , . . . , z n ) ∈ R n | > z > · · · > z i > / > z i +1 > · · · > z n > } ⊂ ∆ n , and whose i -codimensional strata are simply obtained by replacing i symbols " > " by a symbol " = " inthe previous sequence of inequalities. This thinner subdivision is represented on the figures 1, 2 and 3,together with the value of AW on each stratum of the subdivision. Figure 1. The AW-subdivision of ∆ and ∆ (cid:55)−→ × , (cid:55)−→ × . Figure 2. Values of AW on ∆ : the stratum to which AW is applied is colored inred (cid:55)−→ × , (cid:55)−→ × , (cid:55)−→ × . Figure 3. Values of AW on ∆ The polytopal map AW is not coassociative. The Alexander-Whitney coproduct ∆ ∆ n on dg- Z -modules is coassociative. However, the diagonal map AW is not ! This can be checked for the1-simplex ∆ : (AW × id) ◦ AW(2 / 5) = AW × id(0 , / 5) = (0 , , / × AW) ◦ AW(2 / 5) = id × AW(0 , / 5) = (0 , / , . Proposition 3. The polytopal map AW is not coassociative. The polytopal subdivisions that the polytopal maps (AW × id) ◦ AW : ∆ n −→ ∆ n × ∆ n × ∆ n , (id × AW) ◦ AW : ∆ n −→ ∆ n × ∆ n × ∆ n induce on ∆ n are also different. See an instance on figure 4.3.2.3. i -overlapping s -partitions. We defined in subsection 1.3.2 the notion of an overlapping s -partition of a face I of ∆ n . We refine it now : Definition 10. An i -overlapping s -partition of I is a sequence of faces ( I (cid:96) ) (cid:54) (cid:96) (cid:54) s of I such that(i) the union of this sequence of faces is I , i.e. ∪ (cid:54) (cid:96) (cid:54) s I (cid:96) = I ; IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 25 Figure 4. The ( AW × id) ◦ AW -subdivision and the (id × AW) ◦ AW -subdivision of ∆ Figure 5. The first three subdivisions of ∆ (ii) there are exactly i integers (cid:96) such that (cid:54) (cid:96) < s and max( I (cid:96) ) = min( I (cid:96) +1 ) .An overlapping s -partition as defined in definition 5 is then simply a ( s − -overlapping s -partition.A 1-overlapping 3-partition for [0 < < is for instance [0 < < 2] = [0] ∪ [0] ∪ [1 < . Polytopal subdivisions of ∆ n induced by iterations of AW . Definition 11. Define the s -th right iterate of the map AW as AW ◦ s := (id × ( s − × AW) ◦ · · · ◦ (id × AW) ◦ AW : ∆ n −→ (∆ n ) × s . For each s (cid:62) , the map AW ◦ s induces a thinner polytopal subdivision of ∆ n . These subdivisionswill be called the AW ◦ s -subdivisions of ∆ n . They can be described rather simply. While the AW -subdivision is obtained by dividing ∆ n into pieces with all hyperplanes z i = 1 / for (cid:54) i (cid:54) n , the AW ◦ s -subdivision can be constructed as follows : Proposition 4. The AW ◦ s -subdivision of ∆ n is the subdivision obtained after dividing ∆ n by allhyperplanes z i = (1 / k , for (cid:54) i (cid:54) n and (cid:54) k (cid:54) s . The first three subdivisions of ∆ are represented in figure 5. Note that a different choice for AW ◦ s , for instance AW ◦ = (AW × id) ◦ AW , would have yielded a different subdivision of ∆ n .Choices have to be made, because AW is not coassociative. Labeling the AW ◦ s -subdivisions of ∆ n . The image of the map AW ◦ s under the functor C cell −∗ yields the s -th iterate of the Alexander-Whitney coproduct ∆ ∆ n . Hence, the n -dimensional strata ofthe AW ◦ s -subdivision of ∆ n are in one-to-one correspondence with the overlapping ( s + 1) -partitionsof ∆ n . The AW and AW ◦ subdivisions of ∆ are represented in figure 6. We have in fact that : Proposition 5. The codimension i strata of the AW ◦ s -subdivision of ∆ n lying in the interior of ∆ n are in one-to-one correspondence with the ( s − i ) -overlapping ( s +1) -partitions of ∆ n . More generallygiven a face I ⊂ ∆ n , the strata of the AW ◦ s -subdivision of ∆ n which are lying in the interior of I and have codimension i w.r.t. to the dimension of I are in one-to-one correspondence with the ( s − i ) -overlapping ( s + 1) -partitions of I . Figure 6. The AW and AW ◦ subdivisions of ∆ We finally give a recipe to easily label a stratum from the AW ◦ s -subdivision of ∆ n using itsdescription as a set of inequalities and equalities. The top dimensional strata of the AW -subdivisionof ∆ n are given by the sets C i := { ( z , . . . , z n ) ∈ R n | > z > · · · > z i > / > z i +1 > · · · > z n > } ⊂ ∆ n . We check that each stratum C i yields the term [0 < · · · < i ] ⊗ [ i < · · · < n ] under the functor C cell −∗ .Similarly, the top dimensional strata of the AW ◦ s -subdivision of ∆ n are defined by the inequalities · · · > z i k > (1 / k > z i k +1 > · · · , for (cid:54) k (cid:54) s . We write C i ,...,i s for such a stratum. As before, each stratum C i ,...,i s yields the term [0 < · · · < i ] ⊗ [ i < · · · < i ] ⊗ · · · ⊗ [ i s < · · · < n ] under the functor C cell −∗ . Proposition 6. Consider a codimension i stratum of the AW ◦ s -subdivision of ∆ n which is lying inthe interior of ∆ n . This stratum is defined by s − i inequalities of the form · · · > z i k > (1 / k > z i k +1 > · · · , and i equalities of the form · · · > z i k = (1 / k > z i k +1 > · · · . IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 27 The labeling of this stratum can then be obtained under the following simple transformation rules : · · · > z i k > (1 / k > z i k +1 > · · · (cid:55)−→ · · · < i k ] ⊗ [ i k < · · · , · · · > z i k = (1 / k > z i k +1 > · · · (cid:55)−→ · · · < i k − ⊗ [ i k < · · · . This recipe easily carries over to the case of strata lying in the boundary of ∆ n .3.3. The n -multiplihedra n − J m . The multiplihedra. The polytopes encoding A ∞ -morphisms between A ∞ -algebras are the mul-tiplihedra J m , m (cid:62) : they form a collection { J m } m (cid:62) which is a ( { K m } , { K m } ) -operadic bimodulewhose image under the functor C cell −∗ is the ( A ∞ , A ∞ ) -operadic bimodule A ∞ − Morph . The faces ofcodimension i of J m are labeled by all possible broken two-colored trees obtained by blowing-up i times the two-colored m -corolla. See for instance [Maz21] for pictures of the multiplihedra J , J and J . The multiplihedra J m can moreover be realized as the compactifications of moduli spaces of sta-ble two-colored metric ribbon trees CT m , where each CT m is seen as the unique ( m − -dimensionalstratum of CT m .3.3.2. The n -multiplihedra n − J m . Consider the polytope ∆ n × J m for n (cid:62) and m (cid:62) . It is themost natural candidate for a polytope encoding n -morphisms between A ∞ -algebras. However, itdoes not fulfill that property as it is. Indeed, its faces correspond to the data of a face of ∆ n , thatis of some I ⊂ ∆ n , and of a face of J m , that is of a broken two-colored tree obtained by blowing-up several times the two-colored m -corolla. This labeling is too coarse, as it does not contain thefollowing trees, that appear in the A ∞ -equations for n -morphisms I s I . We resolve this issue by constructing a thinner polytopal subdivision of ∆ n × J m . Consider aface F of J m , such that exactly s unbroken two-colored trees appear in the two-colored broken treelabeling it – see an instance above by forgetting the I i . We then refine the polytopal subdivision of ∆ n × F into ∆ n AW ◦ ( s − × F , where ∆ n AW ◦ ( s − denotes ∆ n endowed with its AW ◦ ( s − -subdivision.This refinement process can be done consistently for each face F of J m , in order to obtain a newpolytopal subdivision of ∆ n × J m . Definition 12. The n -multiplihedra are defined to be the polytopes ∆ n × J m endowed with theprevious polytopal subdivision. We denote them n − J m .See some examples in figures 7, 8 and 9. We illustrate definition 12 with the construction of the -multiplihedron ∆ × J depicted on figure 8. The polytope ∆ has one 2-dimensional face labeledby [0 < < and three 1-dimensional faces labeled by [0 < , [1 < and [0 < . The polytope J has one 1-dimensional face labeled by and has two 0-dimensional faces labeled by and. Consider now the product polytope ∆ × J . Its has one unique 3-dimensional face labeledby [0 < < × and five 2-dimensional faces. The faces [0 < × , [1 < × , [0 < × and [0 < < × that are left unchanged under the construction of the previous paragraph, asthey each feature only 1 unbroken two-colored tree. They respectively correspond to the faces A, B, F and G on figure 8. The fifth face is the face [0 < < × . It features 2 unbroken two-coloredtrees : we thus have to refine the polytopal subdivision of ∆ × into ∆ × . This refinementproduces the strata ([0] ⊗ [0 < < × , ([0 < ⊗ [1 < × and ([0 < < ⊗ [2]) × ,which respectively correspond to the labels C, D and E on figure 8. This concludes the constructionof the -multiplihedron ∆ × J . Figure 7. The -multiplihedron ∆ × J Figure 8. The -multiplihedron ∆ × J The n -multiplihedra encode n − A ∞ -morphisms. Now in which sense do these polytopes encode n − A ∞ -morphisms ? Note first that the collection { n − J m } m (cid:62) is not a ( { K m } , { K m } ) -operadicbimodule ! Indeed, a ( { K m } , { K m } ) -operadic bimodule structure would for instance make appear a IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 29 Figure 9. The -multiplihedron ∆ × J stratum labeled by I I s , where I ∪ · · · ∪ I s = ∆ n is an overlapping partition of ∆ n . This stratum does not appear in thepolytopal subdivision of n − J m . Hence these polytopes do not recover the ( A ∞ , A ∞ ) -operadicbimodule n − A ∞ − Morph .However, the polytopal subdivision of n − J m still contains enough combinatorics to recover a n -morphism. This polytope has a unique ( n + m − -dimensional cell [ n − J m ] , which is labeled by ∆ n . By construction : Proposition 7. The boundary of the cell [ n − J m ] is given by ∂ sing [ n − J m ] ∪ (cid:91) h + k = m +11 (cid:54) i (cid:54) kh (cid:62) [ n − J k ] × i [ K h ] ∪ (cid:91) i + ··· + i s = mI ∪···∪ I s =∆ n s (cid:62) [ K s ] × [dim( I ) − J i ] × · · · × [dim( I s ) − J i s ] , where I ∪ · · · ∪ I s = ∆ n is an overlapping partition of ∆ n . Details on the orientation of the top dimensional strata in this boundary are worked out insection 5.3. Note moreover that the collection { n − J m } n (cid:62) is a cosimplicial polytope. This impliesthat the image of each cell [dim( I ) − J m ] under the functor C cell −∗ yields an element whose boundaryis exactly given by the A ∞ -equations for n -morphisms. It is in that sense that the n − J m encode n -morphisms. The previous boundary formula also implies that the n − J m will constitute a goodparametrizing space for constructing moduli spaces in symplectic topology, whose count should giverise to n -morphisms between Floer complexes. n − Ω BAs -morphisms The multiplihedra J m can be realized by compactifying the moduli spaces of stable two-coloredmetric ribbon trees CT m and come with two cell decompositions. The first one consists in consideringeach CT m as a ( m − -dimensional stratum and encodes the operadic bimodule A ∞ − Morph . Thesecond one is obtained by considering the stratification of the moduli spaces CT m by two-coloredstable ribbon tree types, and encodes the operadic bimodule Ω BAs − Morph . The Ω BAs -celldecomposition is moreover a refinement of the A ∞ -cell decomposition. As a consequence, thereexists a morphism of operadic bimodules A ∞ − Morph → Ω BAs − Morph , as shown in [Maz21]. Itis hence sufficient to construct an Ω BAs -morphism between Ω BAs -algebras to then naturally getan A ∞ -morphism between A ∞ -algebras.We define in this section n − Ω BAs -morphisms between Ω BAs -algebras. Building on the viewpointof the previous paragraph, we then explain how, by refining the cell decomposition of the polytope n − J m , we get a new cell decomposition encoding n − Ω BAs -morphisms. This construction yieldsin particular a morphism of operadic bimodules n − A ∞ − Morph → n − Ω BAs − Morph . All signcomputations are moreover postponed to section 5.4.4.1. n − Ω BAs -morphisms. Recollections on Ω BAs -morphisms. Ω BAs -morphisms are the morphisms between Ω BAs -algebras encoded by the quasi-free operadic bimodule generated by all two-colored stable ribbontrees Ω BAs − Morph := F Ω BAs, Ω BAs ( , , , , · · · , sCRT n , · · · ) . A two-colored stable ribbon tree t g whose underlying stable ribbon tree t has e ( t ) inner edges, andsuch that its gauge crosses j vertices of t , has degree | t g | := j − − e ( t ) .The differential of a two-colored stable ribbon tree t g is given by the signed sum of all two-coloredstable ribbon trees obtained from t g under the rule prescribed by the top dimensional strata in theboundary of CT n ( t g ) . : the gauge moves to cross exactly one additional vertex of the underlyingstable ribbon tree (gauge-vertex) ; an internal edge located above the gauge or intersecting it breaksor, when the gauge is below the root, the outgoing edge breaks between the gauge and the root(above-break) ; edges (internal or incoming) that are possibly intersecting the gauge, break below it,such that there is exactly one edge breaking in each non-self crossing path from an incoming edge tothe root (below-break) ; an internal edge that does not intersect the gauge collapses (int-collapse).4.1.2. n − Ω BAs -morphisms. Definition 13. n − Ω BAs -morphisms are the higher morphisms between Ω BAs -algebras encodedby the quasi-free operadic bimodule generated by all pairs (face I ⊂ ∆ n , two-colored stable ribbontree), n − Ω BAs − Morph := F Ω BAs, Ω BAs ( I , I , I , I , · · · , ( I, sCRT n ) , · · · ; I ⊂ ∆ n ) . An operation t I,g := ( I, t g ) is defined to have degree | t I,g | := | I | + | t g | . The differential of t I,g isgiven by the rule prescribed by the top dimensional strata in the boundary of CT m ( t g ) combined IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 31 with the algebraic combinatorics of overlapping partitions, added to the simplicial differential of I ,i.e. ∂t I,g = t ∂ sing I,g + ± ( ∂ CT m t g ) I . We refer to section 5.4 for a more complete definition and sign conventions. The sign computationsare in particular more involved, as we did not describe an ad hoc construction analogous to the shiftedbar construction as in the A ∞ case. We also point out that the symbol I used here is the same asthe one used for the arity 2 generating operation of n − A ∞ − Morph . It will however be clear fromthe context what I stands for in the rest of this paper. We moreover compute the differential inthe following instance | [0 < < | = − ,∂ ( [0 < < ) = ± [1 < ± [0 < ± [0 < ± [0 < < ± [0 < < ± [0 < < ± [0 < < ± [0 < < ± [0] [0 < < ± [0 < 1] [1 < ± [0 < < 2] [2] . From n − Ω BAs -morphisms to n − A ∞ -morphisms. A n − Ω BAs -morphism between two Ω BAs -algebras naturally yields a n − A ∞ -morphism between the induced A ∞ -algebras : Proposition 8. There exists a morphism of ( A ∞ , A ∞ ) -operadic bimodules n − A ∞ − Morph → n − Ω BAs − Morph given on the generating operations of n − A ∞ − Morph by f I,m (cid:55)−→ (cid:88) t g ∈ CBRT m ± f I,t g , where CBRT m denotes the set of two-colored binary ribbon trees of arity m . This proposition is proven in subsection 5.4.7. Note that the collection of operadic bimodules { n − Ω BAs − Morph } n (cid:62) is once again a cosimplicial operadic bimodule, where the cofaces andcodegeneracies are as in subsection 1.3.1. This sequence of morphisms of operadic bimodules definesthen in fact a morphism of cosimplicial operadic bimodules { n − A ∞ − Morph } n (cid:62) −→ { n − Ω BAs − Morph } n (cid:62) . A conjecture on the HOM-simplicial sets HOM ΩBAs − alg ( A, B ) • . Given A and B two Ω BAs -algebras, we define the HOM-simplicial set HOM ΩBAs − alg ( A, B ) n := Hom ( ΩBAs , ΩBAs ) − op . bimod . ( n − Ω BAs − Morph , Hom( A, B )) . Drawing from theorem 1, we conjecture the following result : Conjecture 1. The simplicial sets HOM ΩBAs − alg ( A, B ) • are ∞ -categories. The proof without signs should follow the same lines as the proof without signs of theorem 1,working this time with stable ribbon trees and gauged stable ribbon trees instead of corollae. Thesign computations will however be much more complicated, as we did not describe a constructionanalogous to the shifted bar construction which would yield ad hoc sign conventions. The n -multiplihedra encode n − Ω BAs -morphisms. The n − Ω BAs -cell decomposition of ∆ n × CT m . The polytopes encoding n − A ∞ -morphismshave been defined to be the polytopes ∆ n × J m endowed with a thinner polytopal subdivision inducedby the maps AW ◦ s . These thinner subdivisions incorporate the combinatorics of i -overlapping s -partitions in the boundary of the polytopes ∆ n × J m .Consider now the multiplihedra J m = CT m endowed with its Ω BAs -cell decomposition, i.e. its celldecomposition by broken stable two-colored ribbon tree type. We define a thinner cell decompositionon the product CW-complex ∆ n × CT m as follows. Consider a stratum CT m ( t br,g ) of the modulispace CT m , such that exactly s unbroken two-colored ribbon trees appear in the broken stabletwo-colored ribbon tree t br,g labeling it. We refine the cell decomposition of ∆ n × CT m ( t br,g ) into ∆ n AW ◦ ( s − × CT m ( t br,g ) , where ∆ n AW ◦ ( s − denotes ∆ n endowed with its AW ◦ s -subdivision. Thisrefinement process can again be done consistently for each stratum CT m ( t br,g ) of CT m in order toobtain a thinner cell decomposition of ∆ n × CT m . Definition 14. We define the n − Ω BAs -cell decomposition of the n -multiplihedron ∆ n × CT m tobe the cell decomposition described in the previous paragraph.See some examples in figures 10 and 11. By construction, the n − Ω BAs -cell decomposition of ∆ n × CT m is moreover a refinement of the n − A ∞ -cell decomposition of ∆ n × CT m . Figure 10. The − Ω BAs -cell decomposition of ∆ × CT These CW-complexes encode n − Ω BAs -morphisms. Consider the associahedra K m = T m endowed with their Ω BAs -cell decompositions. We endow moreover the spaces ∆ n × CT m with their n − Ω BAs -cell decompositions. As in the A ∞ case, the collection of CW-complexes { ∆ n × CT m } m (cid:62) is not a ( {T m } , {T m } ) -operadic bimodule. Carrying over the details of subsection 3.3.3, it containshowever enough combinatorics to recover a n − Ω BAs -morphism. What’s more, the collection { ∆ n × CT m } n (cid:62) is again a cosimplicial CW-complex. IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 33 Figure 11. The − Ω BAs -cell decomposition of ∆ × CT Résumé. The higher homotopies or n -morphisms extending the notion of A ∞ -morphisms and A ∞ -homotopies between A ∞ -algebras are defined to be the morphisms of dg-coalgebras ∆ n ⊗ T ( sA ) −→ T ( sB ) . From an operadic viewpoint, they are naturally encoded by the operadic bimodule, n − A ∞ − Morph = F A ∞ ,A ∞ ( I , I , I , I , · · · ; I ⊂ ∆ n ) . where the differential is defined as [ ∂, I ] = k (cid:88) j =1 ( − j ∂ singj I + (cid:88) I ∪···∪ I s = I ± I s I + (cid:88) ± I . The combinatorics of this differential are encoded by new families of polytopes called the n -multipli-hedra, which are the data of the polytopes ∆ n × J m together with a polytopal subdivision inducedby the maps AW ◦ s . They will constitute a good parametrizing space for constructing moduli spacesin symplectic topology, whose count should recover a n -morphism between Floer complexes.On the other side, the natural n -morphisms extending the notion of Ω BAs -morphisms are definedby adapting the operadic viewpoint on n − A ∞ -morphisms. They are naturally encoded by theoperadic bimodule, n − Ω BAs − Morph = F Ω BAs, Ω BAs ( I , I , I , I , · · · , ( I, sCRT m ) , · · · ; I ⊂ ∆ n ) , where the differential is again defined as a signed sum prescribed by a rule on two-colored treescombinatorics combined with the algebraic combinatorics of overlapping partitions, added to thesimplicial differential. This differential is encoded in the data of the polytopes ∆ n × J m endowedwith a thinner cell decomposition induced by two-colored stable ribbon tree types and the maps AW ◦ s . It is moreover sufficient to construct a n − Ω BAs -morphism between Ω BAs -algebras inorder to recover a n − A ∞ -morphism between the induced A ∞ -algebras, thanks to the morphism ofoperadic bimodules n − A ∞ − Morph −→ n − Ω BAs − Morph . We show in part 2 that the previous CW-complexes constitute a good parametrizing space formoduli spaces in Morse theory, whose count will recover a n − Ω BAs -morphism between Morsecochain complexes. 5. Signs for n -morphisms We now work out all the signs left uncomputed in the previous sections of this part. Thesecomputations will be done resorting to the basic conventions on signs and orientations that we werealready using in [Maz21], and that we briefly recall in the first section. In the next two sections,we display and explain the two natural sign conventions for n − A ∞ -morphisms ensuing from thebar construction viewpoint, and then show that one of these conventions is in fact contained in thepolytopes n − J m . We finally give a complete definition of the operadic bimodule n − Ω BAs − Morph and build the morphism of operadic bimodules n − A ∞ − Morph → n − Ω BAs − Morph of proposition 8.5.1. Conventions for signs and orientations. Koszul sign rule. The formulae in this section will be written using the Koszul sign rule. Wewill moreover work exclusively with cohomological conventions.Given A and B two dg Z -modules, the differential on A ⊗ B is defined as ∂ A ⊗ B ( a ⊗ b ) = ∂ A a ⊗ b + ( − | a | a ⊗ ∂ B b . Given A and B two dg Z -modules, we consider the graded Z -module Hom( A, B ) whose degree r component is given by all maps A → B of degree r . We endow it with the differential ∂ Hom( A,B ) ( f ) := ∂ B ◦ f − ( − | f | f ◦ ∂ A =: [ ∂, f ] . Given f : A → A (cid:48) and g : B → B (cid:48) two graded maps between dg- Z -modules, we set ( f ⊗ g )( a ⊗ b ) = ( − | g || a | f ( a ) ⊗ g ( b ) . Finally, given f : A → A (cid:48) , f (cid:48) : A (cid:48) → A (cid:48)(cid:48) , g : B → B (cid:48) and g (cid:48) : B (cid:48) → B (cid:48)(cid:48) , we define ( f (cid:48) ⊗ g (cid:48) ) ◦ ( f ⊗ g ) = ( − | g (cid:48) || f | ( f (cid:48) ◦ f ) ⊗ ( g (cid:48) ◦ g ) . We check in particular that with this sign rule, the differential on a tensor product A ⊗ · · · ⊗ A n isgiven by ∂ A ⊗···⊗ A n = n (cid:88) i =1 id A ⊗ · · · ⊗ ∂ A i ⊗ · · · ⊗ id A n . Tensor product of dg-coalgebras. Given A and B two dg Z -modules, define the twist map τ : A ⊗ B → B ⊗ A , τ ( a ⊗ b ) = ( − | a || b | b ⊗ a . Suppose now that A and B are dg-coalgebras, with respective coproducts ∆ A and ∆ B . The tensorproduct A ⊗ B can then be endowed with a structure of dg-coalgebra whose coproduct is defined as ∆ A ⊗ B := A ⊗ B −→ ∆ A ⊗ ∆ B A ⊗ A ⊗ B ⊗ B −→ id A ⊗ τ ⊗ id B ( A ⊗ B ) ⊗ ( A ⊗ B ) , and whose differential is the product differential ∂ A ⊗ B = ∂ A ⊗ id B + id A ⊗ ∂ B . IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 35 Orientation of the boundary of a manifold with boundary. Let ( M, ∂M ) be an oriented n -manifold with boundary. We choose to orient its boundary ∂M as follows : given x ∈ ∂M , a basis e , . . . , e n − of T x ( ∂M ) , and an outward pointing vector ν ∈ T x M , the basis e , . . . , e n − is positivelyoriented if and only if the basis ν, e , . . . , e n − is a positively oriented basis of T x M .Under this convention, given two manifolds with boundary K and L , the boundary of the productmanifold K × L is then ∂ ( K × L ) = ∂K × L ∪ ( − dim( K ) K × ∂L , where the ( − dim( K ) sign means that the product orientation of K × ∂L differs from its orientationas the boundary of K × L by a ( − dim( K ) sign. This convention also recovers the classical singularand cubical differentials as detailed in [Maz21] : ∂ ∆ n = n (cid:91) i =0 ( − i ∆ n − i and ∂I n = n (cid:91) i =1 ( − i ( I n − i, ∪ − I n − i, ) . Signs for n − A ∞ -morphisms. We now work out the signs in the A ∞ -equations for n − A ∞ -morphisms, thus completing definition 7. More precisely, we will unwind two sign conventions usingthe bar construction viewpoint. The impatient reader can straightaway jump to subsection 5.2.3where the signs used in the rest of this paper are made explicit.5.2.1. Recollections on the bar construction and A ∞ -algebras. Let A be a dg- Z -module. Define thesuspension and desuspension maps s : A −→ sA w : sA → Aa (cid:55)−→ sa sa (cid:55)−→ a , which are respectively of degree − and +1 . We verify that with the Koszul sign rule, w ⊗ m ◦ s ⊗ m = ( − m )id A ⊗ m . Then, note for instance that a degree − m map m m : A ⊗ m → A yields a degree +1 map b m := sm m w ⊗ m : ( sA ) ⊗ m → sA .To the set of operations b m one can associate a unique coderivation D on T ( sA ) . We provedin [Maz21] using this viewpoint that the equation D = 0 yields two sign conventions for the A ∞ -equations [ m , m m ] = − (cid:88) i + i + i = m (cid:54) i (cid:54) n − ( − i i + i m i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) , (A) [ m , m m ] = − (cid:88) i + i + i = m (cid:54) i (cid:54) n − ( − i + i i m i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) , (B)and that these conventions are related by a ( − m ) twist applied to the operation m m , which comesfrom the formula w ⊗ m ◦ s ⊗ m = ( − m )id A ⊗ m .We will adopt the exact same approach to work out two sign conventions for n − A ∞ -morphismsin the following subsection : first by writing A ∞ -equations without signs using the viewpoint of a morphism between bar constructions F : ∆ n ⊗ T ( sA ) → T ( sB ) , and secondly by unfolding the signscoming from the suspension and desuspension maps.5.2.2. The two conventions coming from the bar construction. The two conventions for the A ∞ -equations for n − A ∞ -morphisms are (cid:104) m , f ( m ) I (cid:105) = dim( I ) (cid:88) j =0 ( − j f ( m ) ∂ j I + ( − | I | (cid:88) i + i + i = mi (cid:62) ( − i i + i f ( i +1+ i ) I (id ⊗ i ⊗ m i ⊗ id ⊗ i ) (A) − (cid:88) i + ··· + i s = mI ∪···∪ I s = Is (cid:62) ( − (cid:15) A m s ( f ( i ) I ⊗ · · · ⊗ f ( i s ) I s ) , (cid:104) m , f ( m ) I (cid:105) = dim( I ) (cid:88) j =0 ( − j f ( m ) ∂ j I + ( − | I | (cid:88) i + i + i = mi (cid:62) ( − i + i i f ( i +1+ i ) I (id ⊗ i ⊗ m i ⊗ id ⊗ i ) (B) − (cid:88) i + ··· + i s = mI ∪···∪ I s = Is (cid:62) ( − (cid:15) B m s ( f ( i ) I ⊗ · · · ⊗ f ( i s ) I s ) , which can we rewritten as dim( I ) (cid:88) j =0 ( − j f ( m ) ∂ j I + ( − | I | (cid:88) i + i + i = m ( − i i + i f ( i +1+ i ) I (id ⊗ i ⊗ m i ⊗ id ⊗ i ) (A) = (cid:88) i + ··· + i s = mI ∪···∪ I s = I ( − (cid:15) A m s ( f ( i ) I ⊗ · · · ⊗ f ( i s ) I s ) , dim( I ) (cid:88) j =0 ( − j f ( m ) ∂ j I + ( − | I | (cid:88) i + i + i = m ( − i + i i f ( i +1+ i ) I (id ⊗ i ⊗ m i ⊗ id ⊗ i ) (B) = (cid:88) i + ··· + i s = mI ∪···∪ I s = I ( − (cid:15) B m s ( f ( i ) I ⊗ · · · ⊗ f ( i s ) I s ) , where (cid:15) A = s (cid:88) j =1 ( s − j ) | I j | + s (cid:88) j =1 i j s (cid:88) k = j +1 (1 − i k − | I k | ) ,(cid:15) B = s (cid:88) j =1 i j s (cid:88) k = j +1 | I k | + s (cid:88) j =1 ( s − j )(1 − i j − | I j | ) . These two sign conventions are equivalent : given a sequence of operations m m and f ( m ) I satisfyingequations (A), we check that the operations m (cid:48) m := ( − m ) m m and f (cid:48) ( m ) I := ( − m ) f ( m ) I satisfyequations (B). IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 37 Consider now two dg- Z -modules A and B , together with a collection of degree − m maps m m : A ⊗ m → A and m m : B ⊗ m → B (we use the same notation for sake of readability), and acollection of degree − m + | I | maps f ( m ) I : A ⊗ m → B . We associate to the maps m m the degree +1 maps b m := sm m w ⊗ m , and also associate to the maps f ( m ) I the degree | I | maps F ( m ) I := sf ( m ) I w ⊗ m :( sA ) ⊗ m → sB . We denote D A and D B the unique coderivations coming from the maps b m actingrespectively on T ( sA ) and T ( sB ) , and F : ∆ n ⊗ T ( sA ) → T ( sB ) the unique morphism of coalgebrasassociated to the maps F ( m ) I . The equation F ( ∂ sing ⊗ id T ( sA ) + id ∆ n ⊗ D A ) = D B F is then equivalent to the equations dim( I ) (cid:88) j =0 ( − j F ( m ) ∂ j I +( − | I | (cid:88) i + i + i = m F ( i +1+ i ) I (id ⊗ i ⊗ b i ⊗ id ⊗ i ) = (cid:88) i + ··· + i s = mI ∪···∪ I s = I b s ( F ( i ) I ⊗· · ·⊗ F ( i s ) I s ) . There are now two ways to unravel the signs from these equations, which will lead to conventions(A) and (B).The first way consists in simply replacing the b m and the F ( m ) I by their definition. It yields signconventions (A). The left-hand side transforms as dim( I ) (cid:88) j =0 ( − j F ( m ) ∂ j I + ( − | I | (cid:88) i + i + i = m F ( i +1+ i ) I (id ⊗ i ⊗ b i ⊗ id ⊗ i )= dim( I ) (cid:88) j =0 ( − j sf ( m ) ∂ j I w ⊗ m + ( − | I | (cid:88) i + i + i = m sf ( i +1+ i ) I w ⊗ i +1+ i (id ⊗ i ⊗ sm i w ⊗ i ⊗ id ⊗ i )= dim( I ) (cid:88) j =0 ( − j sf ( m ) ∂ j I w ⊗ m + ( − | I | (cid:88) i + i + i = m ( − i sf ( i +1+ i ) I ( w ⊗ i ⊗ wsm i w ⊗ i ⊗ w ⊗ i )= dim( I ) (cid:88) j =0 ( − j sf ( m ) ∂ j I w ⊗ m + ( − | I | (cid:88) i + i + i = m ( − i i + i sf ( i +1+ i ) I (id ⊗ i ⊗ m i ⊗ id ⊗ i )( w ⊗ i ⊗ w ⊗ i ⊗ w ⊗ i )= s dim( I ) (cid:88) j =0 ( − j f ( m ) ∂ j I + ( − | I | (cid:88) i + i + i = m ( − i i + i f ( i +1+ i ) I (id ⊗ i ⊗ m i ⊗ id ⊗ i ) w ⊗ m , while the right-hand side transforms as (cid:88) i + ··· + i s = mI ∪···∪ I s = I b s ( F ( i ) I ⊗ · · · ⊗ F ( i s ) I s )= (cid:88) i + ··· + i s = mI ∪···∪ I s = I sm s w ⊗ s ( sf ( i ) I w ⊗ i ⊗ · · · ⊗ sf ( i s ) I s w ⊗ i s )= (cid:88) i + ··· + i s = mI ∪···∪ I s = I ( − (cid:80) sj =1 ( s − j ) | I j | sm s ( wsf ( i ) I w ⊗ i ⊗ · · · ⊗ wsf ( i s ) I s w ⊗ i s )= s (cid:88) i + ··· + i s = mI ∪···∪ I s = I ( − (cid:15) A m s ( f ( i ) I ⊗ · · · ⊗ f ( i s ) I s ) w ⊗ m , where (cid:15) A = (cid:80) sj =1 ( s − j ) | I j | + (cid:80) sj =1 i j (cid:16)(cid:80) sk = j +1 (1 − i k − | I k | ) (cid:17) .The second way consists in first composing and post-composing by w and s ⊗ m and then replacingthe b m and F ( m ) I by their definition. It yields the (B) sign conventions. We will denote m (cid:48) m :=( − m ) m m and f (cid:48) ( m ) I := ( − m ) f ( m ) I . The left-hand side then transforms as dim( I ) (cid:88) j =0 ( − j wF ( m ) ∂ j I s ⊗ m + ( − | I | (cid:88) i + i + i = m wF ( i +1+ i ) I (id ⊗ i ⊗ b i ⊗ id ⊗ i ) s ⊗ m = dim( I ) (cid:88) j =0 ( − j f (cid:48) ( m ) ∂ j I + ( − | I | (cid:88) i + i + i = m ( − i f ( i +1+ i ) I w ⊗ i +1+ i ( s ⊗ i ⊗ sm i w ⊗ i s ⊗ i ⊗ s ⊗ i )= dim( I ) (cid:88) j =0 ( − j f (cid:48) ( m ) ∂ j I + ( − | I | (cid:88) i + i + i = m ( − i + i i f ( i +1+ i ) I w ⊗ i +1+ i s ⊗ i +1+ i (id ⊗ i ⊗ m (cid:48) i ⊗ id ⊗ i )= dim( I ) (cid:88) j =0 ( − j f (cid:48) ( m ) ∂ j I + ( − | I | (cid:88) i + i + i = m ( − i + i i f (cid:48) ( i +1+ i ) I (id ⊗ i ⊗ m (cid:48) i ⊗ id ⊗ i ) , IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 39 while the right-hand side transforms as (cid:88) i + ··· + i s = mI ∪···∪ I s = I wb s ( F ( i ) I ⊗ · · · ⊗ F ( i s ) I s ) s ⊗ m = (cid:88) i + ··· + i s = mI ∪···∪ I s = I m s w ⊗ s ( sf ( i ) I w ⊗ i ⊗ · · · ⊗ sf ( i s ) I s w ⊗ i s ) s ⊗ m = (cid:88) i + ··· + i s = mI ∪···∪ I s = I ( − (cid:80) sj =1 ( i j (cid:80) sk = j +1 | I k | ) m s w ⊗ s ( sf ( i ) I w ⊗ i s ⊗ i ⊗ · · · ⊗ sf ( i s ) I s w ⊗ i s s ⊗ i s )= (cid:88) i + ··· + i s = mI ∪···∪ I s = I ( − (cid:15) B m s w ⊗ s s ⊗ s ( f (cid:48) ( i ) I ⊗ · · · ⊗ f (cid:48) ( i s ) I s )= (cid:88) i + ··· + i s = mI ∪···∪ I s = I ( − (cid:15) B m (cid:48) s ( f (cid:48) ( i ) I ⊗ · · · ⊗ f (cid:48) ( i s ) I s ) , where (cid:15) B = (cid:80) sj =1 (cid:16) i j (cid:80) sk = j +1 | I k | (cid:17) + (cid:80) sj =1 ( s − j )(1 − i j − | I j | ) .5.2.3. Choice of convention in this paper. We will work in the rest of this paper with the set ofconventions (B). The operations m m of an A ∞ -algebra will satisfy equations [ ∂, m m ] = − (cid:88) i + i + i = m (cid:54) i (cid:54) n − ( − i + i i m i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) , and a n − A ∞ -morphism between two A ∞ -algebras will satisfy equations (cid:104) ∂, f ( m ) I (cid:105) = dim( I ) (cid:88) j =0 ( − j f ( m ) ∂ j I + ( − | I | (cid:88) i + i + i = ni (cid:62) ( − i + i i f ( i +1+ i ) I (id ⊗ i ⊗ m i ⊗ id ⊗ i ) − (cid:88) i + ··· + i s = mI ∪···∪ I s = Is (cid:62) ( − (cid:15) B m s ( f ( i ) I ⊗ · · · ⊗ f ( i s ) I s ) , where (cid:15) B = (cid:80) sj =1 (cid:16) i j (cid:80) sk = j +1 | I k | (cid:17) + (cid:80) sj =1 ( s − j )(1 − i j − | I j | ) .In [Maz21] we had chosen conventions (B) for A ∞ -algebras and A ∞ -morphisms because they werethe ones naturally arising in the realizations of the associahedra and the multiplihedra à la Loday.We prove a similar result in the following section : these sign conventions are contained in thepolytopes n − J m = ∆ n × J m where J m is a Forcey-Loday realization of the multiplihedron.5.3. Signs and the polytopes n − J m . Loday associahedra and Forcey-Loday multiplihedra. In [Maz21] we introduced explicit poly-topal realizations of the associahedra and the multiplihedra : the weighted Loday realizations K ω of the associahedra from [MTTV19] and the weighted Forcey-Loday realizations J ω of the multi-plihedra from [MV]. We then proved using basic considerations on affine geometry that, under theconvention of section 5.1, their boundaries were equal to ∂K ω = − (cid:91) i + i + i = n (cid:54) i (cid:54) n − ( − i + i i K ω × K (cid:101) ω ,∂J ω = (cid:91) i + i + i = ni (cid:62) ( − i + i i J ω × K (cid:101) ω ∪ − (cid:91) i + ··· + i s = ms (cid:62) ( − ε B K ω × J (cid:101) ω × · · · × J (cid:101) ω s where the weights ω , (cid:101) ω and (cid:101) ω t are derived from the weights ω , and ε B = s (cid:88) j =1 ( s − j )(1 − i j ) . In particular, these polytopes contain sign conventions (B) for A ∞ -algebras and A ∞ -morphisms.5.3.2. The boundary of n − J m . Consider now a n -multiplihedron ∆ n × J ω , where J ω is a Forcey-Lodayrealization of the multiplihedron J m . Forgetting for now about its refined polytopal subdivision, itsboundary reads as ∂ (∆ n × J ω ) = ∂ ∆ n × J ω ∪ ( − n ∆ n × ∂J ω . Note moreover that in the AW ◦ ( s − -polytopal subdivision of ∆ n , each top dimensional cell labeledby an overlapping partition I ∪ · · · ∪ I s = ∆ n is in fact isomorphic to the product I × · · · × I s . Wewrite this as ∆ n AW ◦ ( s − = (cid:91) I ∪···∪ I s =∆ n I × · · · × I s . Proposition 9. The n-multiplihedra ∆ n × J ω endowed with their n − A ∞ -polytopal subdivisioncontain sign conventions (B) for n − A ∞ -morphisms.Proof. The first component of the boundary of ∆ n × J ω is given by ∂ ∆ n × J ω = n (cid:91) i =0 ( − i ∆ n − i × J ω . The second, by the first part of the boundary of ∂J ω , ( − n (cid:91) i + i + i = mi (cid:62) ( − i + i i (∆ n × J ω ) × K (cid:101) ω . IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 41 The third and last component transforms as follows : ( − n ∆ n × ( − (cid:91) i + ··· + i s = ms (cid:62) ( − ε B K ω × J (cid:101) ω × · · · × J (cid:101) ω s =( − n +1 (cid:91) i + ··· + i s = ms (cid:62) ( − ε B ∆ n × K ω × J (cid:101) ω × · · · × J (cid:101) ω s =( − n +1 (cid:91) i + ··· + i s = ms (cid:62) ( − ε B (cid:91) I ∪···∪ I s =∆ n I × · · · × I s × K ω × J (cid:101) ω × · · · × J (cid:101) ω s =( − n +1 (cid:91) i + ··· + i s = ms (cid:62) (cid:91) I ∪···∪ I s =∆ n ( − ε B + s (cid:80) sj =1 | I j | K ω × I × · · · × I s × J (cid:101) ω × · · · × J (cid:101) ω s = − ( − n (cid:91) i + ··· + i s = mI ∪···∪ I s =∆ n s (cid:62) ( − ε B + sn + (cid:80) sj =1 ( i j − ( (cid:80) sk = j +1 | I k | ) K ω × ( I × J (cid:101) ω ) × · · · × ( I s × J (cid:101) ω s ) . We then check that (cid:15) B = n + ε B + sn + (cid:80) sj =1 ( i j − (cid:16)(cid:80) sk = j +1 | I k | (cid:17) modulo 2. Hence, the polytopes n − J m contain indeed sign conventions (B) for n − A ∞ -morphisms. (cid:3) The operadic bimodule n − Ω BAs − Morph . In [Maz21], we computed the signs for Ω BAs -morphisms as follows. Endowing the compactified moduli spaces CT m with their Ω BAs -cell decom-positions, we define the operadic bimodule Ω BAs − Morph to be the realization under the functor C cell −∗ of the operadic bimodule {CT m } m (cid:62) . The signs in the differential are then computed as thesigns arising in the top dimensional strata in the boundary of the moduli spaces CT m ( t g ) . The signsfor the action-composition maps are the signs ensuing from the image under the functor C cell −∗ of theaction-composition maps for the moduli spaces CT m ( t g ) .The goal of this section is to completely state definition 13, with explicit signs and formulae. Wehave however seen in subsection 4.2.2 that there is no operadic bimodule in compactified modulispaces whose image under the functor C cell −∗ could realize the operadic bimodule n − Ω BAs − Morph .We will still compute the signs for the action-composition maps by introducing some suitable spacesof metric trees, which do not define an operadic bimodule but will however carry enough structurefor our computations. The differential will simply be defined by reading the signs arising in the topdimensional strata of the boundary of the CW-complex ∆ n × CT m endowed with its n − Ω BAs -celldecomposition.5.4.1. Notation. As in [Maz21], we choose to use the formalism of orientations on trees to define theoperadic bimodule n − Ω BAs − Morph . Recall that this formalism originates from [MS06]. Definition 15. Given a broken stable ribbon tree t br , an ordering of t br is defined to be an orderingof its i finite internal edges e , . . . , e i . Two orderings are said to be equivalent if one passes fromone ordering to the other by an even permutation. An orientation of t br is then defined to bean equivalence class of orderings, and written ω := e ∧ · · · ∧ e i . Each tree t br has exactly twoorientations. Given an orientation ω of t br we will write − ω for the second orientation on t br , calledits opposite orientation . In this section, we write t br,g for a broken gauged stable ribbon tree, and t g for an unbrokengauged stable ribbon tree. Definition 16. We set to be the unique stable gauged tree of arity 1 and call it the trivialgauged tree . We define the underlying broken stable ribbon tree t br of a t br,g to be the ribbon treeobtained by first deleting all the in t br,g , and then forgetting all the remaining gauges of t br,g .We will moreover refer to a gauge in t br,g which is associated to a non-trivial gauged tree, as a non-trivial gauge of t br,g . An orientation on a broken gauged stable ribbon tree t br,g is then definedto be an orientation ω on t br . An instance of association t br,g (cid:55)→ t br Definition 17. Consider a gauged tree t br,g which has b gauges, trivial or not. A list I := ( I , . . . , I b ) of faces I a ⊂ ∆ n will be called a ∆ n -labeling of t br,g . The tree t br,g endowed with its labeling will bewritten ( I , t br,g ) .We think of ( I , t br,g ) as depicted in the figure below, where trees are represented as corollae for thesake of readability. I b I Definition of the spaces of operations. Definition 18 (Spaces of operations) . Consider the Z -module freely generated by the pairs ( I , t br,g , ω ) ,where ω is an orientation on t br,g and I is a ∆ n -labeling of t br,g . We define the arity m space ofoperations n − Ω BAs − Morph( m ) ∗ to be the quotient of this Z -module under the relation ( I , t br,g , − ω ) = − ( I , t br,g , ω ) . Introducing the notation | I | := (cid:80) ba =1 | I a | , a pair ( I , t br,g , ω ) is then defined to have degree | ( I , t br,g , ω ) | := | I | + | t br,g | . The oriented spaces CT m ( I , t br,g , ω ) . Consider a ∆ n -labeled gauged tree ( I , t br,g ) , togetherwith a choice of orientation ω on t br,g . We define the spaces CT m ( I , t br,g , ω ) := I × · · · × I b × CT m ( t br,g , ω ) . An element of CT m ( I , t br,g , ω ) is thus of the form ( δ , . . . , δ b , λ , . . . , λ g , l e , . . . , l e ( t br ) ) ∈ I × · · · × I b × ] − ∞ , + ∞ [ g × ]0 , + ∞ [ e ( t br ) , where the λ i are the non-trivial gauges of t br,g ordered from left to right, and the l e i are the lengthsof the finite internal edges of t br ordered according to ω . These spaces are then simply oriented bytaking the product orientation of their factors. IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 43 Definition of the action-compositions maps. We may now introduce the "action-composition"maps on the spaces CT m ( I , t br,g ) , that we will use to define the signs of the action-composition mapsfor n − Ω BAs − Morph . Define the maps O i : CT ( I , t br,g , ω ) × T ( t (cid:48) br , ω (cid:48) ) = I × CT ( t br,g , ω ) × T ( t (cid:48) br , ω (cid:48) ) −→ I × CT ( t br,g ◦ i t (cid:48) br , ω ∧ ω (cid:48) ) = CT ( I , t br,g ◦ i t (cid:48) br , ω ∧ ω (cid:48) ) where I stands for the product I × · · · × I b , and the arrow corresponds to the action-compositionmap CT ( t br,g , ω ) × T ( t (cid:48) br , ω (cid:48) ) −→ CT ( t br,g ◦ i t (cid:48) br , ω ∧ ω (cid:48) ) , of the operadic bimodule {CT m } m (cid:62) . Define also the maps M : T ( t br , ω ) × CT ( I , t br,g , ω ) × · · · × CT ( I s , t sbr,g , ω s ) −→ I × · · · × I s × T ( t br , ω ) × CT ( t br,g , ω ) × · · · × CT ( t sbr,g , ω s ) −→ CT ( I ∪ · · · ∪ I s , µ ( t br , t br,g . . . , t sbr,g ) , ω ∧ ω ∧ · · · ∧ ω s ) where the second arrow corresponds to the action-composition map T ( t br , ω ) × CT ( t br,g , ω ) × · · · × CT ( t sbr,g , ω s ) −→ CT ( µ ( t br , t br,g . . . , t sbr,g ) , ω ∧ ω ∧ · · · ∧ ω s ) . The maps O i have sign +1 . The maps M have sign ( − † , where † is defined as follows. Writing g i for the number of non-trivial gauges and j i for the number of gauge-vertex intersections of t ibr,g , i = 1 , . . . , s , and setting t br := t br and g = j = dim( I ) = 0 , † := s (cid:88) i =1 | I i | (cid:32) | t br | + i − (cid:88) l =1 | t lbr,g | (cid:33) + s (cid:88) i =1 g i (cid:32) | t br | + i − (cid:88) l =1 | t lbr | (cid:33) + s (cid:88) i =1 j i (cid:32) | t br | + i − (cid:88) l =1 | t lbr,g | (cid:33) . Definition 19 (Action-composition maps) . The action of the operad Ω BAs on n − Ω BAs − Morph is defined as ( I , t br,g , ω ) ◦ i ( t (cid:48) br , ω (cid:48) ) = ( I , t br,g ◦ i t (cid:48) br , ω ∧ ω (cid:48) ) ,µ (( t br , ω ) , ( I , t br,g , ω ) , . . . , ( I s , t sbr,g , ω s )) = ( − † ( I ∪ · · · ∪ I s , µ ( t br , t br,g . . . , t sbr,g ) , ω ∧ ω ∧ · · · ∧ ω s ) . Using for instance the maps O i and M , and remembering the Koszul sign rules, we can checkthat these action-composition maps satisfy indeed all the associativity conditions for an operadicbimodule. What’s more, choosing a distinguished orientation for every gauged stable ribbon tree t g ∈ sCRT , this definition of the operadic bimodule n − Ω BAs − Morph amounts to defining it asthe free operadic bimodule in graded Z -modules n − Ω BAs − Morph = F Ω BAs, Ω BAs ( I , I , I , I , · · · , ( I, sCRT m ) , · · · ; I ⊂ ∆ n ) . It remains to define a differential on the generating operations ( I, t g , ω ) to recover definition 13.5.4.5. The boundary of the compactified moduli spaces CT m ( t g ) . Before defining the differential onthe operadic bimodule n − Ω BAs − Morph , we recall the signs for the top dimensional strata in theboundary of the compactified moduli spaces CT m ( t g ) that were computed in section I.5.2 in [Maz21]. We fix for the rest of this subsection a gauged stable ribbon tree t g whose gauge intersects j of itsvertices. We also choose an orientation e ∧ · · · ∧ e i on t g and order the j gauge-vertex intersectionsfrom left to right v v j . The (int-collapse) boundary corresponds to the collapsing of an internal edge that does not inter-sect the gauge of the tree t . Suppose that it is the p -th edge e p of t which collapses. Write moreover ( t/e p ) g for the resulting gauged tree and ω p := e ∧ · · · ∧ (cid:98) e p ∧ · · · ∧ e i for the induced orientation onthe edges of t/e p . The boundary component CT m (( t/e p ) g , ω p ) bears a sign ( − p +1+ j ( int-collapse )in the boundary of CT m ( t g , ω ) .The (gauge-vertex) boundary corresponds to the gauge crossing exactly one additional vertex of t .We suppose that this intersection takes place between the k -th and ( k + 1) -th intersections of t g andwrite t g for the resulting gauged tree. If the crossing results from a move , the boundary component CT m ( t g , ω ) has sign ( − j + k ( gauge-vertex A )in the boundary of CT m ( t g , ω ) . If the crossing results from a move , the boundary component CT m ( t g , ω ) has sign ( − j + k +1 ( gauge-vertex B )in the boundary of CT m ( t g , ω ) .The (above-break) boundary corresponds either to the breaking of an internal edge of t , that islocated above the gauge or intersects the gauge, or, when the gauge is below the root, to the outgoingedge breaking between the gauge and the root. Denote e the outgoing edge of t . Suppose that itis the p -th edge e p of t which breaks and write moreover ( t p ) g for the resulting broken gauged tree.The boundary component CT m (( t p ) g , ω p ) bears a sign ( − p + j ( above-break )in the boundary of CT m ( t g , ω ) .The (below-break) boundary corresponds to the breaking of edges of t that are located below thegauge or intersect it, such that there is exactly one edge breaking in each non-self crossing pathfrom an incoming edge to the root. Write ( t br ) g for the resulting broken gauged tree. We order fromleft to right the s non-trivial unbroken gauged trees t g , . . . , t sg of ( t br ) g and denote e j , . . . , e j s theinternal edges of t whose breaking produces the trees t g , . . . , t sg . Beware that we do not necessarilyhave that j < · · · < j s . To this extent, we denote ε ( j , . . . , j s ; ω ) the sign obtained after modifying IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 45 ω by moving e j k to the k -th spot in ω . We write ω br for the induced orientation on ( t br ) g , which isobtained by deleting the edges e j k in ω . The boundary component CT m (( t br ) g , ω br ) has sign ( − ε ( j ,...,j s ; ω )+1+ j ( below-break )in the boundary of CT m ( t g , ω ) .5.4.6. Definition of the differential. Definition 20 (Differential) . The differential of a generating operation ( I, t g , ω ) is defined by readingthe signs of the top dimensional strata in the boundary of the space I × CT m ( t g , ω ) , endowed withits dim( I ) − Ω BAs cell decomposition. It reads as ∂ ( I, t g , ω ) := dim( I ) (cid:88) l =0 ( − l ( ∂ singl I, t g , ω ) + ( − | I | (cid:88) ( − † Ω BAs ( I, int − collapse( t g , ω ))+ ( − | I | (cid:88) ( − † Ω BAs ( I, gauge − vertex( t g , ω )) + ( − | I | (cid:88) ( − † Ω BAs ( I, above − break( t g , ω ))+ ( − | I | (cid:88) I ∪···∪ I b = I ( − † Ω BAs (( I , . . . , I b ) , below − break( t g , ω )) , where b denotes the number of gauges of below − break( t g ) and the signs ( − † Ω BAs denote the Ω BAs − Morph signs listed in the previous subsection.For instance, choosing the orientation e ∧ e on e e , the signs in the computation of subsection 4.1.2 are ∂ (cid:18) [0 < < , e ∧ e (cid:19) = (cid:18) [1 < , e ∧ e (cid:19) − (cid:18) [0 < , e ∧ e (cid:19) + (cid:18) [0 < , e ∧ e (cid:19) − (cid:18) [0 < < , e ∧ e (cid:19) − (cid:18) [0 < < , e ∧ e (cid:19) + (cid:18) [0 < < , e ∧ e (cid:19) − (cid:16) [0] [0 < < , ∅ (cid:17) − (cid:16) [0 < 1] [1 < , ∅ (cid:17) − (cid:16) [0 < < 2] [2] , ∅ (cid:17) + (cid:18) [0 < < , e (cid:19) − (cid:18) [0 < < , e (cid:19) . This concludes the construction of the operadic bimodule n − Ω BAs − Morph .5.4.7. The morphism of operadic bimodules n − A ∞ − Morph → n − Ω BAs − Morph . To conclude,it remains to define the morphism of operadic bimodules n − A ∞ − Morph → n − Ω BAs − Morph .It is enough to define this morphism on the generating operations of n − A ∞ − Morph and to checkthat it is compatible with the differentials. Proposition 8. The map n − A ∞ − Morph → n − Ω BAs − Morph defined on the generating operationsof n − A ∞ − Morph as f I,m (cid:55)−→ (cid:88) t g ∈ CBRT m ( I, t g , ω can ) is a morphism of ( A ∞ , A ∞ ) -operadic bimodules. We refer to section I.5.3 of [Maz21] for the definition of the canonical orientations ω can . It iseasy to check that this map is indeed compatible with the differentials : either making explicitsigns computations, or noting that this morphism corresponds to the refinement of the n − A ∞ -celldecomposition of n − J m to its n − Ω BAs -cell decomposition. IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 47 Part 2 Geometry n -morphisms in Morse theory Let M be a closed oriented Riemannian manifold endowed with a Morse function f together with aMorse-Smale metric. In [Maz21], we explored how to realize the moduli spaces of stable metric ribbontrees T m and the moduli spaces of stable two-colored metric ribbon trees CT m in Morse theory. It wasproven that, upon choosing admissible perturbation data X f on the moduli spaces T m for the function f , the Morse cochains C ∗ ( f ) can be endowed with an Ω BAs -algebra structure whose operations m t for t ∈ sRT m are defined by counting 0-dimensional moduli spaces T X f t ( y ; x , . . . , x m ) . Similarly,choose an additional Morse function g together with admissible perturbation data X g on the modulispaces T m , and admissible perturbation data Y on the moduli spaces CT m which are compatiblewith X f and X g . We can then define an Ω BAs -morphism µ Y : ( C ∗ ( f ) , m X f t ) → ( C ∗ ( g ) , m X g t ) ,whose operations µ t g for t g ∈ sCRT m are defined by counting the 0-dimensional moduli spaces CT Y t g ( y ; x , . . . , x m ) .The goal of this section is to realize the n -multiplihedra n − J m endowed with their n − Ω BAs -celldecomposition in Morse theory. We first introduce the notion of n -simplices of perturbation dataon the moduli spaces CT m (definitions 22 and 23), generalizing the notion of perturbation data onthese moduli spaces defined in [Maz21]. We then use n -simplices of perturbation data to definethe moduli spaces CT I,t g ( y ; x , . . . , x m ) , I ⊂ ∆ n . Under generic assumptions on the simplices ofperturbation data, these moduli spaces are orientable manifolds (proposition 10). Requiring someadditional compatibilities involving the maps AW ◦ s on the simplices of perturbation data, the 1-dimensional moduli spaces CT I,t g ( y ; x , . . . , x m ) can be compactified to 1-dimensional manifolds withboundary, whose boundary is modeled on the boundary of ∆ n ×CT m endowed with its n − Ω BAs -celldecomposition (theorems 3 and 4). We construct as a result a n − Ω BAs -morphism between theMorse cochains C ∗ ( f ) and C ∗ ( g ) (theorem 5), by counting the signed points of the 0-dimensionaloriented manifolds CT I,t g ( y ; x , . . . , x m ) . We finally prove a filling theorem for perturbation dataparametrized by a simplicial subcomplex S ⊂ ∆ n (theorem 6), solving as a corollary the questionthat initially motivated this paper (corollary 1).1.1. Conventions. We will study Morse theory of the Morse function f : M → R using its negativegradient vector field −∇ f . Denote d the dimension of the manifold M and φ s the flow of −∇ f . Fora critical point x define its unstable and stable manifolds W U ( x ) := { z ∈ M, lim s →−∞ φ s ( z ) = x } W S ( x ) := { z ∈ M, lim s → + ∞ φ s ( z ) = x } . Their dimensions are such that dim( W U ( x )) + dim( W S ( x )) = d . We then define the degree of acritical point x to be | x | := dim( W S ( x )) . This degree is often referred to as the coindex of x in thelitterature. We will moreover work with Morse cochains. For two critical point x (cid:54) = y , define T ( y ; x ) := W S ( y ) ∩ W U ( x ) / R to be the moduli space of negative gradient trajectories connecting x to y . Denote moreover T ( x ; x ) = ∅ . Under the Morse-Smale assumption on f and the Riemannian metric on M , for x (cid:54) = y the moduli space T ( y ; x ) has dimension dim ( T ( y ; x )) = | y | − | x | − . The Morse differential ∂ Morse : C ∗ ( f ) → C ∗ ( f ) is then defined to count descending negative gradient trajectories ∂ Morse ( x ) := (cid:88) | y | = | x | +1 T ( y ; x ) · y . n -simplices of perturbation data on a stratum CT m ( t g ) . Fix a gauged stable metricribbon tree T g = ( t g , λ, { l e } e ∈ E ( t ) ) . Let T c = ( t c , L f c ) be its associated two-colored metric ribbontree, E ( t c ) the set of all edges of t c and E ( t c ) ⊂ E ( t c ) the set of internal edges of t c . We pointout that L f c is a linear combination of the parameters λ, { l e } e ∈ E ( t ) and that we should in fact write L f c ( λ, { l e } e ∈ E ( t ) ) . Recall from [Maz21] that : Definition 21 ([Maz21]) . A choice of perturbation data on T g consists of the following data :(i) a vector field [0 , L f c ] × M −→ X fc T M , that vanishes on [1 , L f c − , for every internal edge f c of t c ;(ii) a vector field [0 , + ∞ [ × M −→ X f T M , that vanishes away from [0 , , for the outgoing edge f of t c ;(iii) a vector field ] − ∞ , × M −→ X fi T M , that vanishes away from [ − , , for every incoming edge f i (1 (cid:54) i (cid:54) n ) of t c .In the rest of the paper, we will moreover write D f c for all segments [0 , L f c ] , as well as for allsemi-infinite segments ] − ∞ , and [0 , + ∞ [ . Definition 22. A n -simplex of perturbation data for T g is defined to be a choice of perturbationdata Y δ,T g for every δ ∈ ˚∆ n . Equivalently, it is the datum of a vector field ˚∆ n × D f c × M −→ Y ∆ n,Tg,fc T M for every edge f c ∈ E ( t c ) , abiding by the previous vanishing conditions on D f c . We will denote itas Y ∆ n ,T g := { Y δ,T g } δ ∈ ˚∆ n .Introduce the cone C f c ⊂ CT m ( t g ) × R defined as(i) { (( λ, { l e } e ∈ E ( t ) ) , s ) such that ( λ, { l e } e ∈ E ( t ) ) ∈ CT m ( t g ) and (cid:54) s (cid:54) L f c } if f c is an internaledge ;(ii) { (( λ, { l e } e ∈ E ( t ) ) , s ) such that ( λ, { l e } e ∈ E ( t ) ) ∈ CT m ( t g ) and s (cid:54) } if f c is an incoming edge ;(iii) { (( λ, { l e } e ∈ E ( t ) ) , s ) such that ( λ, { l e } e ∈ E ( t ) ) ∈ CT m ( t g ) and s (cid:62) } if f c is the outgoing edge. IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 49 Definition 23. A n -simplex of perturbation data on CT m ( t g ) , or choice of perturbation data on CT m ( t g ) parametrized by ∆ n , is defined to be the data of a n -simplex of perturbation data Y ∆ n ,T g for every T g ∈ CT m ( t g ) . A n -simplex of perturbation data Y ∆ n ,t g defines maps Y ∆ n ,t g ,f c : ˚∆ n × D f c × M −→ T M , for every edge f c of t c . It is said to be smooth if all these maps are smooth.1.3. The moduli spaces CT I,t g ( y ; x , . . . , x m ) . Recall from [Maz21] that given an admissible choiceof perturbation data Y on the moduli spaces CT m , the moduli spaces CT Y t g ( y ; x , . . . , x m ) are definedas the inverse image of the thin diagonal ∆ ⊂ M × m +1 under the flow map φ Y tg : CT m ( t g ) × W S ( y ) × W U ( x ) × · · · × W U ( x m ) −→ M × m +1 . Definition 24. Let Y ∆ n ,t g be a smooth n -simplex of perturbation data on CT m ( t g ) . Given y ∈ Crit( g ) and x , . . . , x m ∈ Crit( f ) , we define the moduli spaces CT Y ∆ n,tg ∆ n ,t g ( y ; x , . . . , x m ) := (cid:91) δ ∈ ˚∆ n CT Y δ,tg t g ( y ; x , . . . , x m )= (cid:26) ( δ , two-colored perturbed Morse gradient tree associated to ( T g , Y δ,T g ) which connects x , . . . , x m to y ), for T g ∈ CT m ( t g ) and δ ∈ ˚∆ n (cid:27) . x x x x y Figure 12. An example of a perturbed two-colored Morse gradient tree associatedto the perturbation data Y δ for a δ ∈ ˚∆ n . The black segments above the gaugecorrespond to −∇ f and the green ones to −∇ f + Y δ . As for the segments below thegauge, replace f by g in these formulae.An example of a perturbed two-colored Morse gradient tree associated to the perturbation data Y δ for a δ ∈ ˚∆ n is represented on figure 12. Introduce the flow map φ Y ∆ n,tg : ˚∆ n × CT m ( t g ) × W S ( y ) × W U ( x ) × · · · × W U ( x m ) −→ M × m +1 , whose restriction to every δ ∈ ˚∆ n is φ Y δ,tg : CT m ( t g ) × W S ( y ) × W U ( x ) × · · · × W U ( x m ) −→ M × m +1 . Proposition 10. (i) The moduli space CT Y ∆ n,tg ∆ n ,t g ( y ; x , . . . , x m ) can be rewritten as CT Y ∆ n,tg ∆ n ,t g ( y ; x , . . . , x m ) = φ − Y ∆ n,tg (∆) , where ∆ ⊂ M × m +1 is the thin diagonal of M × m +1 . (ii) Given a n -simplex of perturbation data Y ∆ n ,t g making φ Y ∆ n,tg transverse to ∆ , the modulispace CT ∆ n ,t g ( y ; x , . . . , x m ) is an orientable manifold of dimension dim (cid:0) CT ∆ n ,t g ( y ; x , . . . , x m ) (cid:1) = −| t ∆ n ,g | + | y | − m (cid:88) i =1 | x i | . (iii) n -simplices of perturbation data Y ∆ n ,t g such that φ Y ∆ n,tg is transverse to ∆ exist. Replacing ∆ n by any face I ⊂ ∆ n , the moduli spaces CT Y I,tg I,t g ( y ; x , . . . , x m ) can be defined in thesame way and made into orientable manifolds of dimension dim (cid:0) CT I,t g ( y ; x , . . . , x m ) (cid:1) = −| t I,g | + | y | − m (cid:88) i =1 | x i | . We refer to section 2 for the details on transversality and orientability.1.4. Compactifications. The compactified moduli spaces CT I,t g ( y ; x , . . . , x m ) . We now would like to compactify the1-dimensional moduli spaces CT I,t g ( y ; x , . . . , x m ) to 1-dimensional manifolds with boundary. Theyare defined as the inverse image in ˚ I ×CT m ( t g ) × W S ( y ) × W U ( x ) ×· · ·× W U ( x m ) of the thin diagonal ∆ ⊂ M × m +1 under the flow map φ Y I,tg . The boundary components in the compactification shouldcome from those of W S ( y ) , of the W U ( x i ) and of ˚ I × CT m ( t g ) . However, rather than consideringthe boundary components coming from the separate compactifications of ˚ I and CT m ( t g ) , we willconsider the n − Ω BAs -decomposition of I ×CT m ( t g ) and model the remaining boundary componentson this decomposition.Choose admissible perturbation data X f and X g for the functions f and g . Choose moreoversmooth simplices of perturbation data Y I,t g for all t g ∈ sCRT i , (cid:54) i (cid:54) m and I ⊂ ∆ n . We denote ( Y I,m ) I ⊂ ∆ n := ( Y I,t g ) t g ∈ sCRT m I ⊂ ∆ n , and call it a choice of perturbation data on CT m parametrizedby ∆ n . Fixing a two-colored stable ribbon tree type t g ∈ sCRT m and I ⊂ ∆ n we want to compactifythe moduli space CT Y I,tg I,t g ( y ; x , . . . , x m ) using the perturbation data X f , X g and ( Y I,k ) k (cid:54) mI ⊂ ∆ n . Theboundary will be described by the following phenomena :(i) the parameter δ ∈ I tends towards the codimension 1 boudary of I ( ∂ sing I ) ;(ii) an external edge breaks at a critical point (Morse) ;(iii) an internal edge of the tree t collapses (int-collapse) : CT Y I,t (cid:48) g I,t (cid:48) g ( y ; x , . . . , x m ) where t (cid:48) g ∈ sCRT n are all the two-colored trees obtained by collapsing exactly one internaledge, which does not cross the gauge ;(iv) the gauge moves to cross exactly one additional vertex of the underlying stable ribbon tree(gauge-vertex) : CT Y I,t (cid:48) g I,t (cid:48) g ( y ; x , . . . , x m ) where t (cid:48) g ∈ sCRT n are all the two-colored trees obtained by moving the gauge to cross exactlyone additional vertex of t ; IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 51 (v) an internal edge located above the gauge or intersecting it breaks or, when the gauge is belowthe root, the outgoing edge breaks between the gauge and the root (above-break) : CT Y I,t g I,t g ( y ; x , . . . , x i , z, x i + i +1 , . . . , x m ) × T X ft t ( z ; x i +1 , . . . , x i + i ) , where the tree resulting from grafting the outgoing edge of t to the i + 1 -th incoming edgeof t g is t g ;(vi) edges (internal or incoming) that are possibly intersecting the gauge, break below it, suchthat there is exactly one edge breaking in each non-self crossing path from an incoming edgeto the root (below-break) ; the simplex of perturbation data Y I,t g then "breaks" accordingto the combinatorics of the Alexander-Whitney coproduct : T X gt t ( y ; y , . . . , y s ) × CT Y I ,t g I ,t g ( y ; x , . . . ) × · · · × CT Y Is,tsg I s ,t sg ( y s ; . . . , x m ) where the tree resulting from grafting for each r the outgoing edge of t rg to the r -th incomingedge of t is t g , and I ∪ · · · ∪ I s = I is an overlapping partition of I .Note that the (Morse) boundaries are a simple consequence of the fact that external edges are Morsetrajectories away from a length 1 segment.1.4.2. Smooth choice of perturbation data Y I ⊂ ∆ n ,m . We begin by tackling the conditions comingwith the ( ∂ sing I ), (int-collapse) and (gauge-vertex) boundaries. Let t g ∈ sCRT m and denote coll ∪ g − v ( t g ) ⊂ sCRT m the set consisting of all stable gauged trees obtained by collapsing internal edgesof t and/or moving the gauge to cross additional vertices of t . In particular, t g ∈ coll ∪ g − v ( t g ) .We define CT m ( t g ) := (cid:91) t (cid:48) g ∈ coll ∪ g − v ( t g ) CT m ( t (cid:48) g ) for the stratum CT m ( t g ) ⊂ CT m together with its inner boundary components. A choice of pertur-bation data ( Y I,t (cid:48) g ) t (cid:48) g ∈ coll ∪ g − v ( t g ) for a fixed I ⊂ ∆ n corresponds to a dim( I ) -simplex of perturbationdata on CT m ( t g ) . Following section 1.2, such a choice of perturbation data is equivalent to a map ˜ Y I,t g ,f c : ˚ I × ˜ C f c × M −→ T M , for every edge f c of t c , where ˜ C f c ⊂ CT m ( t g ) × R is defined in a similar fashion to C f c . Definition 25. A choice of perturbation data ( Y I,m ) I ⊂ ∆ n is said to be smooth if all maps ˜ Y ∆ n ,t g ,f c : ∆ n × ˜ C f c × M −→ T M , are smooth, where we extended ˚∆ n to ∆ n by defining ˜ Y ∆ n ,t g ,f c := ˜ Y I,t g ,f c on a face I ⊂ ∆ n .1.4.3. The (above-break) boundary. The (above-break) conditions are tackled as in [Maz21]. Write t c for the two-colored ribbon tree associated to t g . The (above-break) boundary corresponds to thebreaking of an internal edge f c of t c located above the set of colored vertices. Denote t c and t thetrees obtained by breaking t c at the edge f c , where t is seen to lie above t c . We have to specify foreach edge e c ∈ E ( t c ) and each δ ∈ ˚ I , what happens to the perturbation Y δ,t c ,e c at the limit.(i) For e c ∈ E ( t ) and (cid:54) = f c , we require that lim Y δ,t c ,e c = X ft ,e c . (ii) For e c ∈ E ( t c ) and (cid:54) = f c , we require that lim Y δ,t c ,e c = Y δ,t c ,e c . (iii) For f c = e c , Y δ,t c ,f c yields two parts at the limit : the part corresponding to the outgoingedge of t and the part corresponding to the incoming edge of t c . We then require that theycoincide respectively with the perturbation X ft and Y δ,t c .An example of each case is illustrated in figure 13. t c Y δ,t c ,e c lim Y δ,t c ,e c = X ft ,e c t c t (above-break) case (i) t c Y δ,t c ,e c lim Y δ,t c ,e c = Y δ,t c ,e c t c t (above-break) case (ii) t c Y δ,t c ,e c lim t c Y δ,t c ,e c = Y δ,t c ,e c lim t Y δ,t c ,e c = X ft ,e c t c t (above-break) case (iii) Figure 13 The (below-break) boundary. Denote t c , . . . , t sc and t the trees obtained by breaking t c belowthe gauge, where t c , . . . , t sc are seen to lie above t . Going back to subsection 3.2.4, consider the map AW ◦ ( s − I : I → I s . It comes with s maps pr r ◦ AW ◦ ( s − I : I → I for (cid:54) r (cid:54) s corresponding to theprojection on the r -th factor of I s . We will simply denote them pr r .We have to specify for each edge e c ∈ E ( t c ) and each δ ∈ ˚ I , what happens to the perturbation Y δ,t c ,e c at the limit. The maps pr r will allow us to produce the overlapping partitions combinatoricson the parameter δ .(i) For e c ∈ E ( t rc ) and not among the breaking edges, we require that lim Y δ,t c ,e c = Y pr r ( δ ) ,t rc ,e c . IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 53 (ii) For e c ∈ E ( t ) and not among the breaking edges, we require that lim Y δ,t c ,e c = X gt ,e c . (iii) For f c among the breaking edges, Y δ,t c ,f c yields two parts at the limit : the part correspondingto the outgoing edge of t rc and the part corresponding to the incoming edge of t . We thenrequire that they coincide respectively with the perturbations Y pr r ( δ ) ,t rc ,e c and X gt ,e c .This is again illustrated in figure 14. t c Y δ,t c ,e c lim Y δ,t c ,e c = Y pr ( δ ) ,t c ,e c t t c t c (below-break) case (i) t c Y δ,t c ,e c lim Y δ,t c ,e c = X gt ,e c t t c t c (below-break) case (ii) t c Y δ,t c ,e c lim t Y δ,t c ,e c = X gt ,e c lim t c Y δ,t c ,e c = Y pr ( δ ) ,t c ,e c t t c t c (below-break) case (iii) Figure 14 Admissible n -simplices of perturbation data. Definition 26. A smooth choice of perturbation data ( Y I,m ) m (cid:62) I ⊂ ∆ n is said to be gluing-compatiblew.r.t. X f and X g if it satisfies the (above-break) and (below-break) conditions described in sub-sections 1.4.3 and 1.4.4. Smooth and gluing-compatible perturbation data ( Y I,m ) m (cid:62) I ⊂ ∆ n such thatall maps φ Y I,tg are transverse to the diagonal ∆ are called admissible w.r.t. X f and X g or simply admissible . Theorem 3. Admissible choices of perturbation data ( Y I,m ) m (cid:62) I ⊂ ∆ n exist. Theorem 4. Let ( Y I,m ) m (cid:62) I ⊂ ∆ n be an admissible choice of perturbation data. The 0-dimensional modulispaces CT I,t g ( y ; x , . . . , x m ) are compact. The 1-dimensional moduli spaces CT I,t g ( y ; x , . . . , x m ) can be compactified to 1-dimensional manifolds with boundary CT I,t g ( y ; x , . . . , x m ) , whose boundary isdescribed in subsection 1.4.1. The proof of theorem 3 is postponed to subsection 2.1.1 and will proceed as in [Maz21]. Theorem 4is a direct consequence of the analysis carried out in chapter 6 of [Mes18]. For this reason, we willnot give details of its proof. We only point out that all spaces T X gt t ( y ; y , . . . , y s ) × CT Y I ,t g I ,t g ( y ; x , . . . ) × · · · × CT Y Is,tsg I s ,t sg ( y s ; . . . , x m ) where I ∪ · · · ∪ I s = I is an i -overlapping s -partition of I , could a priori appear in the boudaryof CT I,t g ( y ; x , . . . , x m ) . The assumption that our choice of perturbation data is admissible ensureshowever in particular that whenever I ∪ · · · ∪ I s = I is not an ( s − -overlapping s -partition of I the previous space is empty, as at least one of its factors then has negative dimension.Theorem 4 implies moreover the existence of gluing maps above − breakT ,MorseI,g ,T ,Morse : [ R, + ∞ ] −→ CT I,t g ( y ; x , . . . , x n ) , below − breakT ,Morse ,T ,MorseI ,g ,...,T s,MorseIs,g : [ R, + ∞ ] −→ CT I,t g ( y ; x , . . . , x n ) , whenever the perturbed Morse trees T ,MorseI,g , T ,Morse and T ,Morse , T ,MorseI ,g , . . . , T s,MorseI s ,g respec-tively lie in a 0-dimensional moduli space, and where notations are as in items (v) and (vi) of subsec-tion 1.4.1. The constructions of explicit gluing maps in subsections II.4.4.3 and II.4.5.4 of [Maz21] inthe case of the moduli spaces CT t g ( y ; x , . . . , x n ) can be adapted without problems to the presentsetting.1.5. n − Ω BAs -morphisms between Morse cochains. Let X f and X g be admissible choices ofperturbation data for the Morse functions f and g . Denote ( C ∗ ( f ) , m X f t ) and ( C ∗ ( g ) , m X g t ) theMorse cochains endowed with their Ω BAs -algebra structures constructed in [Maz21]. Theorem 5. Let ( Y I,m ) m (cid:62) I ⊂ ∆ n be a choice of perturbation that is admissible w.r.t. X f and X g .Defining for every m and t g ∈ sCRT m , and every I ⊂ ∆ n the operations µ I,t g as µ I,t g : C ∗ ( f ) ⊗ · · · ⊗ C ∗ ( f ) −→ C ∗ ( g ) x ⊗ · · · ⊗ x m (cid:55)−→ (cid:88) | y | = (cid:80) mi =1 | x i | + | t I,g | CT Y I,tg I,t g ( y ; x , · · · , x m ) · y , they fit into a n − Ω BAs -morphism ( C ∗ ( f ) , m X f t ) → ( C ∗ ( g ) , m X g t ) The proof is postponed to section 2.4. It boils down to counting the boundary points of the 1-dimensional oriented compactified moduli spaces CT Y I,t g ( y ; x , · · · , x m ) whose boundary is describedin the subsection 1.4.1. As a matter of fact, the set of operations { µ I,t g } does not exactly define a n − Ω BAs -morphism. One of the two differentials ∂ Morse in the bracket [ ∂ Morse , µ I,t g ] appearing inthe n − Ω BAs -equations has to be twisted by a specific sign for the n − Ω BAs -equations to hold.We will speak about a twisted n − Ω BAs -morphism between twisted Ω BAs -algebras. In the casewhere M is odd-dimensional, this twisted n − Ω BAs -morphism is a standard n − Ω BAs -morphism.As explained in subsection 4.1.3 of part 1, if we want moreover to go back to the algebraicframework of A ∞ -algebras, a n − A ∞ -morphism between the induced A ∞ -algebra structures on the IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 55 Morse cochains can simply be obtained under the morphism of operadic bimodules n − A ∞ − Morph → n − Ω BAs − Morph .1.6. Filling inner horns in Morse theory. Consider an inner horn Λ kn ⊂ ∆ n . Definitions 23and 26 can be straightforwardly extended to define an admissible ( k, n ) -inner horn of perturbationdata ( Y I,m ) m (cid:62) I ⊂ Λ kn on the moduli spaces CT m . Shifting to the framework of simplicial sets, while anadmissible n -simplex of perturbation data Y ∆ n yields a morphism of simplicial sets µ Y ∆ n : ∆ n −→ HOM Ω BAs ( C ∗ ( f ) , C ∗ ( g )) • , an admissible ( k, n ) -inner horn of perturbation data Y Λ kn will yield a morphism of simplicial sets µ Y Λ kn : Λ kn −→ HOM Ω BAs ( C ∗ ( f ) , C ∗ ( g )) • , or equivalently a collection of operations µ I,t g : C ∗ ( f ) → C ∗ ( g ) for I ⊂ Λ kn , which satisfy the Ω BAs -equations.Following proposition 8, the morphism of simplicial sets µ Y Λ kn induces a morphism of simplicialsets µ Y Λ kn A ∞ : Λ kn −→ HOM A ∞ ( C ∗ ( f ) , C ∗ ( g )) • . As the simplicial set HOM A ∞ ( C ∗ ( f ) , C ∗ ( g )) • is an ∞ -category, the diagram Λ kn HOM A ∞ ( C ∗ ( f ) , C ∗ ( g )) • ∆ n µ Y Λ knA ∞ µ , can always be filled algebraically , to obtain a n − A ∞ -morphism µ extending µ Y Λ kn A ∞ . This diagramcan in fact be filled geometrically : Proposition 11. For every admissible ( k, n ) -inner horn of perturbation data Y Λ kn , there exists anadmissible n -simplex of perturbation data Y ∆ n extending Y Λ kn . A choice of filler Y ∆ n then defines a n − Ω BAs -morphism µ Y ∆ n filling the diagram Λ kn HOM Ω BAs ( C ∗ ( f ) , C ∗ ( g )) • ∆ n µ Y Λ kn µ Y ∆ n . As a matter of fact, proposition 11 is a corollary to the following theorem, proven in section 2.1 : Theorem 6. For every admissible choice of perturbation data Y S parametrized by a simplicial sub-complex S ⊂ ∆ n , there exists an admissible n -simplex of perturbation data Y ∆ n extending Y S . We restricted ourselves to ( k, n ) -inner horns of perturbation data at first, in order to illustratea geometric instance of the fact that the HOM-simplicial set HOM A ∞ ( C ∗ ( f ) , C ∗ ( g )) • is an ∞ -category. We moreover point out that the morphisms µ Y ∆ n are in fact twisted n − Ω BAs -morphisms,as explained in section 2.4. However, the constructions of this section still hold in that context. Corollary 1. Let Y and Y (cid:48) be two admissible choices of perturbation data on the moduli spaces CT m . The Ω BAs -morphisms µ Y and µ Y (cid:48) are then Ω BAs -homotopic C ∗ ( f ) C ∗ ( g ) µ Y µ Y (cid:48) . Proof. Indeed, these two choices of perturbation data correspond to a choice of perturbation dataparametrized by the simplicial subcomplex of ∆ consisting of its two vertices. This simplicialsubcomplex can then be filled thanks to theorem 6 and yields as a consequence an Ω BAs -homotopybetween the Ω BAs -morphisms µ Y and µ Y (cid:48) . (cid:3) Transversality, signs and orientations Proof of theorems 3 and 6. Proof of theorem 3. We detailed in section II.3. of [Maz21] how to build an admissible choiceof perturbation data ( X n ) n (cid:62) on the moduli spaces T m . Drawing from this construction, we providea sketch of the proof of theorem 3 in this subsection : admissible n -simplices of perturbation data ( Y I,m ) m (cid:62) I ⊂ ∆ n on the moduli spaces CT ( t g ) exist. The proof proceeds again by induction on the integer N = dim( CT ( t g )) + dim( I ) .If N = 0 , dim( I ) = 0 and the gauged tree t g is a corolla whose gauge intersects its root. Let y ∈ Crit( g ) and x , · · · , x m ∈ Crit( f ) and fix an integer l such that l (cid:62) max (cid:32) , | y | − m (cid:88) i =1 | x i | + 1 (cid:33) . Define the parametrization space X lt g := { C l -perturbation data Y t g on CT m ( t g ) } , and introduce the C l -map φ t g : X lt g × CT m ( t g ) × W S ( y ) × W U ( x ) × · · · × W U ( x m ) −→ M × m +1 , such that for every Y t g ∈ X lt g , φ t g ( Y t g , · ) = φ Y tg . Note that we should in fact write φ y,x ,...,x n t g as thedomain of φ t g depends on y, x , . . . , x n . The space X lt g is then a Banach space and the map φ t g is asubmersion. The map φ t g is in particular transverse to the diagonal ∆ ⊂ M × m +1 . The parametrictransversality lemma implies that there exists a subset Y l ; y,x ,...x m t g ⊂ X lt g which is residual in thesense of Baire, and such that for every choice of perturbation data Y t g ∈ Y l ; y,x ,...x m t the map φ Y tg IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 57 is transverse to the diagonal ∆ ⊂ M × m +1 . Any Y t g in the intersection Y lt g := (cid:92) y,x ,...,x m Y l ; y,x ,...x m t g ⊂ X t g then yields a C l -choice of perturbation data on CT ( t g ) such that all maps φ Y tg are transverse tothe diagonal ∆ ⊂ M × m +1 . Using an argument à la Taubes we prove that one can in fact constructa residual set Y t g ⊂ X t g , where X t g is the Fréchet space defined by replacing " C l " by "smooth" inthe definition of X lt g , and such that any Y t g ∈ Y t g yields a smooth choice of perturbation data suchthat all maps φ Y tg are transverse to the diagonal ∆ ⊂ M × m +1 . See subsection II.3.2.2 of [Maz21]for more details on that last point. This wraps up the first step of the induction.Let N (cid:62) and suppose that we have constructed an admissible choice of perturbation data ( Y I,t g ) , where I ⊂ ∆ n and t g ∈ sCRT m are such that dim( CT ( t g )) + dim( I ) (cid:54) N . Let I ⊂ ∆ n and t g ∈ sCRT m be such that dim( CT ( t g )) + dim( I ) = N + 1 . Let y ∈ Crit( g ) and x , · · · , x m ∈ Crit( f ) and fix an integer l such that l (cid:62) max (cid:32) , | y | − m (cid:88) i =1 | x i | − | t I,g | + 1 (cid:33) . We introduce the parametrization space X lI,t g := dim( I ) -simplices of perturbation data Y I,t g on CT m ( t g ) such that the perturbationdata { Y I,t g } ∪ ( Y J,t (cid:48) g ) t (cid:48) g ∈ coll ∪ g − v ( t g ) J ⊂ I are of class C l in the sense of definition 25,and such that Y I,t g is gluing-compatible w.r.t. the perturbation data ( Y I,t g ) . This parametrization space is a Banach affine space. Define again the C l -map φ I,t g : X lI,t g × ˚ I × CT m ( t g ) × W S ( y ) × W U ( x ) × · · · × W U ( x m ) −→ M × m +1 . The map φ I,t g is then transverse to the diagonal ∆ ⊂ M × m +1 . Applying the parametric transver-sality theorem and proceeding as in the case N = 0 , there exists a residual set Y lI,t g ⊂ X lI,t g such that for every choice of perturbation data Y I,t g ∈ Y lI,t g the map φ Y I,tg is transverse to thediagonal ∆ ⊂ M × m +1 . Resorting again to an argument à la Taubes, we can prove the same state-ment in the smooth context. By definition of the parametrization spaces X I,t g this constructionyields an admissible choice of perturbation data ( Y I,t g ) , where the indices I and t g are such that dim( CT ( t g )) + dim( I ) (cid:54) N + 1 . This concludes the proof of theorem 3 by induction.2.1.2. Proof of theorem 6. The proof of theorem 6 proceeds exactly as the previous proof, by re-placing the requirements in the definition of X lI,t g by the conditions prescribed by the simplicialsubcomplex S ⊂ ∆ n .2.2. Orientation and transversality. Signed short exact sequences. Consider a short exact sequence of vector spaces −→ V −→ W −→ V −→ . It induces a direct sum decomposition W = V ⊕ V . Suppose that the vector spaces W , V and V are oriented. We denote ( − ε the sign obtained by comparing the orientation on W to the oneinduced by the direct sum V ⊕ V . We will then say that the short exact sequence has sign ( − ε .In particular, when ( − ε = 1 , we will say that the short exact sequence is positive .2.2.2. Orientation and transversality. Given now two manifolds M, N , a codimension k submanifold S ⊂ N and a smooth map φ : M −→ N which is tranverse to S , the inverse image φ − ( S ) is a codimension k submanifold of M . Moreover,choosing a complementary ν S to T S , the transversality assumption yields the following short exactsequence of vector bundles −→ T φ − ( S ) −→ T M | φ − ( S ) −→ dφ ν S −→ . Suppose now that M , N and S are oriented. The orientations on N and S induce an orientation on ν S . The submanifold φ − ( S ) is then oriented by requiring that the previous short exact sequencebe positive. We will refer to this choice of orientation as the natural orientation on φ − ( S ) .For instance, the moduli space T X t ( y ; x , . . . , x m ) is defined as the inverse image of the diagonal ∆ ⊂ M × m +1 under the map φ X t : T m ( t ) × W S ( y ) × W U ( x ) × · · · × W U ( x m ) −→ M × m +1 . Orienting the domain and codomain of φ X t by taking the product orientation, and orienting thediagonal ∆ ⊂ M × m +1 as M , defines a natural orientation on T t ( y ; x , . . . , x m ) .2.3. Algebraic preliminaries. Reformulating the n − Ω BAs -equations. We set for the rest of this section an orientation ω foreach t g ∈ sCRT n , which endows each moduli space CT n ( t g ) with an orientation. We write moreover µ I,t g for the operations ( I, t g , ω ) of n − Ω BAs − Morph . The Ω BAs -equations for a n − Ω BAs -morphism then read as [ ∂, µ I,t g ] = dim( I ) (cid:88) l =0 ( − l µ ∂ singl I,t g + ( − | I | (cid:88) t t g ,...,t sg )= t g I ∪···∪ I s = I ( − † Ω BAs m t ◦ ( µ I ,t g ⊗ · · · ⊗ µ I s ,t sg )+ (cid:88) t (cid:48) g ∈ coll ( t g ) ( − † Ω BAs µ I,t (cid:48) g + (cid:88) t (cid:48) g ∈ g − vert ( t g ) ( − † Ω BAs µ I,t (cid:48) g + (cid:88) t g i t = t g ( − † Ω BAs µ I,t g ◦ i m t . The signs ( − † Ω BAs need not be made explicit, but can be computed as in section I.5.2 of [Maz21]. IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 59 Twisted n − A ∞ -morphisms and twisted n − Ω BAs -morphisms. Definition 27. (i) A twisted A ∞ -algebra is a dg- Z -module A endowed with two different dif-ferentials ∂ and ∂ , and a collection of degree − m operations m m : A ⊗ m → A suchthat [ ∂, m m ] = − (cid:88) i + i + i = m (cid:54) i (cid:54) m − ( − i + i i m i +1+ i (id ⊗ i ⊗ m i ⊗ id ⊗ i ) , where [ ∂, · ] denotes the bracket for the maps ( A ⊗ m , ∂ ) → ( A, ∂ ) .(ii) Let ( A, ∂ , ∂ , m m ) and ( B, ∂ , ∂ , m m ) be two twisted A ∞ -algebras. A twisted n − A ∞ -morphism from A to B is defined to be a sequence of degree − m + | I | operations f ( m ) I : A ⊗ m → B such that (cid:104) ∂, f ( m ) I (cid:105) = dim( I ) (cid:88) j =0 ( − j f ( m ) ∂ j I + ( − | I | (cid:88) i + i + i = ni (cid:62) ( − i + i i f ( i +1+ i ) I (id ⊗ i ⊗ m i ⊗ id ⊗ i ) − (cid:88) i + ··· + i s = mI ∪···∪ I s = Is (cid:62) ( − (cid:15) B m s ( f ( i ) I ⊗ · · · ⊗ f ( i s ) I s ) , where [ ∂, · ] denotes the bracket for the maps ( A ⊗ m , ∂ ) → ( B, ∂ ) .(iii) A twisted Ω BAs -algebra and a twisted n − Ω BAs -morphism between twisted Ω BAs -algebrasare defined similarly.The explicit formulae obtained by evaluating the n − Ω BAs -equations of a twisted n − Ω BAs -morphism on A ⊗ m then read as follows : − ∂ µ I,t g ( a , . . . , a m ) + ( − | I | + | t g | + (cid:80) i − j =1 | a j | µ I,t g ( a , . . . , a i − , ∂ a i , a i +1 , . . . , a m )+ (cid:88) t g t = t ( − | I | + † Ω BAs + | t | (cid:80) i j =1 | a j | µ I,t g ( a , . . . , a i , m t ( a i +1 , . . . , a i + i ) , a i + i +1 , . . . , a m )+ (cid:88) t t g ,...,t sg )= t g I ∪···∪ I s = I ( − | I | + † Ω BAs + † Koszul m t ( µ I ,t g ( a , . . . , a i ) , . . . , µ I s ,t sg ( a i + ··· + i s − +1 , . . . , a m ))+ (cid:88) t (cid:48) g ∈ coll ( t g ) ( − | I | + † Ω BAs µ I,t (cid:48) g ( a , . . . , a m ) + (cid:88) t (cid:48) g ∈ g − vert ( t g ) ( − | I | + † Ω BAs µ I,t (cid:48) g ( a , . . . , a m )+ dim( I ) (cid:88) l =0 ( − l µ ∂ singl I,t g ( a , . . . , a m ) = 0 , where † Koszul = s (cid:88) r =1 ( | I r | + | t rg | ) r − (cid:88) t =1 i t (cid:88) j =1 | a i + ··· + a it − + j | . As explained in [Maz21], these definitions cannot be phrased using an operadic viewpoint. However,a twisted n − Ω BAs -morphism between twisted Ω BAs -algebras still always descends to a twisted n − A ∞ -morphism between twisted A ∞ -algebras. Proof of theorem 5. Recollections on twisted Ω BAs -algebra structures on the Morse cochains. We prove in [Maz21]that given a Morse function f and an admissible choice of perturbation data X on the moduli spaces T m , the Morse cochains C ∗ ( f ) can be endowed with a twisted Ω BAs -algebra structure by countingthe 0-dimensional moduli spaces T X t t ( y ; x , . . . , x n ) .We twist to this end the natural orientation on the moduli spaces T X t ( y ; x , . . . , x m ) defined insubsection 2.2.2, by a sign of parity σ ( t ; y ; x , . . . , x m ) := dm (1 + | y | + | t | ) + | t || y | + d m (cid:88) i =1 | x i | ( m − i ) , and the orientation on the moduli spaces T ( y ; x ) by a sign of parity σ ( y ; x ) := 1 , where d denotes the dimension of the manifold M . The moduli spaces T X t ( y ; x , . . . , x m ) and T ( y ; x ) endowed with these new orientations are then respectively written (cid:101) T X t ( y ; x , . . . , x m ) and (cid:101) T ( y ; x ) .The operations m t and the differential on C ∗ ( f ) are then defined as m t ( x , . . . , x m ) = (cid:88) | y | = (cid:80) mi =1 | x i | + | t | (cid:101) T X t ( y ; x , . . . , x m ) · y ,∂ Morse ( x ) = (cid:88) | y | = | x | +1 (cid:101) T ( y ; x ) · y . Counting the signed points in the boundary of the oriented 1-dimensional manifolds (cid:101) T t ( y ; x , . . . , x m ) proves that the operations m t define a twisted Ω BAs -algebra structure on ( C ∗ ( f ) , ∂ T wMorse , ∂ Morse ) ,where ( ∂ T wMorse ) k = ( − ( d +1) k ∂ kMorse . In particular, either working with coefficients in Z / , or with coefficients in Z and an odd-dimensionalmanifold M , the operations m t define an Ω BAs -algebra structure on the Morse cochains.2.4.2. Twisted n − Ω BAs -morphisms between the Morse cochains. Let X f and X g be admissiblechoices of perturbation data for the Morse functions f and g . Denote ( C ∗ ( f ) , m X f t ) and ( C ∗ ( g ) , m X g t ) the Morse cochains endowed with their Ω BAs -algebra structures. Given an admissible n -simplex ofperturbation data ( Y I,m ) m (cid:62) I ⊂ ∆ n , we now construct a twisted n − Ω BAs -morphism µ I,t g : ( C ∗ ( f ) , ∂ T wMorse , ∂ Morse ) −→ ( C ∗ ( g ) , ∂ T wMorse , ∂ Morse ) , I ⊂ ∆ n , t g ∈ sCRT , which completes the proof of theorem 5.The moduli space CT Y I,tg I,t g ( y ; x , . . . , x m ) is defined as the inverse image of the diagonal ∆ ⊂ M × m +1 under the map φ Y I,tg : ˚ I × CT m ( t g ) × W S ( y ) × W U ( x ) × · · · × W U ( x m ) −→ M × m +1 . Orienting the domain and codomain of φ Y I,tg with the product orientation, and orienting the di-agonal ∆ ⊂ M × m +1 as M , defines a natural orientation on CT I,t g ( y ; x , . . . , x m ) as explained insubsection 2.2.2. IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 61 Definition 28. We define (cid:102) CT Y I,tg I,t g ( y ; x , . . . , x m ) to be the oriented manifold CT Y I,tg I,t g ( y ; x , . . . , x m ) whose natural orientation has been twisted by a sign of parity σ ( t I,g ; y ; x , . . . , x m ) := dm (1 + | y | + | t I,g | ) + | t I,g || y | + d m (cid:88) i =1 | x i | ( m − i ) . Proposition 12. If the moduli space (cid:102) CT I,t g ( y ; x , . . . , x m ) is 1-dimensional, its boundary decom-poses as the disjoint union of the following components(i) ( − | y | + | I | + † Ω BAs + | t | (cid:80) i i =1 | x i | (cid:102) CT I,t g ( y ; x , . . . , x i , z, x i + i +1 , . . . , x m ) × (cid:101) T t ( z ; x i +1 , . . . , x i + i ); (ii) ( − | y | + | I | + † Ω BAs + † Koszul (cid:101) T t ( y ; y , . . . , y s ) × (cid:102) CT I ,t g ( y ; x , . . . ) × · · · × (cid:102) CT I s ,t sg ( y s ; . . . , x m ) ;(iii) ( − | y | + | I | + † Ω BAs (cid:102) CT I,t (cid:48) g ( y ; x , . . . , x m ) for t (cid:48) g ∈ coll ( t ) ;(iv) ( − | y | + | I | + † Ω BAs (cid:102) CT I,t (cid:48) g ( y ; x , . . . , x m ) for t (cid:48) g ∈ g − vert ( t ) ;(v) ( − | y | + † Koszul +( m +1) | x i | (cid:102) CT I,t g ( y ; x , . . . , z, . . . , x m ) × (cid:101) T ( z ; x i ) where we have set † Koszul = | I | + | t g | + (cid:80) i − j =1 | x j | ;(vi) ( − | y | +1 (cid:101) T ( y ; z ) × (cid:102) CT I,t g ( z ; x , . . . , x m ) ;(vii) ( − | y | + l (cid:102) CT ∂ singl I,t g ( y ; x , . . . , x m ) . Define the operations µ I,t g : C ∗ ( f ) ⊗ m → C ∗ ( g ) as µ I,t g ( x , . . . , x m ) = (cid:88) | y | = (cid:80) mi =1 | x i | + | t I,g | (cid:102) CT Y I,t g ( y ; x , . . . , x m ) · y . Counting the points in the boundary of the oriented 1-dimensional manifolds (cid:102) CT I,t g ( y ; x , . . . , x m ) finally proves that : Theorem 5. The operations µ I,t g define a twisted n − Ω BAs -morphism between the Morse cochains ( C ∗ ( f ) , ∂ T wMorse , ∂ Morse ) and ( C ∗ ( g ) , ∂ T wMorse , ∂ Morse ) . We send the reader back to [Maz21] for the complete check of signs in the case of the operations m t , which easily transports to the case of the operations µ I,t g . Again, either working with coefficientsin Z / , or with coefficients in Z and an odd-dimensional manifold M , the operations µ I,t g fit into astandard n − Ω BAs -morphism between Ω BAs -algebras.3. Towards the problem of the composition At the end of [Maz21] we stated two main questions. The first was the motivational questionsolved in this article and the second one came as follows : Problem 2. Given three Morse functions f , f , f , choices of perturbation data X i , and choices ofperturbation data Y ij defining morphisms µ Y : ( C ∗ ( f ) , m X t ) −→ ( C ∗ ( f ) , m X t ) ,µ Y : ( C ∗ ( f ) , m X t ) −→ ( C ∗ ( f ) , m X t ) ,µ Y : ( C ∗ ( f ) , m X t ) −→ ( C ∗ ( f ) , m X t ) , can we construct an Ω BAs -homotopy such that µ Y ◦ µ Y (cid:39) µ Y through this homotopy ? Thatis, can the following cone be filled in the Ω BAs realm C ∗ ( f ) C ∗ ( f ) C ∗ ( f ) µ Y µ Y µ Y ? The author plans to prove in an upcoming article that the answer to this question is positive.This simple problem will in fact again generalize to a wider range of constructions in Morse theory,involving the n -morphisms introduced in this article as well as some new interesting combinatorics. IGHER ALGEBRA OF A ∞ AND Ω BAs -ALGEBRAS IN MORSE THEORY II 63 References [Abo11] Mohammed Abouzaid. 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