These operations were later generalized by McClure and Smith in [MS04], using the framework of operadic algebra, to an operad S called the sequence operad

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THIBAUT MAZUIR

Relectures nales (se faire une liste des points à vérier à chaque relecture), puis en ligne The Alexander-Whitney map, dened by the formula

AW([i1 <· · ·< ik]) :=

k

X

j=1

[i1 <· · ·< ij]⊗[ij <· · ·< ik],

endows the singular chains of a topological space S∗(X) with a structure of coassociative dg- coalgebra, and its dual, the cup product, endows the singular cochains of a topological space with a structure of associative dg-algebra. The cup product was proven in 1947 by Steenrod ([Ste47]) to be part of a larger set of operations, called the cup-i-products. These operations were later generalized by McClure and Smith in [MS04], using the framework of operadic algebra, to an operad S called the sequence operad.

Now switching viewpoint, we see AW as a map between the coalgebra S∗(X) and the product coalgebra S∗(X)⊗S∗(X): this map then preserves the dierential, but does not preserve the coproduct (see section 3.1). However, it preserves the coproduct up to homotopy. The theory of A∞-coalgebras provides a suitable framework to study this problem. Indeed, the correct homotopy notion of "dierential map that preserves the coproduct up to homotopy" is given by that of an A∞-morphism.

The goal of this short note is to prove that the map AW can be extended to an A∞-morphism in characteristic 2 (and upon working out the signs, in characteristic 0 too). To this end, we will introduce in the rst two parts basic results concerning the sequence operad S and obstruction theory forA∞-morphisms, and the proof of the theorem will be detailed in the last part.

As we work in characteristic 2, the signs −and +will be equal, but we will sometimes prefer one over the other to illustrate some particular property.

1. ∆ⁿ and the sequence operad S 1.1. The dg-coalgebra ∆ⁿ.

1.1.1. Denition. Dene ∆ⁿ to be the graded Z2 -module generated by the faces of the standard n-simplex ∆ⁿ,

∆ⁿ= M

06i1<···<i_k6n

Z2[i₁ <· · ·< i_k],

where the grading is|I|:= dim(I)for I a face of∆ⁿ. This graded Z2 -module can be endowed with a dg-coalgebra structure, whose dierential is the simplicial dierential

∂_∆ⁿ([i₁<· · ·< i_k]) :=

k

X

j=1

[i₁<· · ·<iˆ_j <· · ·< i_k],

1

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and whose coproduct is the Alexander-Whitney coproduct

∆_∆ⁿ([i₁ <· · ·< i_k]) :=

k

X

j=1

[i₁<· · ·< i_j]⊗[i_j <· · ·< i_k].

These dg-coalgebras are to be seen as the natural realizations of the simplices∆ⁿin the category of dg-coalgebras.

1.1.2. Overlapping partitions. Let I be a face of ∆ⁿ. Following McClure-Smith [MS04], we dene an overlapping partition ofI to be a sequence of faces(I_l)_16l6s ofI, such that

(i) the union of this sequence of faces is I, that is ∪_16l6sI_l=I ; (ii) for all 16l < s,max(Il) = min(Il+1) ;

(iii) min(I₁) = min(I) andmax(I_s) = max(I).

These sequences of faces are those which naturally arise when applying several times the Alexander- Whitney coproduct to a faceI. For instance, the Alexander-Whitney coproduct corresponds to the signed sum of all overlapping 2-partitions ofI. Iteratingntimes the Alexander-Whitney coproduct, we get the signed sum of all overlapping (n+ 1)-partitions of I. See below for an instance of an overlapping partition :

[0<1<2] = [0]∪[0]∪[0<1]∪[1]∪[1<2]∪[2].

1.2. The sequence operad. Most of the material in this section is derived from [MS04].

1.2.1. Motivations. Consider two copies ∆ⁿ₁ and ∆ⁿ₂ of ∆ⁿ. We now see AW := ∆∆ⁿ :∆ⁿ−→∆ⁿ₁ ⊗∆ⁿ₂ ,

as a map between the dg-coalgebras∆ⁿand∆ⁿ₁⊗∆ⁿ₂, where∆ⁿ₁⊗∆ⁿ₂ is endowed with the following dierential and coproduct :

∂_∆ⁿ

1⊗∆ⁿ₂ :=∂_∆ⁿ

1 ⊗id + id⊗∂_∆ⁿ

2 ∆_∆ⁿ

1⊗∆ⁿ₂ := (id⊗τ⊗id)∆_∆ⁿ

1 ⊗∆_∆ⁿ

2 , whereτ(x⊗y) =y⊗x. Introduce also the map

AW :˜ ∆ⁿ−→∆ⁿ₁ ⊗∆ⁿ₂

I 7−→ X

I1∪I2=I

I2⊗I1 .

Steenrod pointed out in [Ste47] that AW˜ andAW are homotopic through the map

H(I) = X

I1∪I2∪I3=I

I1∪I3⊗I2 .

The ideas used to dene these three maps can in fact be generalized, to associate to each overlapping partitionI₁∪· · ·∪I_s=Ian element of(∆ⁿ)^⊗sby sending each set of the partition to its corresponding component in the tensor product. The algebra underlying these operations is proven in [MS04] to be controlled by an operad S, called the sequence operad, that we now proceed to describe.

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1.2.2. The sequence operad. The operadS is more precisely a dg-operad, which means that we have to dene the spaces of graded operations of arity k, for eachk >1, that are dg-modules S(k)•, as well as the compositions between them. We dene theS(k)_• in this section and refer to [MS04] for the denition of the composition - that we are not going to need in this paper.

Begin by dening a surjection s : {1, . . . , m} → {1, . . . , k} to be non-degenerate if for all i ∈ {1, . . . , m},s(i)6=s(i+ 1). These non-degenerate surjections are simply to be thought as sequences of numbers in{1, . . . , k}, of lengthm, such that all1, . . . , kappear in the sequence, and such that two adjacent numbers are always dierent. For instance, for m = 4and k= 3, the following sequences are non-degenerate : 1232, 1321, 1323 ; and the following ones are degenerate : 1123, 1233, 1212, 3232.

The moduleS(k)_q, fork>1 and q>0, is then dened to be theZ2-module freely generated by the non-degenerate surjections s : {1, . . . , q+k} → {1, . . . , k}. For s: {1, . . . , m} → {1, . . . , k} a non-degenerate surjection, the dierential is dened as

∂S(s) =

m

X

i=1

s|_{1,...,_ˆ_i,...,m} ,

where the maps s|_{1,...,_ˆ_i,...,m} which are not surjections or which become degenerate surjections are taken to 0. For instance,

∂(123212) = 23212 + 13212 + 12312 + 12321 ,

∂(1321) = 321 + 132.

1.2.3. The operad S is contractible. One of the main properties of the operad S is that it is contractible, i.e. for allk, the dg-moduleS(k)_• has the homology of a point (in fact this operad is even a model for an E∞-operad).

We prove by induction that S(k)_• has the same homology asS(1), which has the homology of a point. Consider the following three maps :

i:S(k−1)q−→ S(k)q , r:S(k)_q−→ S(k−1)q , h:S(k)_q−→ S(k)_q+1 .

The mapitakes a sequence, adds +1 to each of its entries and then concatenates a 1 at its beginning.

It is a chain map. If a sequence contains only one 1, and this 1 is its rst entry, then the map r deletes the 1, and lowers the remaining entries by 1 ; the map r takes the sequence to 0 otherwise.

The map r is not a chain map. Finally the maph simply concatenates a 1 at the beginning of the sequence. These three maps satisfy

[∂, h] = id−ir .

A few simple arguments then prove that i∗ :H∗(S(k−1))→H∗(S(k))is surjective, and knowing that S(1)has the homology of a point, this concludes the proof.

Givenx a cycle in S(k), the formula

x=∂h(x) +ir(x),

can in fact be iterated for r(x) to construct explicitly an element y ∈ S(k) such that x = ∂y. Explicitly, y=P|x|

k=0i^khr^k(x).

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1.3. ∆ⁿ is a S-coalgebra. We nally endow ∆ⁿ with a structure of S-coalgebra. We explicit in this section the operations associated to the non-degenerate surjections and refer to [MS04] for the remaining part of the arguments.

Let s be a non-degenerate surjection {1, . . . , q+k} → {1, . . . , k}. We want to associate a map

∆ⁿ→(∆ⁿ)^⊗kto s. The sequence of numbers associated to this surjection is to be read as follows : thei-th entry corresponds to thei-th setI_i of a(q+k)-overlapping partition of I, while the number s(i)it comes with means thatIi is sent to thes(i)-th component of(∆ⁿ)^⊗k. This is better illustrated on the following instances :

AW(I) = 12(I) = X

I1∪I₂=I

I₁⊗I₂ ,

AW(I˜ ) = 21(I) = X

I1∪I2=I

I2⊗I1 ,

121(I) = X

I1∪I2∪I3=I

I1∪I3⊗I2 ,

13234(I) = X

I1∪I2∪I3∪I4∪I5=I

I₁⊗I₃⊗I₂∪I₄⊗I₅ .

2. Obstruction theory for A∞-morphisms

This part is a free adaptation from material from the PhD thesis of Kenji Lefèvre-Hasegawa [LH03].

2.1. A∞-coalgebras and A∞-morphisms.

2.1.1. A∞-coalgebras. LetA be a dg-vector space over Z2 with dierential ∆1. We choose to work with homological convention, so it has degree −1. A structure ofA∞-coalgebra onA is the data of a collection of maps of degreen−2

∆_n:A−→A^⊗n, which satisfy the following equations, called the A∞-equations

[∆1,∆n] = X

i1+i2+i3=n i2>2

(id^⊗i¹ ⊗∆i2 ⊗id^⊗i³)∆i1+1+i3.

In particular,

∆₂∆₁= (∆₁⊗id + id⊗∆₁)∆₂ , [∆₁,∆₃] = (id⊗∆₂−∆₂⊗id)∆₂ ,

DeningH∗(A)to be the homology ofArelative to∆₁, the last two equations show that∆₂descends to a coassociative coproduct on H∗(A). Hence, anA∞-coalgebra is just the right homotopy notion of a dg-coalgebra whose coproduct is coassociative up to homotopy. Indeed to dene such a notion, we have to keep track of all the higher homotopies coming with the fact that the coproduct is coassociative up to homotopy : these higher homotopies are exactly the∆_n.

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2.1.2. A∞-morphisms. Denote from now on∆₁as∂. AnA∞-morphism between twoA∞-coalgebras A and B is dened to be a family of maps of degreen−1

fn:A→B^⊗n , satisfying for alln>1,

[∂, fn] = X

i1+i2+i3=n i2>2

(id^⊗i¹ ⊗∆^A_i₂ ⊗id^⊗i³)fi1+1+i3+ X

i1+···+is=n s>2

(fi1 ⊗ · · · ⊗fis)∆^B_s .

We check in particular thatf1is a chain map and we recover equation[∂, f2] = (f1⊗f₁)∆^A₂−∆^B₂f1, meaning that f₁ preserves the coproduct up to the homotopy f₂. As a result, an A∞-morphism between A∞-coalgebras induces a morphism of coassociative coalgebras on the level of homology.

Thefnkeep again track of all higher homotopies coming with the property "preserving the coproduct up to homotopy".

2.2. Obstruction theory for A∞-morphisms. LetAandB be twoA∞-coalgebras. A collection of maps (fm : A → B^⊗m)_16m6n satisfying the rst n A∞-equations for A∞-morphisms, will be called an A_n-morphism betweenAand B. It is then natural to ask if thisA_n-morphism extends to an An+1-morphism. In other words, does there exist a mapfn+1:A→B^⊗n+1 such that

[∂, fn+1] = X

i1+i2+i3=n+1 i2>2

(id^⊗i¹ ⊗∆i2 ⊗id^⊗i³)fi1+1+i3 + X

i1+···+is=n+1 s>2

(fi1⊗ · · · ⊗fis)∆^B_s ?

Consider now the dg-module (Homdg−mod(A, B^⊗n+1),[∂,·]) where a degree r element is a morphism of dg-modules of degreer. WriteRHS_nfor the right-hand side of the above equation. This is an element ofHomdg−mod(A, B^⊗n+1)of degreen−1, and proving that theAn-morphism extends to an An+1morphism hence amounts to proving thatRHSnis a boundary in(Homdg−mod(A, B^⊗n+1),[∂,·]). While the previous statement is obviously not always true, the following always holds :

Proposition. [LH03] The morphismRHS_n is a cycle in(Homdg−mod(A, B^⊗n+1),[∂,·]).

This proposition gives a recipe to constructA∞-morphisms by induction : we begin with a chain map f₁ and have to check at each induction step that the cycle RHS_n is a boundary. This is the crux of our forthcoming proof.

3. AW extends to an A∞-morphism

3.1. The theorem. Consider again the map AW between the dg-coalgebras ∆ⁿ and ∆ⁿ₁ ⊗∆ⁿ₂. This map may preserve the dierentials, but it does not preserve the coproduct ! Indeed, consider the following diagram

∆ⁿ ∆ⁿ₁ ⊗∆ⁿ₂ ∆ⁿ₁ ⊗∆ⁿ₁ ⊗∆ⁿ₂ ⊗∆ⁿ₂

∆ⁿ⊗∆ⁿ (∆ⁿ₁ ⊗∆ⁿ₂)⊗(∆ⁿ₁ ⊗∆ⁿ₂)

AW

∆∆n

∆_∆ⁿ

1⊗∆_∆n 2

id⊗τ⊗id AW⊗AW

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In the language of the sequence operad S, the upper path is equal to1324, while the lower path is equal to1234, hence they are not equal. However, this diagram commutes up to homotopy. Indeed, we notice that∂(13234) = 1234−1324, hence the map

13234 :∆ⁿ−→(∆ⁿ₁ ⊗∆ⁿ₂)⊗(∆ⁿ₁ ⊗∆ⁿ₂) is an homotopy making the previous diagram homotopy commutative.

In light of section 2.1.1, consider∆ⁿand∆ⁿ₁⊗∆ⁿ₂ asA∞-coalgebras all of whose higher coproducts are null. The map AW preserves the coproduct up to homotopy : can it be extended to an A∞- morphism, that is does there exist an A∞-morphism whose rst term is equal toAW ?

Theorem. The map AW seen as a map between the dg-coalgebra ∆ⁿ and the product dg-coalgebra

∆ⁿ₁ ⊗∆ⁿ₂, can be extended to an A∞-morphism.

The higher coproducts will be built inductively, using obstruction theory for A∞-morphisms and the fact that the operadS is contractible.

3.2. Proof. We write in the remaining part of this sectionAW1:= AW to emphasize that we want it to be the rst term of an A∞-morphism.

We see above that the map AW2:= 13234from ∆ⁿ to (∆ⁿ₁ ⊗∆ⁿ₂)⊗(∆ⁿ₁ ⊗∆ⁿ₂) satises [∂,AW₂] = ∆_∆ⁿ

1⊗∆ⁿ₂AW₁−(AW₁⊗AW₁)∆_∆ⁿ .

This is the second A∞-equation for A∞-morphisms between the dg-coalgebras∆ⁿ and ∆ⁿ₁ ⊗∆ⁿ₂. Hence AW1 andAW2 are constructed.

Now notice that, if A and B are dg-coalgebras and {f_n:A→B^⊗n}_n>1 is anA∞-morphism, the A∞-equation writes as

[∂, fn] = X

i1+i2+2=n

(id^⊗i¹⊗∆2⊗id^⊗i²)fn−1+ X

k+l=n

(f_k⊗f_l)∆2 ,

as the∆sfors>3are all null. We also point out that all the maps we have considered so far, namely

∆_∆ⁿ₁⊗∆ⁿ₂,∆_∆ⁿ,AW1andAW2are operations from the sequence operadS. Hence, composing them together or applying the dierential [∂,·]always yields a linear combination of operations from the operad S.

Given m > 2, suppose that we have constructed maps AW_k for all 1 6 k 6 m such that they dene an Am-morphism and such that for all k>1,AW_k is an operation fromS. We then set

RHSm := X

i1+i2+2=m+1

(id^⊗i¹ ⊗∆∆ⁿ₁⊗∆ⁿ₂ ⊗id^⊗i²)AWm+ X

k+l=m+1

(AWk⊗AWl)∆∆ⁿ .

The maps AWk,∆∆ⁿ₁⊗∆ⁿ₂ and ∆∆ⁿ being operations of the sequence operadS,RHSm is again an operation from this operad. Moreover, is it a cycle as stated in the proposition, and it has degree m−1 which is greater than 1. As H_i(S(k)) = 0 for i>1, this cycle is a boundary, which means that there existsAWn+1 ∈ S extending theAW_k to anAn+1-morphism. By induction, we conclude that the Alexander-Whitney map can indeed be extended to an A∞-morphism of dg-coalgebras !

In fact, knowing formulae for theAW_m m6n, we could compute an explicit formula for AW_n+1 using the remark at the end of section 1.2.3. The author however did not manage to conjecture an explicit formula for theAW_n.

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References

[LH03] Kenji Lefèvre-Hasegawa. Sur les A [inni]-catégories. PhD thesis, 2003. Thèse de doctorat dirigée par Keller, Bernhard Mathématiques Paris 7 2003.

[MS04] James E. McClure and Jerey H. Smith. Cosimplicial objects and little n-cubes. I. Amer. J. Math., 126(5):11091153, 2004.

[Ste47] N. E. Steenrod. Products of cocycles and extensions of mappings. Ann. of Math. (2), 48:290320, 1947.