A computer algebra system for the study of commutativity up-to-coherent homotopies
aa r X i v : . [ m a t h . A T ] F e b A COMPUTER ALGEBRA SYSTEM FOR THE STUDY OF COMMUTATIVITYUP-TO-COHERENT HOMOTOPIES
ANIBAL M. MEDINA-MARDONES
Abstract.
The Python package
ComCH is a lightweight specialized computer algebra system that providesmodels for well known objects, the surjection and Barratt-Eccles operads, parameterizing the product structureof algebras that are commutative in a derived sense. The primary examples of such algebras treated by
ComCH are the cochain complexes of spaces, for which it provides effective constructions of Steenrod cohomologyoperations at all prime. Introduction
All the basic notions of number, from the integers to the complex, are equipped with a commutative product,and it was believed until Hamilton’s introduction of the quaternions, that the product of any number systemmust be commutative. Hamilton’s discovery encouraged the consideration of other algebraic structures wherecommutativity was not demanded, and the effect this had on algebra is only comparable to that of non-euclidean geometries on the study of spaces. Around a century later, after the development of topology andhomotopy, commutativity was revisited and additional levels enriching the basic dichotomy were identified.These correspond to coherent systems correcting homotopically the lack of strict commutativity, and constitutethe focus of much current research on theoretical and applied topology.After the pioneering work of Steenrod [2, 3], Adem [4], Serre [5], Cartan [6], Araki-Kudo [7], Dyer-Lashof [8],Stashef [9], Boardman-Vogt [10], May [11, 12], and many others, today there is a rich theory of commutativityup-to-coherent-homotopies whose modern framework is provided by operads and PROPs, and where E n -operadsplay a central role parameterizing the different levels of homotopical commutativity. In ComCH , we focus onthe category of chain complexes, and consider two models of the E ∞ -operad equipped with filtrations by E n -operads. These are respectively due to McClure-Smith [13] and Berger-Fresse [14] and are known as thesurjection and Barratt-Eccles operads.The homology of algebras over E n -operads are equipped not only with an induced commutative product butalso with homology operations when the coefficient ring is the field F p = Z /p Z . The study of these operationsat the chain level has become an important issue in topological data analysis [15], condensed matter physics[16], category theory [17] and others areas. To provide researchers with effective tools for their study, ComCH implements the constructions of [18], making available for the first time chain level representations of theseinvariants for spaces presented simplicially or cubically. When the prime is 2, describing Steenrod operationsat the chain level is classical and there are implementations for the simplicial [19] and cubical [20] cases in
Sage [21] and
ChainCon [22] respectively. For odd primes, we do not know of any previous implementationeither in the simplicial or cubical contexts. In the former case, a different effective approach was developed byGonzales-Diaz and Real [23, 24] based on the Eilenberg–Zilber contraction.
Acknowledgment
We gratefully acknowledge contributions from Djian Post, Wojciech Reise and Michelle Smith. We thankDennis Sullivan, Kathryn Hess, John Morgan, Greg Brumfiel, Ralph Kaufmann, Paolo Salvatore, UmbertoLupo, Guillaume Tauzin, and Lewis Tunstall for insightful conversations. We also thank the Laboratory forTopology and Neuroscience at EPFL for its support and hospitality while part of this work was developed.
Mathematics Subject Classification.
Primary 55-04, 18M60; Secondary 55S05, 18M70, 55N31.
Key words and phrases.
Computer algebra system, Python, homotopical algebra, operads, cohomology operations, cup product,simplicial set, cubical set.The author acknowledges financial support from Innosuisse grant 32875.1 IP-ICT - 1. See for example Chapter V of [1].
Contents
1. Introduction 1Acknowledgment 12. Overview of
ComCH E ∞ -operads 32.5. Surjection operad 32.6. Barratt-Eccles operad 43. Steenrod operations 53.1. Steenrod-Adem structures 53.2. Steenrod operations 53.3. Surjections as linear maps 63.4. Examples 74. Outlook 7References 72. Overview of
ComCH
In this section we describe the overall structure and main functionalities of
ComCH , referring to its documen-tation for a description of all of its classes and their methods. Free modules and symmetric groups.
Let R be the ring of integers or one of its quotients. In ComCH the class
FreeModuleElement serves to model elements in free R -modules, where R is specified by the attribute torsion . Let S r be the set of self-bijection of { , . . . , r } regarded as a group by composition. An element σ ∈ S r will be represented by the sequence of its values ( σ (1) , . . . , σ ( r )) and it is modeled in ComCH using theclass
SymmetricGroupElement .2.2.
Operads.
Operads parameterize algebraic structures on chain complexes. The precise although lengthydefinition can be found for example in [25]. We will present a key example from which the definition can beabstracted. Let C be a chain complex of R -module, and consider the set End C ( r ) = Hom ( C, C ⊗ r ) of R -linearmaps as a chain complex in the usual way. The set End C = (cid:8) End C ( r ) (cid:9) r ≥ is equipped with the following structure: a left action of S r on End C ( r ) and composition chain maps ◦ i : End C ( r ) ⊗ End C ( s ) End C ( r + s − f ⊗ g (id ⊗ · · · ⊗ g ⊗ · · · ⊗ id) ◦ f satisfying forms of equivariance, associativity, and unitality.An O -coalgebra structure on C is a structure preserving morphism from O to End C . We remark that it isalso common to consider the operad End A obtained from the complexes End A ( r ) = Hom ( A ⊗ r , A ), referringto operad morphisms O →
End A as O -algebra structures on A . The linear duality functor induces from an O -coalgebra in C an O -algebra structure on A = Hom ( C, R ). Currently hosted at https://comch.readthedocs.io/en/latest/
C.A.S. FOR THE STUDY OF COMMUTATIVITY UP-TO-COHERENT HOMOTOPIES 3
Symmetric ring operad.
Let us consider R [S] = { R [S r ] } r ≥ with R [S r ] the group ring of S r thoughtof as a dg R -module concentrated in degree 0. It has the structure of an operad with left action induced fromleft multiplication, and compositions induced from the maps(1) ◦ i : S r × S s → S r + s − sending a pair ( x, y ) to the bijection x ◦ i y represented diagrammatically by1 · · · ( y z }| { i · · · i + s − · · · r + s − | {z } x . More precisely, x ◦ i y is the sequence obtained by applying the following three steps: 1) shift up by s − x greater than s , 2) shift up by i − y and, 3) replace the i -th value of x withthe shifted y . We model elements in R [S] using the class SymmetricRingElement which combines the classes
FreeModuleElement and
SymmetricGroupElement . For example, we have >>> x = SymmetricRingElement({(2,3,1): -1, (1,3,2): 1})>>> y = SymmetricRingElement({(1,3,2): 1, (1,2,3): 2})>>> print(x * y)- (2,1,3) - 2(2,3,1) + (1,2,3) + 2(1,3,2)>>> print(x.compose(y, 2))- (2,4,3,5,1) - 2(2,3,4,5,1) + (1,5,2,4,3) + 2(1,5,2,3,4) E ∞ -operads. An important class of operads are those defining resolutions of the ground ring R as an R [S r ]-module. Such operads are called E ∞ -operads. They typically come equipped with a filtration by socalled E n -operads parameterizing different levels of derived commutativity, with E corresponding to the lackof any assumed commutativity, and E ∞ to the largest possible degree of homotopical commutativity. ComCH implements models of two well known E ∞ -operads equipped with filtrations by E n -operads which we nowdescribe.2.5. Surjection operad.
For a positive integer r let X ( r ) d be the free R -module generated by all functions x : { , . . . , d + r } → { , . . . , r } modulo the R -submodule generated by degenerate functions, i.e., those whichare either non-surjective or have a pair of equal consecutive values. There is a left action of S r on X ( r ) whichis up to signs defined on basis elements by π · x = π ◦ x . We represent a surjection x as the sequences of itsvalues (cid:0) x (1) , . . . , x ( n + r ) (cid:1) . The boundary map in this complex is defined up to signs by ∂x = r + d X i =1 ± (cid:0) x (1) , . . . , d x ( i ) , . . . , x ( n + r ) (cid:1) , and the i -th composition x ◦ i y of x ∈ X ( r ) and y ∈ X ( s ) is defined, up to signs, as follows. Let w be thecardinality of x − ( i ). For every collection of ordered indices (2) 1 = j ≤ j ≤ j ≤ · · · ≤ j w − ≤ j w = s we construct an associated splitting of y ( y ( j ) , . . . , y ( j )); ( y ( j ) , . . . , y ( j )); · · · ; ( y ( j w − ) , . . . , y ( j w )) . The element x ◦ i y ∈ X ( r + s −
1) is represented as the sum over the set of order indices (2) of the sequenceobtained in the following three steps: 1) shift up by s − x greater than i , then 2) shift up by i − y , and finally 3) replace in order the occurrencesof i in x by the corresponding sequence in the splitting.The elements in this operad are modeled using the class SurjectionElement . For example,
ANIBAL M. MEDINA-MARDONES >>> x = SurjectionElement({(1,2,1,3): 1})>>> print(x.boundary())(2,1,3) - (1,2,3)>>> y = SurjectionElement({(1,2,1): 1})>>> print({x.compose(y, 1))(1,3,1,2,1,4) - (1,2,3,2,1,4) - (1,2,1,3,1,4)
The signs appearing in these constructions are determined by the attribute convention with possible valuesthe strings
McClure-Smith and
Berger-Fresse . We refer to [13] and [14] for details on these distinct signconventions.We will now review the definition of the complexity of a surjection element. The importance of this conceptis that the set of surjection elements with complexity less than n defines an E n -suboperad of X [13].The complexity of a finite binary sequence (i.e. a sequence of two distinct values) is defined as the number ofconsecutive distinct elements in it. For example, (1,2,2,1) and (1,1,1,2) have complexities 2 and 1 respectively.The complexity of a basis surjection element is defined as the maximum value of the complexities of its binarysubsequences. Notice that for elements in X (2), complexity and degree agree. The class SurjectionElement models this concept with the attribute complexity . For example, >>> x = SurjectionElement({(1,2,1,3,1): 1})>>> print(x.complexity)1
Barratt-Eccles operad.
For a non-negative integer r define the simplicial set E (S r ) by E (S r ) n = { ( σ , . . . , σ n ) | σ i ∈ S r } ,d i ( σ , . . . , σ n ) = ( σ , . . . , b σ i , . . . , σ n ) ,s i ( σ , . . . , σ n ) = ( σ , . . . , σ i , σ i , . . . , σ n ) . It is equipped with a left S r -action defined on basis elements by σ ( σ , . . . , σ n ) = ( σσ , . . . , σσ n ) . The chain complex resulting from applying the functor of normalized R -chains to it is denoted E ( r ), and theunderlying set of the Barratt-Eccles operad is E = {E ( r ) } r ≥ . To define its composition structure we use theEilenberg-Zilber map. Let us notice that at the level of the simplicial sets E (S r ) we have compositions ◦ i : E ( r ) × E ( s ) → E ( r + s − E by precomposing N • ( ◦ i ) : N • ( E ( r ) × E ( s )) −→ N • ( E ( r + s − E ( r + s − E ( r ) ⊗ E ( s ) = N • ( E ( r )) ⊗ N • ( E ( s )) −→ N • ( E ( r ) × E ( s )) . For example, >>> x = BarrattEcclesElement({((1,2),(2,1)):1, ((2,1),(1,2)):2})>>> print(x.boundary())((1,2),) - ((2,1),)>>> y = BarrattEcclesElement({((2,1,3),):3})>>> print(x.compose(y, 2))3((1,3,2,4),(3,2,4,1)) + 6((3,2,4,1),(1,3,2,4))
The complexity of a Barratt-Eccles element is define analogously to that of surjection elements. In this casetoo the subset of elements with complexity less than n defines an E n -suboperad [14]. C.A.S. FOR THE STUDY OF COMMUTATIVITY UP-TO-COHERENT HOMOTOPIES 5
An important structure present in the Barratt-Eccles operad missing in the surjection operad is a diagonalchain map compatible with compositions. It is given by the Alexander-Whitney diagonal which on basisBarratt-Eccles element is ∆( σ , . . . , σ n ) = n X i =1 ( σ , . . . , σ i ) ⊗ ( σ i , . . . , σ n ) . For example, >>> x = BarrattEcclesElement({((1,2), (2,1)): 1})>>> print(x.diagonal())(((1, 2),), ((1, 2), (2, 1))) + (((1, 2), (2, 1)), ((2, 1),)) Steenrod operations
In this section we effectively describe how to compute Steenrod operations on the mod- p cohomology ofspaces presented as simplicial or cubical sets. The implementation of these algorithms is one of the main novelcontributions of ComCH to the available software used in algebraic topology.3.1.
Steenrod-Adem structures.
Let C r be the cyclic group of order r thought of as the subgroup of S r generated by an element ρ . The elements T = ρ − N = 1 + ρ + · · · + ρ r − in R [ C r ] define a minimal resolution of R by free R [ C r ]-modules W ( r ) = R [ C r ] T ←− R [ C r ] N ←− R [ C r ] T ←− · · · . We denote a preferred basis element of W ( r ) i by e i .A Steenrod structure on an operad O is a collection, indexed by r >
0, of C r -equivariant chain maps W ( r ) ψ r −→ O ( r ) for which there exists a factorization through an E ∞ -operad W ( r ) → R ( r ) → O ( r ) such thatthe first map is a quasi-isomorphism and the second is an S r -equivariant chain map. If the maps R ( r ) → O ( r )define a morphism of operads R → O , we say the Steenrod structure is a Steenrod-Adem structure. For anypair of integers r and i , a Steenrod structure produces a preferred element ψ r ( e i ) in O ( r ) i .Steenrod-Adem structures for the surjection and Barratt-Eccles operads are implemented in ComCH followingtheir introduction in [18]. Some examples of ψ r ( e i ) are >>> r, i = 3, 2>>> y = Surjection.steenrod_adem_structure(r, i)>>> print(y)(1,2,3,1,2) + (1,3,1,2,3) + (1,2,3,2,3)>>> x = BarrattEccles.steenrod_adem_structure(r, i)>>> print(x)((1,2,3),(2,3,1),(3,1,2)) + ((1,2,3),(3,1,2),(1,2,3)) Steenrod operations.
Let A be a chain complex of Z -modules. Let us assume the operad End A isequipped with a Steenrod structure ψ : W →
End A . For any prime p , define the linear map D d : A ⊗ F p → A ⊗ F p by D d ( a ) = ( ψ ( e d )( a ⊗ p ) d ≥ , d < . As in [11], for any integer s the Steenrod operations P s : H • ( A ; F ) → H • + s ( A ; F )and, for p > P s : H • ( A ; F p ) → H • +2 s ( p − ( A ; F p ) ,βP s : H • ( A ; F p ) → H • +2 s ( p − − ( A ; F p ) , ANIBAL M. MEDINA-MARDONES are defined for a class [ a ] of degree q respectively by P s (cid:0) [ a ] (cid:1) = (cid:2) D s − q ( a ) (cid:3) and P s (cid:0) [ a ] (cid:1) = (cid:2) ( − s ν ( q ) D (2 s − q )( p − ( a ) (cid:3) ,βP s (cid:0) [ a ] (cid:1) = (cid:2) ( − s ν ( q ) D (2 s − q )( p − − ( a ) (cid:3) , where ν ( q ) = ( − q ( q − m/ ( m !) q and m = ( p − / Surjections as linear maps.
In this subsection we describe a Steenrod structure on the cochains of sim-plicial and cubical sets, which defines, by the previous subsection, Steenrod operations on their mod- p cohomol-ogy. Using the linear duality functor, it suffices to define a Steenrod structure on chains. Furthermore, by nat-urality, it suffices to define it on the universal endomorphism operad End C • with End C • ( r ) = Hom ( C • , C ⊗ r • ),the chain complex of natural transformations from the functor of normalized simplicial or cubical chains toan r -iterated tensor product of itself. An element f in this abstract chain complex can be more concretelydescribed as a set { f n } n ≥ of elements in C • (∆ ∞ ) ⊗ r and C • ( I ∞ ) ⊗ r respectively, with f n being the element f (cid:0) [0 , . . . , n ] (cid:1) or f (cid:0) [0 , × n (cid:1) respectively. In ComCH , elements in C • (∆ ∞ ) ⊗ r and C • ( I ∞ ) ⊗ r are modeled using the SimplicialElement and
CubicalElement classes. For example, >>> x = SimplicialElement({((0,1), (1,2,3), (2,3)): 1})>>> print(x._latex_())[0,1] \otimes [1,2,3] \otimes [2,3]>>> y = CubicalElement({((0,1), (2,1), (2,2)): 1})>>> print(y._latex_())[0][1] \otimes [01][1] \otimes [01][01]
To define the Steenrod structure on the universal operad
End C • it suffices to construct a collection of naturalS r -equivariant chains maps X ( r ) → Hom ( C • , C ⊗ r • ) since we already have a Steenrod-Adem structure on X . Todescribe the S r -equivariant chain map X ( r ) → Hom ( C • , C ⊗ r • ) we follow [26, 27]. Represent a basis surjectionelement in X ( r ) n as the labeled directed graph
11 2 ... k ... · · ·· · · r ... k r ...
11 2 3 n + r · · · ... where there are no hidden vertices and the strands at the top are joined to the strands at the bottom using theinformation prescribed by the surjection. Any such graph gives rise to a map in Hom ( C • , C ⊗ r • ) after associatingappropriate maps to the generating pieces and in Hom ( C • , C ⊗ • ) and Hom ( C ⊗ • , C • ) respectively. Suchmaps where associated to these generating pieces in [26] for the simplicial case and [18] for the cubical one. In ComCH we implement these constructions allowing for surjection elements to act on simplicial and cubicalchains. For example, we have We remark for the interested reader that both of these structures are induced from an E ∞ -bialgebra structure on the cellularchains of the interval. C.A.S. FOR THE STUDY OF COMMUTATIVITY UP-TO-COHERENT HOMOTOPIES 7 >>> x = SurjectionElement({(1,2,1): 1}, convention=’McClure-Smith’)>>> a = Simplicial.standard_element(2)>>> print(x(a))- ((0,1,2),(0,1)) + ((0,2),(0,1,2)) - ((0,1,2),(1,2))>>> b = Cubical.standard_element(2)>>> print(x(b))- ((2,2),(1,2)) + ((2,1),(2,2)) + ((0,2),(2,2)) - ((2,2),(2,0))
We remark that in the simplicial context the action of the unique non-degenerate surjection s i : { , . . . , i + 2 } →{ , } with s i (1) = 1 agree up to sign with the cup- i coproducts originally introduced by Steenrod [2] andaxiomatized in [28].3.4. Examples.
We remark that cochains, being defined as
Hom ( C • , R ), are concentrated in non-positivedegrees. We will only give examples in the simplicial context since, in general, the corresponding cubicalexpressions involve many more terms. We refer to the documentation of ComCH and its Jupyter notebooks forexamples in the cubical context.1) Let us consider the prime 2. The value P − ( x ) (cid:0) [0 , , , , (cid:1) for x homogeneous of degree − x ⊗ on the following output >>> p, s, q = 2, -1, -3>>> print(Surjection.steenrod_chain(p, s, q))((0,1,2,3),(0,1,3,4)) + ((0,2,3,4),(0,1,2,4)) +((0,1,2,3),(1,2,3,4)) + ((0,1,3,4),(1,2,3,4))
2) Let us consider the prime 3. The value βP − ( x ) (cid:0) [0 , , . . . , (cid:1) for x homogeneous of degree − x ⊗ on the following output >>> p, s, q = 3, -1, -3>>> print(Surjection.steenrod_chain(p, s, q, bockstein=True))2((0,6,7,8),(0,1,2,3),(3,4,5,6)) + ((0,1,7,8),(1,2,3,4),(4,5,6,7))+ 2((0,1,2,8),(2,3,4,5),(5,6,7,8))
3) Let us consider the prime 3 again. The value P − ( x ) (cid:0) [0 , , . . . , (cid:1) for x homogeneous of degree − x ⊗ on the following output >>> p, s, q = 3, -2, -4>>> print(Surjection.steenrod_chain(p, s, q, bockstein=False))((0,1,2,3,4),(4,5,6,7,8),(8,9,10,11,12)) Outlook
Operations on the homology of algebras that are only E n for a finite n are well understood homologically[29] but not at the chain level for n >
2. The case n = 2 has been studied by Tourtchine [30] and will beimplemented in ComCH .Secondary cohomology operations result from relations among primary. The Cartan and Adem Relationsare of particular importance, and constructing cochains enforcing them is open problem for p >
2. The case p = 2 was treated by Brumfiel, Morgan and the author [31, 32] and will soon be implemented in ComCH .Cubical chains appear naturally from simplicial sets through the cobar construction [33] and in
ComCH wehave implemented an E ∞ -structure on them. To model the double cobar construction one needs to studypermutahedral chains [34]. In forthcoming work we describe an E ∞ -structure on permutahedral sets suitablefor implementation on ComCH . References [1] M. Kline,
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Max Plank Institute for Mathematics, Bonn, Germany
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