Boardman--Vogt tensor products of absolutely free operads
aa r X i v : . [ m a t h . K T ] M a y BOARDMAN–VOGT TENSOR PRODUCTSOF ABSOLUTELY FREE OPERADS
MURRAY BREMNER AND VLADIMIR DOTSENKO
To the memory of Trevor Evans (1925–1991), the pioneer of interchange laws in universal algebra A bstract . We establish a combinatorial model for the Boardman–Vogt tensorproduct of several absolutely free operads, that is free symmetric operads thatare also free as S -modules. Our results imply that such a tensor product is alwaysa free S -module, in contrast with the results of Kock and Bremner–Madariagaon hidden commutativity for the Boardman–Vogt tensor square of the operad ofnon-unital associative algebras. I ntroduction Interchange law.
Consider two binary operations → and ↑ on the same set X .These operations are said to satisfy the interchange law if (for all x , . . . , x ∈ X )(1) ( x ↑ x ) → ( x ↑ x ) = ( x → x ) ↑ ( x → x ) . Note that this relation does not require X to possess any extra structure, e.g. it isnot required to be an Abelian group, or a vector space. Unlike familiar relationslike associativity, each term in this relation involves three operation symbols, sothis is, in the language of algebraic operads, a cubic relation.Geometrically, the interchange law expresses the equivalence of the two se-quences of bisections which partition a square into four equal squares:1 x → x −−−−−→ ( x ↑ x ) → x −−−−−−−−→
12 3 ( x ↑ x ) → ( x ↑ x ) −−−−−−−−−−−−→
12 341 x ↑ x −−−−−→ x ↑ ( x → x ) −−−−−−−−→
12 4 ( x → x ) ↑ ( x → x ) −−−−−−−−−−−−→
12 34An important toy model of an interchange law is that between the two operationson PROPs of endomorphisms. Recall that the collection of setsEnd X = (cid:8) End X ( p , q ) (cid:9) p , q ≥ = (cid:8) Map( X p , X q ) (cid:9) p , q ≥ is equipped with two associative operations: − The vertical composition from End X ( p , q ) × End X ( q , r ) to End X ( p , r ) which is thecomposition of linear maps: if f : X p → X q and g : X q → X r then f ↑ g : X p g ◦ f −−−→ X r . Mathematics Subject Classification.
Primary 18D50. Secondary 05A15, 05E15, 18D05, 18G10,52C22.
Key words and phrases. absolutely free operad, algebraic operad, interchange law, nonassociativealgebra, Boardman-Vogt tensor product of symmetric operads, operad of little d -rectangles, rectangularpartitions of the unit d -dimensional cube.The research of Murray Bremner was supported by a Discovery Grant from NSERC, the NaturalSciences and Engineering Research Council of Canada. − The horizontal composition from End X ( p , q ) × End X ( r , s ) to End X ( p + r , q + s ) whichis induced by the direct product of maps: if f : X p → X q and g : X r → X s then f → g : X p + r (cid:27) X p × X r f × g −−−→ X q × X s (cid:27) X q + s . These two operations are related by the interchange law (1).Two binary operations satisfying the interchange law seem to have first ap-peared explicitly in the mathematical literature in Godement’s “five rules of func-torial calculus” [9, Appendix §
1, equation (V)]. More generally, one can talk aboutinterchange for operations of arbitrary arities. The corresponding definition ap-peared, independently, in work of Evans [7] and of Boardman and Vogt [2]. Thelatter reference has become the definitive source on interchange of algebraic struc-tures, encoding it under the name of Boardman–Vogt tensor product of operads;its influence on algebraic topology and higher category theory is hard to over-estimate. By contrast, the former reference remained mostly unnoticed (even byMathematical Reviews).
Geometry of interchange.
The geometric model of the interchange law for twobinary operations that we mentioned above utilises subdivisions of the unit squareinto several pieces which are obtained by iterated bisections orthogonal to thecoordinate axes. This geometric model admits a straightforward generalisation to d dimensions. In this case the combinatorial objects of interest are subdivisions ofthe unit cube into d -dimensional rectangles with disjoint interiors by a sequence ofbisections orthogonal to the coordinate axes. The d interchanging binary operationsare represented by bisections orthogonal to the d coordinate hyperplanes.Let us remark that such subdivisions of the unit cube are subsets of the com-ponents of the operad of little d -cubes (or, more precisely, little d -rectangles), andin fact form a suboperad. However, these subsets are discrete, and therefore ex-hibit rigidity that renders the connection somewhat superficial; in particular, in thehomology of the operad of little d -cubes the corresponding operad collapses into C om , the operad of commutative associative algebras. It is also worth noting thatthe notion of a subdivision we are working with is di ff erent from the commonlyconsidered partitions of the unit cube in the combinatorics literature; the closestbut still di ff erent notion is that of the so called “guillotine partitions”, or “slicingfloorplans”; see the recent paper [1] of Asinowski, Barequet, Mansour and Pinterand references therein.The subdivisions of the unit d -cube have d interchanging binary products, thatis, the action of the Boardman–Vogt tensor product of d copies of the absolutelyfree operad on one binary generator. There exists a similar geometric model forany d -fold Boardman–Vogt tensor product T ( X ) ⊗ · · · ⊗ T ( X d )of several absolutely free operads. Namely, to encode a generator of arity m >
2, wemay also consider subdivisions into m equal parts using m − T ( X i ), we may assign tohyperplane cuts of the same directions labels which distinguish them from oneanother. This leads to the general notion of ( X , . . . , X d )-subdivisions of the unitcube, and to the structure of an operad on those subdivisions which we call the cutoperad . Homological methods.
To establish that our geometric model encodes Boardman–Vogt tensor products faithfully (Theorem 3.2), it turns out to be crucial to movefrom combinatorics to homological algebra. In a way, all key results of this paperare connected through a conceptual result on right module resolutions. To state
OARDMAN–VOGT TENSOR PRODUCTS OF ABSOLUTELY FREE OPERADS 3 that result, recall that the category of S -modules has a monoidal structure (cid:3) , calledthe matrix product by Dwyer and Hess in [5], or the arithmetic product by Maiaand M´endez in [14], which categorifies the product of Dirichlet series. Theorem (Th. 3.5) . Let T ( X ) , . . . , T ( X d ) be reduced connected absolutely free setoperads. There exists a minimal resolution (cid:16) ( I ⊕ k X ) (cid:3) · · · (cid:3) ( I ⊕ k X d ) (cid:17) ◦ (cid:16) T ( X ) ⊗ · · · ⊗ T ( X d ) (cid:17) of the augmentation module I over (the linearised version of) the d-fold Boardman–Vogttensor product T ( X ) ⊗ · · · ⊗ T ( X d ) by free right modules. Here the homological degree ofall factors I is equal to zero, and the homological degree of X k is equal to for all ≤ k ≤ d. Organisation of the paper.
The paper is organised as follows. In Section 1, werecall the key relevant definitions of the theory of operads. In Section 2 we create,in three easy steps, a combinatorial set-up for modelling interchange, the X • -subdivisions of the unit d -cube. In Section 3, we establish that X • -subdivisionsencode the Boardman–Vogt product faithfully, in other words, that the cut operaddetermined by the datum ( X , . . . , X d ) is isomorphic to the d -fold tensor product T ( X ) ⊗ · · · ⊗ T ( X d ), and prove Theorem 3.5 stated above. In Section 4, we discussa possible generalisation of that theorem and its limitations. Acknowledgements.
We would like to thank Sara Madariaga for useful discus-sions at an early stage of work on this project. The second author is grateful toKathryn Hess for the reference [5] where the matrix product of collections is re-lated to the Boardman–Vogt tensor product and especially for sharing her workin progress with William Dwyer and Ben Knudsen which gave us an a posteriori intuitive explanation of our homological result.1. R ecollections
We refer the reader to the comprehensive monograph [13] by Loday and Val-lette for background on algebraic operads, and only recall some of the notions ofparticular importance for this paper.We denote by
Fin the category of nonempty finite sets (with bijections as mor-phisms); we use the “topologist’s notation” n = { , . . . , n } . Underlying objects ofall operads of these paper will be objects of one of the following three symmetricmonoidal categories: the category Set of finite sets (with all maps as morphisms),the category
Vect of finite-dimensional vector spaces (with all linear maps asmorphisms), or the category Ch of nonnegatively graded chain complexes withfinite-dimensional components (with all chain maps as morphisms). Denote one ofthose categories by C . Recall that a ( C -valued) symmetric collection (or an S -module )is a contravariant functor from the category Fin to C . The category S -mod ofsymmetric collections has symmetric collections as objects, and natural transfor-mations of functors as morphisms. An immediate consequence of functoriality isthat for every S -module F the object F ( n ) acquires a right action of S n , the group ofautomorphisms of n (which explains the terminology); we denote by v .σ the resultof the action of σ ∈ S n on v ∈ F ( n ). We say that a symmetric collection M is free iffor each n the action of S n on M ( n ) is free.1.1. Composition of symmetric collections.
We begin by recalling one well knownmonoidal structure on S -mod. MURRAY BREMNER AND VLADIMIR DOTSENKO
Definition 1.1.
Let P and Q be two symmetric collections. The (symmetric) compo-sition P ◦ Q is defined by the formula( P ◦ Q )( X ) : = G k P ( k ) O S k G f : X ։ k Q ( f − (1)) ⊗ . . . ⊗ Q ( f − ( k )) , where the sum is taken over all surjections f .Recall that the unit collection I is defined as follows: I ( X ) = , | X | = , , | X | , , where 0 is the initial object of C . It is well known that the operation ◦ makes S -mod into a monoidal category with the unit object I . Monoids in ( S -mod , ◦ , I )are known as symmetric operads . The structure map O ◦ O → O is denoted by γ O ,or simply γ where there is no ambiguity. A k -linear symmetric operad O is said tobe augmented if it is equipped with a morphism ǫ : O → I satisfying ǫη = id, where η : I → O is the unit of the monoid O .Unless otherwise stated, all operads we work with are reduced (that is, O (0) = connected (that is, O (1) = ). In the linear context, such operads are automat-ically augmented, with augmentation being the quotient by the ideal of elementsof arity greater than one.We say that an operad is absolutely free if it is generated by elements that possessno symmetries and satisfy no relations. In other words, an absolutely free operadis a free operad generated by a free symmetric collection.1.2. Matrix product of symmetric collections.
The next definition we recall hereis much less known. It was first proposed by Maia and M´endez in [14] underthe name “arithmetic product” in order to categorify the Dirichlet product of twosequences of numbers, and then rediscovered by Dwyer and Hess in [5] under thename “matrix monoidal structure”. We shall keep the latter name because we feelthat it serves as a better illustration of the underlying combinatorics.
Definition 1.2.
Let X and Y be two symmetric collections. The matrix product X (cid:3) Y is defined by the formula( X (cid:3) Y )( X ) : = G ( π,τ ) X ( π ) ⊗ Y ( τ ) , where the sum is taken over all pairs of orthogonal set partitions π = { π , . . . , π k } , τ = { τ , . . . , τ l } of X , so that X = π ⊔ · · · ⊔ π k = τ ⊔ · · · ⊔ τ l and | π i ∩ τ j | = i = , . . . , k and j = , . . . , l .It is known that the operation (cid:3) makes S -mod into a monoidal category withthe unit object I , see [14]. More amusingly (although not immediately important)for the purpose of this paper, Dwyer and Hess established in [5, Prop. 1.20] thatthere exists a natural transformation σ : ( V ◦ W ) (cid:3) ( Y ◦ Z ) → ( V (cid:3) Y ) ◦ ( W (cid:3) Z ) , so the interchange law manifests itself once again! OARDMAN–VOGT TENSOR PRODUCTS OF ABSOLUTELY FREE OPERADS 5
Boardman–Vogt tensor product of operads.
The third monoidal structurethat we define here is the monoidal structure on the category of symmetric setoperads, introduced by Boardman and Vogt in [2], and extensively used in algebraictopology since then. Throughout this section, all operads are assumed to beoperads in
Set . Definition 1.3.
Let P and Q be two symmetric operads. The Boardman–Vogt tensorproduct P ⊗ Q is defined by the formula P ⊗ Q = ( P ⊔ Q ) / I , where I is the ideal in the coproduct (free product) of P and Q generated by allelements of P ⊔ Q of the form(2) γ ( p ; q , . . . , q ) − γ ( q ; p , . . . , p ) .σ k , l , where p ∈ P ( k ), and q ∈ Q ( l ), and σ ∈ S kl which “exchanges rows and columns”,that is for each 1 ≤ ( i − l + j ≤ kl with 1 ≤ i ≤ k and 1 ≤ j ≤ l , we have σ k , l (( i − l + j ) = ( j − k + i . Algebras over the operad P ⊗ Q are called algebras with interchanging P - and Q -actions .The following rather obvious result on Boardman–Vogt tensor products is oftenuseful. The closest reference for it that we could find is a particular case P = Q ,see the work of Dunn [4, Prop. 1.6]. Proposition 1.4.
Suppose that P = T ( X ) / ( R ) and Q = T ( Y ) / ( S ) are presentations ofthe operads P and Q by generators and relations. Then P ⊗ Q = T ( X ⊔ Y ) / ( R ⊔ S ⊔ IC ) , where IC are the relations γ ( x ; y , . . . , y ) − γ ( y ; x , . . . , x ) .σ k , l where x ∈ X ( k ) and y ∈ Y ( l ) are generators of P and Q respectively. In plain words, actions of two operads interchangeif and only if the actions of their generators interchange.Proof. It is su ffi cient to prove that if p ∈ P ( k ) satisfies (2) with both q ∈ Q ( l ), q ′ ∈ Q ( l ′ ), then p .α satisfies (2) with q .β for all permutations α ∈ S k , β ∈ S l , and p satisfies (2) with q ◦ s q ′ for any 1 ≤ s ≤ l . Both of these are easily checked by directinspection. (cid:3) In the presence of constants, Boardman–Vogt tensor products exhibit variouscollapsing properties, which are variations of the Eckmann–Hilton argument [6]in algebraic topology. Namely, the following result holds.
Proposition 1.5 (Fiedorowicz and Vogt [8, Prop. 3.8]) . Suppose that the operads P and Q are such that P (1) = Q (1) = { id } , and that the four components P (0) , Q (0) , P (2) , Q (2) are nonempty. We have P ⊗ Q (cid:27) u C om , where u C om is the operad of unital commutative associative algebras. In particular, for theoperad u A ss of unital associative algebras, we haveu A ss ⊗ u A ss (cid:27) u C om . Even in the set-up of this paper where constant operations are not allowed, un-expected phenomena arise. Let us consider the Boardman–Vogt square A ss ⊗ A ss of the operad A ss of non-unital associative algebras. It is generated by two asso-ciative products · and ⋆ satisfying the interchange law( a · a ) ⋆ ( a · a ) = ( a ⋆ a ) · ( a ⋆ a ) . MURRAY BREMNER AND VLADIMIR DOTSENKO
In [12], it was observed that an unexpected “commutativity” property holds in theoperad A ss ⊗ A ss . Proposition 1.6 (Kock [12, Prop. 2.3]) . In the Boardman–Vogt tensor product A ss ⊗ A ss,the following holds in arity : ( a ⋆ a ⋆ a ⋆ a ) · ( a ⋆ a ⋆ a ⋆ a ) · ( a ⋆ a ⋆ a ⋆ a ) · ( a ⋆ a ⋆ a ⋆ a ) = ( a ⋆ a ⋆ a ⋆ a ) · ( a ⋆ a ⋆ a ⋆ a ) · ( a ⋆ a ⋆ a ⋆ a ) · ( a ⋆ a ⋆ a ⋆ a ) . In particular, the underlying S -module of ( A ss ⊗ A ss )(16) is not free. The latter result was improved by the first author in his recent work withMadariaga [3].
Proposition 1.7 ( [3, Prop. 3.4 and Th. 4.2]) . In each arity n ≤ the underlying S n -module of ( A ss ⊗ A ss )( n ) is free. The underlying S -module of ( A ss ⊗ A ss )(9) is not free.In particular, the following relation implying that of Proposition 1.6 holds: ( a ⋆ a ) · ( a ⋆ a ⋆ a ⋆ a ) · ( a ⋆ a ⋆ a ) = ( a ⋆ a ) · ( a ⋆ a ⋆ a ⋆ a ) · ( a ⋆ a ⋆ a ) . Remark 1.8.
It is natural to ask what triggers the non-freeness of the underlying S -module of ( A ss ⊗ A ss )(9). One natural guess which is suggested by the resultsof this paper is that 9 = ·
3, where 3 is the smallest arity in which the operad A ss has a nontrivial relation. It would be interesting to determine whether ornot it is true that for operads P and Q whose underlying S -modules are free, theBoardman–Vogt tensor product P ⊗ Q has a free underlying S -module up to arity kl −
1, where k and l are, respectively, the smallest arities where P and Q haverelations.2. A geometric model for interchange of absolutely free operads It turns out that interchanging d absolutely free operads admits a remarkablegeometric representation. We describe it in three steps. First, we consider a partic-ular case when each of these operads is generated by one (not necessarily binary)generator, and define a map from the corresponding Boardman–Vogt tensor prod-uct into the operad of little d -rectangles. Next, we present a geometric constructionof an arbitrary absolutely free operad in terms of subdivisions of the unit intervalwith some extra labelling data. Finally, we consider a certain superposition ofthese two constructions to represent arbitrary Boardman–Vogt tensor products.Note that at this stage we do not claim this representation to be faithful; the proofof its faithfulness is one of the key results of this paper which appears in Section 3.2.1. Interchanging one-generated absolutely free operads.
Suppose that T ( X ), T ( X ), . . . , T ( X d ) are (reduced connected) absolutely free operads, and supposethat for each collection X k there exists an integer a k > X k ( a ) = S a k , a = a k , ∅ , a , a k , in other words, X k is freely generated by one element of arity a k .Let us consider a version of the little d -cubes operad which we shall call theoperad of little d -rectangles, and denote R ect d . By definition, its component ofarity n parametrises all possible ways to place n rectangular boxes of dimension d labelled 1 , . . . , n inside the unit cube so that their interiors are disjoint and theirfaces are parallel to the faces of the cube. The operadic composition γ ( c ; c , . . . , c m )of such configurations shrinks each of the configurations c i in the directions ofthe coordinate axes to ensure that the ambient unit cube fits exactly into the i -threctangle of c , and then glues the configuration of rectangles thus obtained in placeof that rectangle, adjusting the labels in the usual way. OARDMAN–VOGT TENSOR PRODUCTS OF ABSOLUTELY FREE OPERADS 7
Definition 2.1.
For collections X , . . . , X d as above, the cut operad C ( d ) X • is the sub-operad of R ect d generated by the operations ω k , 1 ≤ k ≤ d , where ω k is the configu-ration of rectangles [0 , k − × [0 , / a k ] × [0 , d − k , [0 , k − × [1 / a k , / a k ] × [0 , d − k ,... [0 , k − × [( a k − / a k , × [0 , d − k , numbered 1 , . . . , a k in the order they are listed here.Let us show that these operations interchange, which by Proposition 1.4 impliesthat there exists a surjective homomorphism T ( X ) ⊗ T ( X ) ⊗ · · · ⊗ T ( X d ) → C ( d ) X • . Lemma 2.2.
The operations ω i pairwise interchange.Proof. We see that the operation γ ( ω k ; ω l , . . . , ω l ) is obtained by first cutting the unitcube into a k equal parts in the direction of the k -th coordinate hyperplane, andthen cutting each of the parts thus obtained into a l equal parts in the directionsof the l -th coordinate hyperplane. The operation γ ( ω l ; ω k , . . . , ω k ) is obtained byfirst cutting the unit cube into a l equal parts in the direction of the l -th coordinatehyperplane, and then cutting each of the parts thus obtained into a k equal partsin the directions of the k -th coordinate hyperplane. The only di ff erence betweenthe two is the labelling of the interiors of the a k a l parts thus obtained, and thatdi ff erence is fixed by the permutation σ k , l . (cid:3) A geometric model for an absolutely free operad.
In this section, we presenta geometric model for a (reduced connected) absolutely free operad. Let us assumethat X is a free symmetric collection of finite sets with X (0) = X (1) = ∅ . Definition 2.3. An X -subdivision of arity n of a line segment [ a , b ] ⊂ R is definedby the following recursive rule: • The trivial subdivision consisting just of the segment [ a , b ] without any extradata is the only X -subdivision of arity 1. • Choose an integer m ≥
2, a partition n = n + · · · + n m , and an element w ∈ X ( m ).Let t k = (( m − k ) a + kb ) / m with k = , . . . , m − m − a , b ] into m equal parts. Let us label each of these m − w ,and impose arbitrary X -subdivisions of arities n , . . . , n m on the m segments[ a , t ] , [ t , t ] , . . . , [ t m − , b ] . All X -subdivisions of arity n are obtained in this way.In other words, we cut the segment into several equal parts, cut each of the partsin several equal parts, etc., each time labelling the cuts by a generator of the freeoperad of appropriate arity.This definition trivially implies that X -subdivisions of arity n of the unit interval[0 ,
1] are in one-to-one correspondence with elements of what is known as theabsolutely free algebra (or term algebra) for the signature X . In order to modeloperads, we should label interiors of the segments into which we subdivide theunit interval by integers 1 , . . . , n in all possible ways. The operad compositioncomes from substitution of subdivisions in the same way as in Definition 2.1; theonly di ff erence is that when inserting subdivisions we must also copy the labelsof cuts. The operad thus obtained is immediately seen to be isomorphic to theabsolutely free operad T ( X ). MURRAY BREMNER AND VLADIMIR DOTSENKO
Interchanging several absolutely free operads.
We shall combine the pre-vious two constructions to represent arbitrary tensor products of absolutely freeoperads. Let us now assume that X , . . . , X d are free symmetric collections of finitesets with X k (0) = X k (1) = ∅ for all 1 ≤ k ≤ d . Definition 2.4. An X • -subdivision of arity n of a d -dimensional rectangle R : = [ a , b ] × · · · × [ a d , b d ] ⊂ R d is defined by the following recursive rule: • The trivial subdivision consisting just of the rectangle R without any extra datais the only X • -subdivision of arity 1. • Choose an integer 1 ≤ k ≤ d , an integer m ≥
2, a partition n = n + · · · + n m , andan element w ∈ X k ( m ). Let t k , l = (( m − l ) a k + lb k ) / m , 1 ≤ l ≤ m −
1, and let β l : = R ∩ { x k = t k , l } be the m − k -th direction that divide R into m equal parts. Let us label points of each of these cuts by the element w , andimpose arbitrary X • -subdivisions of arities n , . . . , n m on the m parts[ a , b ] × · · · × [ a k , t k , ] , × · · · × [ a d , b d ] , [ a , b ] × · · · × [ t k , , t k , ] , × · · · × [ a d , b d ] ,... [ a , b ] × · · · × [ t k , m − , b k ] , × · · · × [ a d , b d ] . All X • -subdivisions of arity n are obtained in this way.In other words, we cut R into several equal parts in one of the directions ofcoordinate hyperplanes, cut each of the parts in several equal parts, etc., eachtime labelling the cuts by a generator of appropriate arity.Let us use this geometric construction to define an operad. This generalisesDefinition 2.1; the cut operad from that definition is tautologically isomorphic tothe cut operad below when all operads T ( X i ) are one-generated. Definition 2.5.
The d -dimensional cut operad C ( d ) X • has, as its arity n component, the X • -subdivisions of arity n of the unit d -cube [0 , d where interiors of the rectanglesinto which we subdivide the cube are labelled by integers 1 , . . . , n in all possibleways. The operad composition comes from substitution of labelled subdivisionsin the same way as in the paragraph following Definition 2.3.Let us establish that this construction gives a representation of the d -fold Boardman–Vogt tensor product T ( X ) ⊗ · · · ⊗ T ( X d ). Proposition 2.6.
Let us consider, for each x ∈ X k ( a k ) , the operation ω k , x ∈ C ( d ) X • ( a k ) thatcorresponds to the X • -subdivision of the unit cube [0 , k − × [0 , / a k ] × [0 , d − k , [0 , k − × [1 / a k , / a k ] × [0 , d − k ,. . . [0 , k − × [( a k − / a k , × [0 , d − k , where the parts are numbered , . . . , a k in the order they are listed here and all the a k − cuts are labelled x. The operations ω k , x for various choices of k and x generate the operad C ( d ) X • . Moreover, the operations ω k , x and ω l , y interchange for k , l. OARDMAN–VOGT TENSOR PRODUCTS OF ABSOLUTELY FREE OPERADS 9
Proof.
The first statement follows from the definition of the operad C ( d ) X • . The secondone is proved completely analogously to Lemma 2.2. (cid:3) Corollary 2.7.
There exists a surjective homomorphism T ( X ) ⊗ T ( X ) ⊗ · · · ⊗ T ( X d ) ։ C ( d ) X • .
3. P roof of the main theorem
In the previous section, we established that the cut operad C ( d ) X • is a homomorphicimage of T ( X ) ⊗ · · · ⊗ T ( X d ). We shall now establish that these operads areisomorphic. The proof of this result is obtained through an indirect argument. Tomake that argument more transparent, we start sketch a proof of the recurrencerelation for the numbers of elements in the cut operad representing d interchangingbinary operations, and then leave the combinatorics universe that was su ffi cientthus far and encode the more general recurrence relation homologically. The mainresult then follows from general properties of minimal resolutions of right modulesover operads.3.1. Sketch of enumeration of binary cuts in d dimensions. Counting binary cutsof the unit square is fairly straightforward. Let C (2) n be the number of distinct sub-divisions of the unit square into n pieces which are obtained by iterated bisectionsorthogonal to the coordinate axes. Since there are two di ff erent directions, a firstapproximation to the recurrence relation is the same as for the Catalan numbersbut with two di ff erent types of parentheses; namely, C (2) n = n − X i = C (2) i C (2) n − i , as we need to choose the direction of the first cut, and then subdivide the tworesulting rectangles. This involves double counting when we examine “full” bi-sections in two orthogonal directions corresponding to the interchange law. Thisdouble counting is easy to correct, and the actual recurrence relation is C (2) n = n − X i = C (2) i C (2) n − i − X n + n + n + n = n C (2) n C (2) n C (2) n C (2) n , which formalises the na¨ıve idea that the doubly counted subdivisions are thosewhere we make two perpendicular cuts, and then subdivide the four resultingsquares. If we denote by f ( t ) the generating function for the numbers C (2) n , thisrecurrence relation can be written in a concise form f ( t ) − f ( t ) + f ( t ) = t , which takes into account the initial condition C (2)1 =
1. These numbers are doc-umented in the OEIS entry “association types in 2-dimensional algebra” [15, Se-quence A236339].This argument easily generalises to the d -dimensional case. The correspondingrecurrence relation for the numbers C ( d ) n of distinct subdivisions of the unit cubeinto n parts becomes, by a similar inclusion-exclusion argument, C ( d ) n = d X k = " ( − k − dk ! X n ,..., n k ≥ n + ··· + n k = n k Y i = C ( d ) n i , or, in terms of the generating function f d ( t ) for the numbers C ( d ) n , d X k = dk ! f d ( t ) k = t . A rigorous proof of this relation follows from a more general result obtained byhomological methods, see Corollary 3.6 below.3.2.
A minimal resolution of the augmentation module.
In this section, we give ahomological statement which formalises the inclusion-exclusion argument abovefor the general cut operad. For that, we have to leave the set-theoretic context, andwork with linearisations of the corresponding set operads. Below, the notation C ( d ) X • is used for the linearised cut operad; we hope that it does not lead to a confusion. Lemma 3.1.
There exists a minimal resolution (cid:16) ( I ⊕ k X ) (cid:3) · · · (cid:3) ( I ⊕ k X d ) (cid:17) ◦ C ( d ) X • of the augmentation C ( d ) X • -module I by free right modules. Here the homological degree of allfactors I is equal to zero, and the homological degree of k X k is equal to for all ≤ k ≤ d.Proof. Let us denote, for brevity,(3) H ( d ) X • = ( I ⊕ k X ) (cid:3) · · · (cid:3) ( I ⊕ k X d ) . We shall place the collection H ( d ) X • in the same context as the d -dimensional cutoperad. Namely, for each term k X i (cid:3) · · · (cid:3) k X i s with i < · · · < i s obtained byexpanding the product (3), we choose a basis of elements w ⊗ · · · ⊗ w s , w j ∈ X i j ( π ( j ) ) , and associate with such element the X -subdivision of the unit cube into n = | π (1) | parts with hyperplanes parallel to { x i = } , then subdivision of each of the partsthus obtained into n = | π (2) | parts with hyperplanes parallel to { x i = } , etc. Welabel the j -th cut by w j , and also label the interiors of the d -dimensional rectanglesthus obtained using the orthogonal partitions: for 1 ≤ m ≤ n , . . . 1 ≤ m s ≤ n s , the( m , . . . , m s )-rectangle obtains the label which is the only element of π (1) m ∩ · · · ∩ π ( s ) m s .The collection H ( d ) X • ◦ C ( d ) X • can now be viewed as follows. Its basis elements areindexed by X • -subdivisions of the unit cube, where we take a “full” subdivisionfrom H ( d ) X • , and then insert inside each of its boxes a X • -subdivision of the unit d -cube. To make the distinction between two types of cuts clear, we shall refer tocuts coming from H ( d ) X • as black cuts, and the cuts coming from C ( d ) X • as white cuts,so that the X • -subdivisions we use are now two-coloured.Suppose that c is a basis element of H ( d ) X • ◦ C ( d ) X • . We shall call, for i = , . . . , k , thehyperplane α i = { x i = } a cut-through direction for c if there exists an integer n i ≥ v ∈ X i ( n i ) for which the hyperplane pieces parallel to α i which cutthe unit cube into n i equal parts are fully covered by cuts of c , and all the points ofthose cuts are labelled by the element v .We now define a structure of a chain complex on H ( d ) X • ◦ C ( d ) X • . For that, it isconvenient to assign to a two-coloured X • -subdivision c a basis element C = ( x v , i x v , i · · · · · x v r , i r ⊗ ξ w , j ∧ · · · ∧ ξ w s , j s ) c of H ( d ) X • ◦ C ( k ) X • , where α i , . . . , α i r are the cut-through directions for c with the respectivelabels v , . . . , v r , and α j , . . . , α j s are the black cuts of c with the respective labels w , . . . , w s . Here x v , i , v ∈ X i , are formal commuting variables, and ξ w , j , w ∈ X j , areformal anti-commuting variables. OARDMAN–VOGT TENSOR PRODUCTS OF ABSOLUTELY FREE OPERADS 11
We define a linear map d of homological degree − H ( d ) X • ◦ C ( d ) X • as follows. Fora basis element C as above, we put d ( C ) = s X p = ( − p − ( x v , i x v , i · · · · · x v r , i r x w p , j p ⊗ ξ w , j ∧ · · · ∧ ˆ ξ w p , j p ∧ · · · ∧ ξ w s , j s ) c ( p ) , where c ( p ) is the X • -subdivision for which the colour of the black cuts in the directionof the hyperplane α j p is changed from black to white. By a direct computation, d =
0, so H ( d ) X • ◦ C ( d ) X • acquires a chain complex structure.We also define a linear map h of homological degree 1 on H ( d ) X • ◦ C ( d ) X • as follows.For a basis element C as above, we put h ( C ) = r X q = ( x v , i x v , i · · · · · ˆ x v q , i q · · · · · x v r , i r ⊗ ξ v q , i q ∧ ξ w , j ∧ · · · ∧ ξ w s , j s ) c ( q ) , where c ( q ) is the X • -subdivision for which the colour of the hyperplance in the q -thcut-through direction for c is changed from white to black. By a direct computation,( dh + hd )( C ) = ( n b ( c ) + n w ( c )) C , where n b ( c ) is the number of the black cuts in c and n w ( c ) is the number of cut-through directions for c ; in fact, the formulas for the di ff erential and the map h are designed in such a way that they mimic the classical Koszul complex (thepolynomial de Rham complex). Note that for every p the subcollection (cid:16) H ( d ) X • ◦ C ( d ) X • (cid:17) p spanned by all basis elements for which n b ( c ) + n w ( c ) = p is closed under both d and h . For n >
0, let us define a map h ′ on (cid:16) H ( d ) X • ◦ C ( d ) X • (cid:17) p by the formula h ′ = p h .Clearly, dh ′ + h ′ d = id, and hence the chain complex (cid:16) H ( d ) X • ◦ C ( d ) X • (cid:17) p is acyclic. Also,we have (cid:16) H ( d ) X • ◦ C ( d ) X • (cid:17) (cid:27) I , as for all non-unary elements there is either at least oneblack cut, or at least one cut-through direction (or both).Finally, it is obvious that this resolution is minimal, as the di ff erential creates atleast one white cut, thus landing in the augmentation ideal. (cid:3) Faithfulness of the combinatorial representation of interchange.
We arefinally able to establish that the cut operad represents the Boardman–Vogt tensorproduct faithfully.
Theorem 3.2.
We have C ( d ) X • (cid:27) T ( X ) ⊗ · · · ⊗ T ( X d ) . Proof.
Let us move to the linear context, and replace the set operads C ( d ) X • and T ( X ) ⊗ · · · ⊗ T ( X d ) by their linearisations (keeping the same notation). FromLemma 3.1, we know that(( I ⊕ X ) (cid:3) · · · (cid:3) ( I ⊕ X d )) ◦ C ( d ) X • is a minimal resolution of I , the augmentation module for C ( d ) X • by free right C ( d ) X • -modules. It is well known that minimal resolutions of modules are defineduniquely up to an isomorphism, and that for a reduced connected k -linear op-erad O the generators of a minimal resolution of the augmentation O -module I in low homological degrees have easy interpretations in terms of that operad: thegenerators of homological degree 1 correspond to the minimal set of generators Y for O and the generators of homological degree 2 correspond to the minimal set ofrelations (minimal set of generators of the kernel of the surjection T ( Y ) ։ O ). Inour case, elements of homological degree 1 are indexed by choices of direction of [simultaneous] black cuts, and a label for such cut, which is not surprising: as weknow, the operad C ( d ) X • is generated by X • . Elements of homological degree 2 areindexed by choices of two directions of [simultaneous] black cuts and their labels,say p and q . The di ff erential of such an element is the di ff erence of two elementswhere the simultaneous black cuts in one of the two directions are made white.Such an element encodes a relation in the operad: its di ff erential is the di ff erenceof two equal elements where all the black cuts are made white; thus such an el-ement represents the corresponding interchange law between p and q . Thus, allrelations of C ( d ) X • follow from interchange laws between the generating operations.We now refer to the presentations of Boardman–Vogt tensor products given byProposition 1.4 to complete the proof. (cid:3) The following result shows that, by contrast with Propositions 1.6 and 1.7, nounexpected symmetries arise for interchanging absolutely free structures.
Corollary 3.3.
The underlying S n -module of ( T ( X ) ⊗ · · · ⊗ T ( X d ))( n ) is free.Proof. This follows from the trivial observation that the underlying S n -module of C ( d ) X • ( n ) is free, since all possible labelling of rectangles are allowed. (cid:3) Remark 3.4.
Let us mention an application of Theorem 3.2 to a more “classical”question stated in terms of varieties of algebras. It follows immediately fromthat theorem that in the variety of nonassociative algebras defined by d binaryoperations ⋆ , . . . , ⋆ d with no symmetry satisfying the d ( d − / X has a ”monomial basis” consisting of allsubdivisions of the unit d -cube into smaller d -rectangles with disjoint interiors byiterated bisections orthogonal to coordinate axes with additional labelling: eachof those d -rectangles should be given a label from X . The multiplication of these“labelled subdivisions” may then be defined geometrically as follows: If p and q are labelled subdivisions, then for 1 ≤ i ≤ d , the product p ⋆ i q is the labelledsubdivision (cid:16) p ∪ ( e i + q ) (cid:17) , where e i is the unit vector in the i th direction.Combining all the results we proved, we can now establish the key conceptualresult of this paper. Theorem 3.5.
Let T ( X ) , . . . , T ( X d ) be reduced connected absolutely free set operads.There exists a minimal resolution (cid:16) ( I ⊕ k X ) (cid:3) · · · (cid:3) ( I ⊕ k X d ) (cid:17) ◦ (cid:16) T ( X ) ⊗ · · · ⊗ T ( X d ) (cid:17) of the augmentation module I over (the linearised version of) the d-fold Boardman–Vogttensor product T ( X ) ⊗ · · · ⊗ T ( X d ) by free right modules. Here the homological degree ofall factors I is equal to zero, and the homological degree of X k is equal to for all ≤ k ≤ d.Proof. By Theorem 3.2, we have C ( d ) X • (cid:27) T ( X ) ⊗ · · · ⊗ T ( X d ), so the operad C ( d ) X • inLemma 3.1 can be replaced by T ( X ) ⊗ · · · ⊗ T ( X d ) , the d -fold Boardman–Vogt tensor product. (cid:3) For completeness, we state the general version of the inclusion-exclusion func-tional equation discussed in the introduction. To that end, we shall need the linearmap N from the algebra of Dirichlet series to the algebra of formal power seriesfor which N ( n − s ) = x n . OARDMAN–VOGT TENSOR PRODUCTS OF ABSOLUTELY FREE OPERADS 13
Corollary 3.6.
Let D ( d ) X • ( s ) be the Dirichlet generating function of Euler characteristics ofthe collection ( I ⊕ X ) (cid:3) · · · (cid:3) ( I ⊕ X d ) with the homological grading as described in Lemma 3.1 and Theorem 3.5. We haveD ( d ) X • ( s ) = d Y k = − X n ≥ dim X k ( n ) n ! n s . Furthermore, the power series g ( d ) X ( x ) = N ( D ( d ) X • ( s )) is the compositional inverse of the generating functionf ( d ) X • ( x ) = X n ≥ | T ( X ) ⊗ · · · ⊗ T ( X d )( n ) | n ! x n . Proof.
The first statement follows from the fact that the operation (cid:3) categorifies theproduct of Dirichlet series. The second statement is obtained by computing theEuler characteristics of the complex(( I ⊕ X ) (cid:3) · · · (cid:3) ( I ⊕ X d )) ◦ ( T ( X ) ⊗ · · · ⊗ T ( X d ))in two di ff erent ways, directly via the composition of collections and via the ho-mology (which is I in degree zero and arity one, and zero otherwise). (cid:3)
4. C oncluding remarks
The statement of Theorem 3.5 is aesthetically appealing, and, if we note that (cid:16) X ◦ T ( X ) → T ( X ) (cid:17) (cid:27) (cid:16) I ⊕ k X (cid:17) ◦ T ( X )is the minimal resolution of the augmentation T ( X )-module I by free right mod-ules, admits the following obvious generalisation to arbitrary set operads. Definition 4.1.
Let P , . . . , P d be reduced connected set operads, and suppose that V k ◦ P k is the underlying S -module of the minimal resolution of the augmentationmodule I over (the linearised version of) P k by right modules. We say that this d -tuple of operads has (cid:3) -multiplicative homology if there exists a minimal resolution( V (cid:3) · · · (cid:3) V d ) ◦ ( P ⊗ · · · ⊗ P d )of the augmentation module I over (the linearised version of) the d -fold Boardman–Vogt tensor product P ⊗ · · · ⊗ P d by free right modules.In this section, we discuss intuition behind this property, present two examplesshowing that it should not be expected to hold in general, and make a conjecturegeneralising Theorem 3.5 to a slightly wider class of examples.Let us first o ff er some intuition behind (cid:3) -multiplicativity. For that, let us con-sider the right module version of the Boardman–Vogt tensor product ˜ ⊗ (obtainedfrom the bimodule tensor product introduced by Dwyer and Hess [5]), whichsatisfies the following two properties crucial for us. Proposition 4.2.
Assume that we consider operads and modules in simplicial sets.1 (Dwyer and Hess [5, Th. 1.14]). For any S -modules V , . . . , V d we have ( V ◦ P ) ˜ ⊗ · · · ˜ ⊗ ( V d ◦ P d ) (cid:27) ( V (cid:3) · · · (cid:3) V d ) ◦ ( P ⊗ · · · ⊗ P d ) . P and Q be operads, and let G be a right Q -module. If G is cofibrant in the projective model structure, then the functor − ⊗ G (fromright P -modules to right P ⊗ Q -modules) is a left Quillen functor. Let us try to proceed, for the sake of the argument, as if these results wereavailable in the k -linear context. We consider, for each 1 ≤ k ≤ d , the dg module V k ◦ P k which is the minimal resolution of the augmentation P k -module I by freeright modules. By a result of Fresse [10, Prop. 14.2.2], a minimal resolution iscofibrant whenever V k and P k are cofibrant as S -modules. Thus, under this extraassumption it would follow from the left Quillen property that( V ◦ P ) ˜ ⊗ · · · ˜ ⊗ ( V d ◦ P d ) (cid:27) ( V (cid:3) · · · (cid:3) V d ) ◦ ( P ⊗ · · · ⊗ P d )is quasi-isomorphic to I , so the d -tuple of operads P , . . . , P d have (cid:3) -multiplicativehomology.There is, however, a big problem with this argument (and hence it is only goodas an intuitive explanation of (cid:3) -multiplicativity): Proposition 4.2 is not availablein the linear setting, and there is nothing on the level of simplicial sets for us tolinearise: for operads in simplicial sets there is no notion of augmentation. In fact,the following example shows that cofibrancy as S -modules is certainly not enough. Example 4.3.
Consider the symmetric operad A ss of non-unital associative alge-bras. Note that the underlying S -module of A ss is free, and that we have the Koszul(minimal) resolution A ss ¡ ◦ A ss of the augmentation module. However, it is clearthat the minimal resolution of the augmentation module for A ss ⊗ A ss cannot beof the form (cid:16) A ss ¡ (cid:3) A ss ¡ (cid:17) ◦ (cid:16) A ss ⊗ A ss (cid:17) , as computing Euler characteristics would have immediately implied freeness of theunderlying S -module of A ss ⊗ A ss , contradicting Propositions 1.6 and 1.7. Thus, (cid:3) -multiplicativity of homology fails in this case.The following example of failure of (cid:3) -multiplicativity for homology is lesssurprising, since the corresponding operads are not Σ -cofibrant on the level of sets. Example 4.4.
Let us take T ( X ) (cid:27) T ( X ) to be the free operad generated by one commutative binary operation. We have the minimal resolutions (cid:16) I ⊕ k X (cid:17) ◦ T ( X ) and (cid:16) I ⊕ k X (cid:17) ◦ T ( X )for the respective augmentation modules, but the minimal resolution of the aug-mentation module for T ( X ) ⊗ T ( X ) cannot be of the form (cid:16) ( I ⊕ X ) (cid:3) ( I ⊕ X ) (cid:17) ◦ (cid:16) T ( X ) ⊗ T ( X ) (cid:17) , since by a direct computation the space (cid:16) ( I ⊕ X ) (cid:3) ( I ⊕ X ) (cid:17) (4) is six-dimensional,and the space of generators of the minimal resolution in arity 4 is five-dimensional.Thus, (cid:3) -multiplicativity of homology fails in this case as well.We conclude with a conjecture that slightly strengthens Theorem 3.5. Conjecture 4.5.
Suppose that the reduced connected set operad O is free as an S -module,and that F = T ( X ) is a reduced connected absolutely free set operad. The pair of operads O and F has (cid:3) -multiplicative homology. R eferences [1] A. A sinowski , G. B arequet , T. M ansour , R. P inter : Cut equivalence of d -dimensional guillotinepartitions. Discrete Math.
331 (2014), 165–174.[2] J. B oardman , R. M. V ogt : Homotopy Invariant Algebraic Structures on Topological Spaces . LectureNotes in Mathematics, 347. Springer-Verlag, Berlin-New York, 1973.[3] M. B remner , S. M adariaga : Permutation of elements in double semigroups.
Semigroup Forum unn : Tensor product of operads and iterated loop spaces.
J. Pure Appl. Algebra
50 (1988), no. 3,237–258.
OARDMAN–VOGT TENSOR PRODUCTS OF ABSOLUTELY FREE OPERADS 15 [5] W. D wyer , K. H ess : The Boardman-Vogt tensor product of operadic bimodules.
Algebraic Topol-ogy: Applications and New Directions , pages 71–98. Contemporary Mathematics, 620. AmericanMathematical Society, Providence, RI, 2014.[6] B. E ckmann , P. H ilton : Group-like structures in general categories, I. Multiplications and comul-tiplications.
Mathematische Annalen
145 (1961 / vans : Endomorphisms of abstract algebras. Proceedings of the Royal Society of Edinburgh, SectionA: Mathematics
66 (1962), no. 1, 53–64.[8] Z. F iedorowicz , R. M. V ogt : An additivity theorem for the interchange of E n -structures. Advancesin Math.
273 (2015), 421–484.[9] R. G odement : Topologie Alg´ebrique et Th´eorie des Faisceaux , Vol. 1. Actualit´es Scientifiques et In-dustrielles, 1252, Publications de l’lnstitut de Math´ematique de I’Universit´e de Strasbourg XIII,Hermann, Paris, 1958, 283 pp.[10] B. F resse : Modules Over Operads and Functors . Lecture Notes in Mathematics, 1967, Springer Verlag,2009, 314 pp.[11] K. H ess : Private communication, April 7, 2017.[12] J. K ock : Note on commutativity in double semigroups and two-fold monoidal categories.
Journalof Homotopy and Related Structures oday , B. V allette : Algebraic Operads . Grundlehren der Mathematischen Wissenschaften, 346.Springer-Verlag, Berlin-Heidelberg, 2012.[14] M. M aia , M. M´ endez : On the arithmetic product of combinatorial species.
Discrete Math. loane , editor:
The On-Line Encyclopedia of Integer Sequences , published electronically at https://oeis.org .D epartment of M athematics and S tatistics , U niversity of S askatchewan , S askatoon , C anada E-mail address : [email protected] S chool of M athematics , T rinity C ollege D ublin , I reland , and D epartamento de M atem ´ aticas ,CINVESTAV-IPN, C ol . S an P edro Z acatenco , M´ exico , D.F., CP 07360, M exico E-mail address ::