Graded medial n -ary algebras and polyadic tensor categories
aa r X i v : . [ m a t h . R A ] J a n G RADED MEDIAL n - ARY ALGEBRAS AND POLYADIC TENSOR CATEGORIES S TEVEN D UPLIJ
Center for Information Technology (WWU IT), Universit¨at M¨unster, R¨ontgenstrasse 7-13D-48149 M¨unster, Deutschland A BSTRACT . Algebraic structures in which the property of commutativity is substituted by themediality property are introduced. We consider (associative) graded algebras and instead of al-most commutativity (generalized commutativity or ε -commutativity) we introduce almost mediality(”commutativity-to-mediality” ansatz). Higher graded twisted products and “deforming” brackets(being the medial analog of Lie brackets) are defined. Toyoda’s theorem which connects (univer-sal) medial algebras with abelian algebras is proven for the almost medial graded algebras introducedhere. In a similar way we generalize tensor categories and braided tensor categories. A polyadic(non-strict) tensor category has an n -ary tensor product as an additional multiplication with n ´ associators of the arity n ´ satisfying a ` n ` ˘ -gon relation, which is a polyadic analog of thepentagon axiom. Polyadic monoidal categories may contain several unit objects, and it is also possiblethat all objects are units. A new kind of polyadic categories (called groupal) is defined: they are closeto monoidal categories, but may not contain units: instead the querfunctor and (natural) functorialisomorphisms, the quertors, are considered (by analogy with the querelements in n -ary groups). Thearity-nonreducible n -ary braiding is introduced and the equation for it is derived, which for n “ co-incides with the Yang-Baxter equation. Then, analogously to the first part of the paper, we introduce“medialing” instead of braiding and construct “medialed” polyadic tensor categories. E-mail address : [email protected]; http://ivv5hpp.uni-muenster.de/u/douplii . Date : of start July 17, 2019.
Date : of completion January 11, 2020.
Total : 107 references, 20 diagrams.2010
Mathematics Subject Classification.
RADED MEDIAL n - ARY ALGEBRAS AND POLYADIC TENSOR CATEGORIES C ONTENTS
1. I
NTRODUCTION
22. P
RELIMINARIES
LMOST MEDIAL BINARY GRADED ALGEBRAS
EDIAL N - ARY ALGEBRAS
LMOST MEDIAL N - ARY GRADED ALGEBRAS n -ary brackets 146. T OYODA ’ S THEOREM FOR ALMOST MEDIAL ALGEBRAS
INARY TENSOR CATEGORIES
OLYADIC TENSOR CATEGORIES N -ary coherence 239. N - ARY UNITS , UNITORS AND QUERTORS
RAIDED TENSOR CATEGORIES
EDIALED POLYADIC TENSOR CATEGORIES
ONCLUSIONS
EFERENCES
NTRODUCTION
The commutativity property and its “breaking” are quite obvious and unique for binary algebraicstructures, because the permutation group S has only one non-identity element. If the operationis n -ary however, then one has n ! ´ non-identity permutations from S n , and the uniqueness islost. The standard way to bring uniqueness to an n -ary structure is by restricting to a particular n -ary commutation by fixing one chosen permutation using external (sometimes artificial) criteria.We introduce a different, canonical approach: to use another property which would be unique bydefinition, but which can give commutativity in special cases. Mediality M URDOCH [1939] (actingon n elements) is such a property which can be substituted for commutativity (acting on n elements)in the generators/relations description of n -ary structures. For n “ , any medial magma is acommutative monoid, and moreover for binary groups commutativity immediately follows frommediality. - 2 -inary gradation 2. I NTRODUCTION
In the first part of our paper we consider n -ary graded algebras and propose the fol-lowing idea: instead of considering the non-unique commutativity property and its “break-ing”, to investigate the unique property of mediality and its “breaking”. We exploit this“commutativity-to-mediality” ansatz to introduce and study almost medial n -ary graded alge-bras by analogy with almost commutative algebras (generalized or ε -commutative graded al-gebras) R ITTENBERG AND W YLER [1978], S
CHEUNERT [1979], and β -commutative algebrasB AHTURIN ET AL . [2003] (see, also, B
ONGAARTS AND P IJLS [1994], C
OVOLO AND M ICHEL [2016], M
ORIER -G ENOUD AND O VSIENKO [2010]). We prove an analogue of Toyoda’s theorem,which originally connected medial algebras with abelian algebras T
OYODA [1941], for almost me-dial n -ary graded algebras, which we introduce. Note that almost co-mediality for polyadic bialge-bras was introduced earlier in D UPLIJ [2018b]. For other (binary) generalizations of grading, see,e.g. D
ALZOTTO AND S BARRA [2008], E
LDUQUE [2006], N
YSTEDT [2017].The second part of the paper is devoted to a similar consideration of tensor categories M AC L ANE [1971], E
TINGOF ET AL . [2015]. We define polyadic tensor categories by considering an n -ary ten-sor product (which may not be iterated from binary tensor products) and n -ary coherence conditionsfor the corresponding associators. The peculiarities of polyadic semigroupal and monoidal cate-gories are studied and the differences from the corresponding binary tensor categories are outlined.We introduce a new kind of tensor categories, polyadic nonunital “groupal” categories, which con-tain a “querfunctor” and “quertors” (similar to querelements in n -ary groups D ¨ ORNTE [1929], P
OST [1940]). We introduce arity-nonreducible n -ary braidings and find the equation for them that in thebinary case turns into the Yang-Baxter equation in the tensor product form. Finally, we apply the“commutativity-to-mediality” ansatz to braided tensor categories J OYAL AND S TREET [1993] andintroduce “medialing” and corresponding “medialed” tensor categories.The proposed “commutativity-to-mediality” ansatz can lead to medial n -ary superalgebras andLie superalgebras, as well as to a medial analog of noncommutative geometry.2. P RELIMINARIES
The standard way to generalize the commutativity is using graded vector spaces and cor-responding algebras together with the commutation factor defined on some abelian gradinggroup (see, e.g. R
ITTENBERG AND W YLER [1978], S
CHEUNERT [1979] and B
OURBAKI [1998],N
ASTASESCU AND V AN O YSTAEYEN [2004]). First, recall this concept from a slightly differentviewpoint.2.1.
Binary gradation.
Let A ” A p q “ x A | µ , ν ; λ y be an associative (binary) algebra over afield k (having unit P k and zero P k ) with unit e (i.e. it is a unital k -algebra) and zero z P A .Here A is the underling set and µ : A b A Ñ A is the (bilinear) binary multiplication (which wewrite as µ r a, b s , a, b P A ), usually in the binary case denoted by dot µ ” p¨q , and ν : A b A Ñ A isthe (binary) addition denoted by p`q , and a third (linear) operation λ is the action λ : K b A Ñ A (widely called a “scalar multiplication”, but this is not always true, as can be seen from the polyadiccase D UPLIJ [2019]).Informally, if A as a vector space can be decomposed into a direct sum, then one can introducethe binary gradation concept : each element a P A is endowed by an additional characteristic, its gradation denoted by a prime a showing to which subspace it belongs, such that a belongs to adiscrete abelian group (initially N simply to “enumerate” the subspaces, and this can be furthergeneralized to a commutative semigroup). This group is called the binary grading group G “x G, ν y , and usually its operation is written as plus ν ” p` q , and the neutral element by . Denotethe subset of homogeneous elements of degree a P G by A a S CHEUNERT [1979], D
ADE [1980].- 3 -. P
RELIMINARIES
Almost commutativity
Definition 2.1.
An associative algebra A is called a binary graded algebra over k (or G -algebra A G ), if the algebra multiplication µ respects the gradation i.e. µ r A a , A b s ” A a ¨ A b Ď A a ` b , @ a , b P G, ( )where equality corresponds to strong gradation .If there exist invertible elements of each degree a P G , then A is called a cross product , and ifall non-zero homogeneous elements are invertible, A is a graded division algebra D ADE [1980].Homogeneous (binary) morphisms ϕ : A G Ñ B G preserve the grading ϕ p A a q Ă B a , @ a P G , and the kernel of ϕ is an homogeneous ideal. The corresponding class of G -algebras and thehomogeneous morphisms form a category of G -algebras G - Alg (for details, see, e.g. B
OURBAKI [1998], D
ADE [1980]).2.2.
Almost commutativity.
The graded algebras have a rich multiplicative structure, because ofthe possibility to deform (or twist) the algebra product µ by a function depending on the gradation.Let us consider the twisting function ( twist factor ) τ : G ˆ G Ñ k . Definition 2.2. A twisted graded product µ p τ q is defined for homogeneous elements by µ p τ q r a, b s “ τ p a , b q µ r a, b s , a, b P A ; a , b P G. ( ) Proposition 2.3.
If the twisted algebra A A | µ p τ q E is associative, then the twisting function becomesa 2-cocycle τ ÞÑ σ : G ˆ G Ñ k ˆ on the abelian group G satisfying σ p a , b q σ p a ` b , c q “ σ p a , b ` c q σ p b , c q , a , b , c P G. ( ) Proof.
The result follows from the binary associativity condition for µ p σ q . (cid:3) Example . An example of a solution to the functional equation ( ) is σ p a , b q “ p exp p a qq b .The classes of σ form the (Schur) multiplier group S CHEUNERT [1979], and for further propertiesof σ and a connection with the cohomology classes H p G, k q , see, e.g., C OVOLO AND M ICHEL [2016].In general, the twisted product ( ) can be any polynomial in algebra elements. Nevertheless, thespecial cases where µ p ε q r a, b s becomes a fixed expression for elements a, b P A are important. Definition 2.5.
If the twisted product coincides with the opposite product for all a, b P A , we callthe twisting function a τ ÞÑ ε : G ˆ G Ñ k ˆ , such that µ p ε q r a, b s “ µ r b, a s , or ε p a , b q a ¨ b “ b ¨ a, @ a, b P A, a , b P G. ( ) Definition 2.6.
A binary algebra A p ε q for which the twisted product coincides with the oppositeproduct ( ), is called -level almost commutative ( ε -commutative). Assertion 2.7.
If the algebra for which ( ) takes place is associative, the 0-level commutationfactor ε satisfies the relations ε p a , b q ε p b , a q “ , ( ) ε p a ` b , c q “ ε p a , c q ε p b , c q , ( ) ε p a , b ` c q “ ε p a , b q ε p a , c q , a , b , c , d P G. ( ) Proof.
The first relation ( ) follows from permutation in ( ) twice. The next ones follow frompermutation in two ways: for ( ) a ¨ b ¨ c ÞÑ a ¨ c ¨ b ÞÑ c ¨ a ¨ b and p a ¨ b q ¨ c ÞÑ c ¨ p a ¨ b q , and for( ) a ¨ b ¨ c ÞÑ b ¨ a ¨ c ÞÑ b ¨ c ¨ a and a ¨ p b ¨ c q ÞÑ p b ¨ c q ¨ a , using ( ). (cid:3) - 4 -ower of higher level commutation brackets 2. P RELIMINARIES
In a more symmetric form this is ε p a ` b , c ` d q “ ε p a , c q ε p b , c q ε p a , d q ε p b , d q . ( )The following general expression ε ˜ j a ÿ i a “ a i a , j b ÿ i b “ b i b ¸ “ j a ź i a “ j b ź i b “ ε ` a i a , b i b ˘ , a i a , b i b P G, i a , i b , j a , j b P N , ( )can be written. In the case of equal elements we have ε p j a a , j b b q “ p ε p a , b qq j a j b . ( ) Remark . Recall that the standard commutation factor ε : G ˆ G Ñ k ˆ of an almostcommutative ( ε -commutative or ε -symmetric) associative algebra is defined in a different wayR ITTENBERG AND W YLER [1978], S
CHEUNERT [1979] ε p a , b q b ¨ a “ a ¨ b. ( )Comparing with ( ) we have ε p a , b q “ ε p b , a q , @ a , b P G. ( )2.3. Tower of higher level commutation brackets.
Let us now construct the tower of higher levelcommutation factors and brackets using the following informal reasoning. We “deform” the almostcommutativity relation ( ) by a function L : A ˆ A Ñ A as ε p a , b q a ¨ b “ b ¨ a ` L p ε q p a, b q , @ a, b P A, a , b P G, ( )where ε p a , b q is the -level commuting factor satisfying ( )–( ).Consider the function (bracket) L p ε q p a, b q as a multiplication of a new algebra A L “ A A | µ p ε ,L q “ L p ε q p a, b q E ( )called a -level bracket algebra . Then ( ) can be treated as its “representation” by the associativealgebra A . Proposition 2.9.
The algebra A L is almost commutative with the commutation factor ` ´ ε ´ ˘ .Proof. Using ( ) and ( )–( ) we get ε p b , a q L p ε q p a, b q ` L p ε q p b, a q “ , which can berewritten in the almost commutativity form ( ) as p´ ε p b , a qq L p ε q p a, b q “ L p ε q p b, a q . It followsfrom ( ) that ` ´ ε ´ p a , b q ˘ L p ε q p a, b q “ L p ε q p b, a q . ( ) (cid:3) The triple identity for L p ε q p a, b q can be obtained using ( )–( ), ( ) and ( ) ε p c , a q L p ε q ´ L p ε q p a, b q , c ¯ ` ε p a , b q L p ε q ´ L p ε q p b, c q , a ¯ ( ) ` ε p b , c q L p ε q ´ L p ε q p c, a q , b ¯ “ , @ a, b, c P A, a , b , c P G. In the more symmetric form using ( ) we have ε ` c , b ˘ ε ` d , a ˘ L p ε q ´ L p ε q p a, b q , L p ε q p c, d q ¯ ` ε ` d , c ˘ ε ` a , b ˘ L p ε q ´ L p ε q p b, c q , L p ε q p d, a q ¯ ε ` a , d ˘ ε ` b , c ˘ L p ε q ´ L p ε q p c, d q , L p ε q p a, b q ¯ ` ε ` b , a ˘ ε ` c , d ˘ L p ε q ´ L p ε q p d, a q , L p ε q p b, c q ¯ “ . ( ) - 5 -. P RELIMINARIES
Tower of higher level commutation bracketsBy analogy with ( ) we successively further “deform” ( ) then introduce “deforming” func-tions and higher level commutation factors in the following way.
Definition 2.10.
The k -level commutation factor ε k p a , b q is defined by the following “difference-like” equations ε p a , b q L p ε q p a, b q “ L p ε q p b, a q ` L p ε ,ε q p a, b q , ( ) ε p a , b q L p ε ,ε q p a, b q “ L p ε ,ε q p b, a q ` L p ε ,ε ,ε q p a, b q , ( )... ε k p a , b q L p ε ,ε ,...,ε k ´ q k ´ p a, b q “ L p ε ,ε ,...,ε k ´ q k ´ p b, a q ` L p ε ,ε ,...,ε k q k p a, b q . ( ) Definition 2.11. k -level almost commutativity is defined by the vanishing of the last “deforming”function L p ε ,ε ,...,ε k q k p a, b q “ , @ a, b P A, ( )and can be expressed in a form analogous to ( ) ε k p a , b q L p ε ,ε ,...,ε k ´ q k ´ p a, b q “ L p ε ,ε ,...,ε k ´ q k ´ p b, a q . ( ) Proposition 2.12.
All higher level “deforming” functions L p ε ,ε ,...,ε i q i , i “ , . . . , k can be expressedthrough L p ε q p a, b q from ( ) multiplied by a combination of the lower level commutation factors ε i p a , b q , i “ , . . . , k .Proof. This follows from the equations ( )–( ). (cid:3) The first such expressions are L p ε ,ε q p a, b q “ r ε p a , b q ` ε p b , a qs L p ε q p a, b q , ( ) L p ε ,ε ,ε q p a, b q “ r ε p a , b q p ε p a , b q ` ε p b , a qq ` ε p a , b q ε p b , a q ` s L p ε q p a, b q , ( )...Recall the definition of the ε -Lie bracket S CHEUNERT [1979] r a, b s ε “ a ¨ b ´ ε p a , b q b ¨ a, @ a, b P A, a , b P G. ( ) Assertion 2.13.
The -level “deforming” function L p ε q p a, b q is the ε -twisted ε -Lie bracket L p ε q p a, b q “ ε p a , b q r a, b s ε “ ε . ( ) Proof.
This follows from ( ) and ( ). (cid:3) Remark . The relations ( ) and ( ) are analogs of the ε -Jacobi identity of the ε -Lie algebraS CHEUNERT [1979].
Corollary 2.15.
All higher level “deforming” functions L p ε ,ε ,...,ε i q i , i “ , . . . , k can be expressedthrough the twisted ε -Lie bracket ( ) with twisting coefficients.In search of a polyadic analog of almost commutativity, we will need some additional concepts,beyond the permutation of two elements (in the binary case), called commutativity, and various sumsof permutations (of n elements, in n -ary case, which are usually non-unique).Instead we propose to consider a new concept, polyadic mediality (which gives a unique relationbetween n elements in n -ary case), as a polyadic inductive generalization of commutativity. We- 6 -edial binary magmas and quasigroups 2. P RELIMINARIES then twist the multiplication by a gradation (as in the binary case above) to obtain the polyadicversion of almost commutativity as almost mediality . However, let us first recall the binary andpolyadic versions of the mediality property.2.4.
Medial binary magmas and quasigroups.
The mediality property was introduced as a gener-alization of the associative law for quasigroups, which are a direct generalization of abelian groupsM
URDOCH [1939]. Other names for mediality are entropicity, bisymmetry, alternaton and abelian-ness (see, e.g., A CZ ´ EL [1948], J EEK AND K EPKA [1983], E
VANS [1963]).Let M “ x M | µ y be a binary magma (a closed set M with one binary operation µ without anyadditional properties, also called a (Hausmann-Ore) groupoid ). Definition 2.16.
A (binary) magma M is called medial , if µ r µ r a, b s , µ r c, d ss “ µ r µ r a, c s , µ r b, d ss , @ a, b, c, d P M. ( ) Definition 2.17.
We call the product of elements in the r.h.s. of ( ) medially symmetric to thel.h.s. product.Obviously, if a magma M contains a neutral element (identity) e P M , such that µ r a, e s “ µ r e, a s “ a , @ a P M , then M is commutative µ r a, b s “ µ r b, a s , @ a, b P M . Therefore,any commutative monoid is an example of a medial magma. Numerous different kinds of magmaand their classification are given in J EEK AND K EPKA [1983]. If a magma M is cancellative( µ r a, b s “ µ r a, c s ñ b “ c , µ r a, c s “ µ r b, c s ñ a “ b , @ a, b, c P M ), it is a binaryquasigroup Q “ x Q | µ y for which the equations µ r a, x s “ b , µ r y, a s “ b , @ a, b P Q , havea unique solution H OWROYD [1973]. Moreover S
HOLANDER [1949], every medial cancellativemagma can be embedded in a medial quasigroup (satisfying ( )), and the reverse statement isalso true J E ˇ ZEK AND K EPKA [1993]. For a recent comprehensive review on quasigroups(includingmedial and n -ary ones), see, e.g. S HCHERBACOV [2017], and references therein.The structure of medial quasigroups is determined by the (Bruck-Murdoch-)Toyoda theoremB
RUCK [1944], M
URDOCH [1941], T
OYODA [1941].
Theorem 2.18 (Toyoda theorem) . Any medial quasigroup Q medial “ x Q | µ y can be presented inthe linear (functional) form µ r a, b s “ ν r ν r ϕ p a q , ψ p b qs , c s “ ϕ p a q ` ψ p b q ` c, @ a, b, c P Q, ( ) where x Q | ν ” p`qy is an abelian group and ϕ, ψ : Q Ñ Q are commuting automophisms ϕ ˝ ψ “ ψ ˝ ϕ , and c P Q is fixed. If Q medial has an idempotent element (denoted by ), then µ r a, b s “ ν r ϕ p a q , ψ p b qs “ ϕ p a q ` ψ p b q , @ a, b P Q, ( )It follows from the Toyoda theorem, that medial quasigroups are isotopic to abelian groups, and theirstructure theories are very close B RUCK [1944], M
URDOCH [1941].The mediality property ( ) for binary semigroups leads to various consequences C
HRISLOCK [1969], N
ORDAHL [1974]. Indeed, every medial semigroup S medial “ x S | µ y is a Putcha semi-group ( b P S aS ñ b m P S a S , @ a, b P S , m P N , S “ S Y t u ), and therefore S medial canbe decomposed into the semilattice ( a “ a ^ ab “ ba , @ a, b P S ) of Archimedean semigroups ( @ a, b P S , D m, k P N , a m “ S bS ^ b k “ S aS ^ ab “ ba ). If a medial semigroup S medial isleft (right) cancellative , ab “ ac ñ b “ c ( ba “ ca ñ b “ c ), then it is left (right) commutative This should not be confused with the Brandt groupoid or virtual group. - 7 -. P
RELIMINARIES abc “ bac ( abc “ acb ), @ a, b, c P S and left (right) separative , ab “ a ^ ba “ b , @ a, b P S ( ab “ b ^ ba “ a ) (for a review, see, N AGY [2001]).For a binary group x G | µ y mediality implies commutativity, because, obviously, abcd “ acbd ñ bc “ cb , @ a, b, c, d P G . This is not the case for polyadic groups, where mediality implies semicom-mutativity only (see e.g., G ŁAZEK AND G LEICHGEWICHT [1982], D OG [2016]).Let A “ x A | µ , ν ; λ y be a binary k -algebra, not necessarily unital, cancellative and associa-tive. Then mediality provides the corresponding behavior which depends on the properties of the“vector multiplication” µ . For instance, for unital cancellative and associative algebras, medialityimplies commutativity, as for groups G ŁAZEK AND G LEICHGEWICHT [1982].3. A
LMOST MEDIAL BINARY GRADED ALGEBRAS
Consider an associative binary algebra A over a field k . We introduce a weaker version of grada-tion than in ( ). Definition 3.1.
An associative algebra A is called a binary higher graded algebra over k , if thealgebra multiplication of four ( “ ) elements respects the gradation µ r A a , A b , A c , A d s ” A a ¨ A b ¨ A c ¨ A d Ď A a ` b , @ a , b , c , d P G, ( )where equality corresponds to strong higher gradation .Instead of ( ) let us introduce the higher twisting function ( higher twist factor ) for four ( “ )elements τ : G ˆ Ñ k . Definition 3.2. A twisted (binary) higher graded product µ p τ q is defined for homogeneous elementsby µ p τ q r a, b, c, d s “ τ p a , b , c , d q a ¨ b ¨ c ¨ d, a, b, c, d P A ; a , b , c , d P G. ( )An analog of (total) associativity for the twisted binary higher graded product operation µ p τ q isthe following condition on seven elements ( “ ¨ ´ ) for all a, b, c, d, t, u, v P A µ p τ q ” µ p τ q r a, b, c, d s , t, u, v ı “ µ p τ q ” a, µ p τ q r b, c, d, t s , u, v ı “ µ p τ q ” a, b, µ p τ q r c, d, t, u s , v ı “ µ p τ q ” a, b, c, µ p τ q r d, t, u, v s ı . ( ) Proposition 3.3.
If the twisted higher graded product satisfies the higher analog of associativitygiven by ( ), then the twisting function becomes a higher analog of the cocycle ( ) τ ÞÑ σ : G ˆ Ñ k ˆ on the abelian group G satisfying for all a , b , c , d , t , u , v P G σ p a , b , c , d q σ p a ` b ` c ` d , t , u , v q “ σ p b , c , d , t q σ p a , b ` c ` d ` t , u , v q ( ) “ σ p c , d , t , u q σ p a , b , c ` d ` t ` u , v q “ σ p d , t , u , v q σ p a , b , c , d ` t ` u ` v q . Next we propose a medial analog of almost commutativity as follows. Instead of deformingcommutativity by the grading twist factor ε as in ( ), we deform the mediality ( ) by thehigher twisting function τ ( ). Definition 3.4.
If the higher twisted product coincides with the medially symmetric product (see( )) for all a, b P A , we call the twisting function a - level mediality factor τ ÞÑ ρ : G ˆ Ñ k ˆ ,such that (cf. ( )) µ p ρ q r a, b, c, d s “ µ r a, c, b, d s , or ( ) ρ p a , b , c , d q a ¨ b ¨ c ¨ d “ a ¨ c ¨ b ¨ d, @ a, b, c, d P A, a , b , c , d P G. ( )- 8 -ower of higher binary mediality brackets 3. A LMOST MEDIAL BINARY GRADED ALGEBRAS
From ( ) follows the normalization condition for the mediality factor ρ p a , a , a , a q “ , @ a P G. ( ) Definition 3.5.
A binary algebra A p ρ q “ x A | µ , ν y for which the higher twisted product coin-cides with the medially symmetric product µ p ρ q r a, b, c, d s “ µ r a, c, b, d s ( ), is called a -levelalmost medial ( ρ -commutative) algebra. Proposition 3.6.
If the algebra for which ( ) holds is associative, the 0-level mediality factor ρ satisfies the relations ρ p a , b , c , d q ρ p a , c , b , d q “ , a , b , c , d , f , g , h P G, P k , ( ) ρ p a , c ` d ` f ` g , b , h q “ ρ p a , c , b , d q ρ p c , d , b , f q ρ p d , f , b , g q ρ p f , g , b , h q , ( ) ρ p a , g , b ` c ` d ` f , h q “ ρ p a , g , b c q ρ p c , g , d , f q ρ p d , g , f , h q ρ p b , g , c , d q . ( ) Proof.
As in ( ), the relation ( ) follows from applying ( ) twice. The next ones follow frompermutation in two ways using ( ): for ( ) a ¨ b ¨ p c ¨ d ¨ f ¨ g q ¨ h ÞÑ a ¨ p c ¨ d ¨ f ¨ g q ¨ b ¨ h, a, c, d, f, g, b, h P A, ( ) a ¨ b ¨ c ¨ d ¨ f ¨ g ¨ h ÞÑ a ¨ c ¨ b ¨ d ¨ f ¨ g ¨ h ÞÑ a ¨ c ¨ d ¨ b ¨ f ¨ g ¨ h ÞÑ a ¨ c ¨ d ¨ f ¨ b ¨ g ¨ h ÞÑ a ¨ c ¨ d ¨ f ¨ g ¨ b ¨ h, ( )and for ( ) a ¨ p b ¨ c ¨ d ¨ f q ¨ g ¨ h ÞÑ a ¨ g ¨ p b ¨ c ¨ d ¨ f q ¨ h, ( ) a ¨ b ¨ c ¨ d ¨ f ¨ g ¨ h ÞÑ a ¨ b ¨ c ¨ d ¨ g ¨ f ¨ h ÞÑ a ¨ b ¨ c ¨ g ¨ d ¨ f ¨ h ÞÑ a ¨ b ¨ g ¨ c ¨ d ¨ f ¨ h ÞÑ a ¨ g ¨ b ¨ c ¨ d ¨ f ¨ h. ( ) (cid:3) Assertion 3.7.
If the -level almost medial algebra A p ρ q is cancellative, then it is isomorphic to analmost commutative algebra.Proof. After cancellation by a and d in ( ), we obtain ε p b , c q b ¨ c “ c ¨ b , where ε p b , c q “ ρ p a , b , c , d q . ( )In case A p ρ q is unital, one can choose ε p b , c q “ ρ p e , b , c , e q ” ρ p , b , c , q , since theidentity e P A is zero graded. (cid:3) Tower of higher binary mediality brackets.
By analogy with ( ), let us deform the medialtwisted product µ p ρ q ( ) by the function M p ρ q : A ˆ A ˆ A ˆ A Ñ A as follows ρ p a , b , c , d q a ¨ b ¨ c ¨ d “ a ¨ c ¨ b ¨ d ` M p ρ q p a, b, c, d q , @ a, b, c, d P A, a , b , c , d P G, ( )where ρ is a -level mediality factor ( ) which satisfies ( )–( ).Let us next introduce a -ary multiplication µ p ρ ,M q r a, b, c, d s “ M p ρ q p a, b, c, d q , @ a, b, c, d P A . Definition 3.8. A -ary algebra A p ρ ,M q “ A A | µ p ρ ,M q E ( )is called a -level medial bracket algebra . - 9 -. A LMOST MEDIAL BINARY GRADED ALGEBRAS
Proposition 3.9.
The -ary algebra A p ρ ,M q is almost medial with the mediality factor ` ´ ρ ´ ˘ .Proof. Using ( ) and ( )–( ) we get ρ p a , c , b , d q M p ρ q p a, b, c, d q ` M p ρ q p a, c, b, d q “ ,which can be rewritten in the almost medial form ( ) as p´ ρ p a , c , b , d qq M p ρ q p a, b, c, d q “ M p ρ q p a, c, b, d q . From ( ) we get ` ´ ρ ´ p a , b , c , d q ˘ M p ρ q p a, b, c, d q “ M p ρ q p a, c, b, d q . ( ) (cid:3) Let us “deform” ( ) again successively by introducing further “deforming” functions M k andhigher level mediality factors ρ k : G ˆ G ˆ G ˆ G Ñ k in the following way. Definition 3.10.
The k -level mediality factor ρ k p a , b , c , d q is defined by the following “difference-like” equations ρ p a , b , c , d q M p ρ q p a, b, c, d q “ M p ρ q p a, c, b, d q ` M p ρ ,ρ q p a, b, c, d q , ( ) ρ p a , b , c , d q M p ρ q p a, b, c, d q “ M p ρ q p a, c, b, d q ` M p ρ ,ρ ,ρ q p a, b, c, d q , ( )... ρ k p a , b , c , d q M p ρ ,ρ ,...,ρ k ´ q k ´ p a, b, c, d q “ M p ρ ,ρ ,...,ρ k ´ q k ´ p a, c, b, d q ` M p ρ ,ρ ,...,ρ k q k p a, b, c, d q , ( ) @ a, b, c, d P A, a , b , c , d P G. Definition 3.11. k -level almost mediality is defined by the vanishing of the last “deforming” medialfunction M p ρ ,ρ ,...,ρ k q k p a, b, c, d q “ , @ a, b, c, d P A, ( )and can be expressed in a form analogous to ( ) and ( ) ρ k p a , b , c , d q M p ρ ,ρ ,...,ρ k ´ q k ´ p a, b, c, d q “ M p ρ ,ρ ,...,ρ k ´ q k ´ p a, c, b, d q . ( ) Proposition 3.12.
The higher level “deforming” functions M p ρ ,ρ ,...,ρ i q i p a, b, c, d q , i “ , . . . , k canbe expressed through M p ρ q p a, b, c, d q from ( ) multiplied by a combination of the lower levelmediality factors ρ i p a , b , c , d q , i “ , . . . , k .Proof. It follows from the equations ( )–( ). (cid:3)
4. M
EDIAL N - ARY ALGEBRAS
We now extend the concept of almost mediality from binary to polyadic ( n -ary) algebras in theunique way which uses the construction from the previous section.Let A p n q “ x A | µ n , ν y be an associative n -ary algebra (with n -ary linear multiplication A b n Ñ A ) over a field k with (possible) polyadic unit e (then A p n q a unital k -algebra) definedby µ n r e n ´ , a s “ a , @ a P A (where a can be on any place) and (binary) zero z P A . We restrictourselves (as in M ICHOR AND V INOGRADOV [1996], G
OZE ET AL . [2010]) by the binary addition ν : A b A Ñ A which is denoted by p`q (for more general cases, see D UPLIJ [2019]). Now polyadic (total) associativity G OZE ET AL . [2010] can be defined as a kind of invariance D
UPLIJ [2018a] µ n “ a , µ n “ b p n q ‰ , c ‰ “ invariant, ( )- 10 -. M EDIAL N - ARY ALGEBRAS where a, c are ( linear ) polyads (sequences of elements from A ) of the necessary length P OST [1940], b p n q is a polyad of the length n , and the internal multiplication can be on any place. To describethe mediality for arbitrary arity n we need the following matrix generalization of polyads (as wasimplicitly used in D UPLIJ [2018a, 2019]).
Definition 4.1. A matrix ( n -ary) polyad ˆ A p n q ” ˆ A p n ˆ n q of size n ˆ n is the sequence of n elements ˆ A p n ˆ n q “ p a ij q P A b n , i, j “ , . . . , n , and their product A p µ q n : A b n Ñ A contains n ` of n -arymultiplications µ n , which can be written as (we use hat for matrices of arguments, even informally) A p µ q n ” p µ n q ˝p n ` q ” ˆ A p n q ı “ µ n »——– µ n r a , a , . . . , a n s ,µ n r a , a , . . . , a n s , ... µ n r a n , a n , . . . , a nn s fiffiffifl P A ( )due to the total associativity ( ) (by “omitting brackets”).This construction is the stack reshape of a matrix or row-major order of an array. Example . In terms of matrix polyads the (binary) mediality property ( ) becomes p µ q ˝ ” ˆ A p q ı “ p µ q ˝ ” ˆ A T p q ı , or A p µ q “ A T p µ q ( ) ˆ A p q “ ˆ a a a a ˙ ñ p a , a , a , a q P A b , ( )where ˆ A T p q is the transposed polyad matrix representing the sequence p a , a , a , a q P A b , A p µ q “ pp a ¨ a q ¨ p a ¨ a qq P A and A T p µ q “ pp a ¨ a q ¨ p a ¨ a qq P A with p¨q ” µ . Definition 4.3. A polyadic ( n -ary) mediality property is defined by the relation p µ n q ˝ n ` ” ˆ A p n q ı “ p µ n q ˝ n ` ” ˆ A T p n q ı , or A p µ q n “ A T p µ q n , ( ) ˆ A p n q “ p a ij q P A b n . ( ) Definition 4.4. A polyadic medial twist map χ p n q medial is defined on the matrix polyads as D UPLIJ [2018b] ˆ A p n q χ p n q medial ÞÑ ˆ A T p n q . ( ) Definition 4.5. A n -ary algebra A p n q is called medial , if it satisfies the n -ary mediality property( ) for all a ij P A .It follows from ( ), that not all medial binary algebras are abelian. Corollary 4.6.
If a binary medial algebra A p q is cancellative, it is abelian. Assertion 4.7.
If a n -ary medial algebra A p n q is cancellative, each matrix polyad ˆ A p n q satisfies n ´ commutativity-like relations. - 11 -. A LMOST MEDIAL N - ARY GRADED ALGEBRAS
5. A
LMOST MEDIAL N - ARY GRADED ALGEBRAS
The gradation for associative n -ary algebras was considered in M ICHOR AND V INOGRADOV [1996], G
NEDBAYE [1995]. Here we introduce a weaker version of gradation, because we needto define the grading twist not for n -ary multiplication, i. e. the polyads of the length n , but only forthe matrix polyads ( ) of the length n (for the binary case, see ( )). Definition 5.1.
An associative n -ary algebra A p n q is called a higher graded n -ary algebra over k , ifthe algebra multiplication of n elements respects the gradation i.e. p µ n q ˝ n ` ” A p a ij q ı ” µ n »———– µ n “ A a , A a , . . . , A a n ‰ ,µ n “ A a , A a , . . . , A a ‰ , ... µ n “ A a n , A a n , . . . , A a nn ‰ fiffiffiffifl Ď A a ` ... ` a nn , @ a ij P G, i, j “ , . . . , n ( )where equality corresponds to strong higher gradation .Let us define the higher twisting function ( higher twist factor ) for n elements τ n : G ˆ n Ñ k by using matrix polyads (for n “ see ( )). Definition 5.2. A n -ary higher graded twisted product µ p τ q n is defined for homogeneous elementsby µ p τ q n ” ˆ A p n q ı “ τ n ´ ˆ A n q ¯ A p µ q n , a ij P A ; a ij P G, i, j “ , . . . , n, ( )where ˆ A p n q “ p a ij q P A b n is the matrix polyad of elements ( ), and ˆ A n q “ ` a ij ˘ P G b n is thematrix polyad of their gradings .A medial analog of n -ary almost mediality can be introduced in a way analogous to the binarycase ( ). Definition 5.3.
If the higher twisted product coincides with the medially symmetric product (see( )) for all a ij P A , we call the twisting function a - level n - ary mediality factor τ n ÞÑ ρ p n q : G ˆ n Ñ k ˆ , such that (cf. ( )) µ p ρ q n ” ˆ A p n q ı “ A T p µ q n , or ( ) ρ p n q ´ ˆ A n q ¯ A p µ q n “ A T p µ q n , a ij P A ; a ij P G, i, j “ , . . . , n. ( )It follows from ( ) that the normalization condition for the n -ary mediality factor is ρ p n q ¨˚˝ n hkkkikkkj a , . . . , a ˛‹‚ “ , @ a P G. ( ) Assertion 5.4.
The -level n -ary mediality factor ρ p n q satisfies ρ p n q ´ ˆ A n q ¯ ρ p n q ˆ´ ˆ A n q ¯ T ˙ “ . ( ) Proof.
It follows from ( ) and its transpose together with the relation ` B T ˘ T “ B for any matrixover k . (cid:3) - 12 -. A LMOST MEDIAL N - ARY GRADED ALGEBRAS
Definition 5.5. An n -ary algebra for which the higher twisted product coincides with the mediallysymmetric product ( ), is called a -level almost medial ( ρ -commutative) n -ary algebra A p ρ q n .Recall B OURBAKI [1998], that a tensor product of binary algebras can be naturally endowedwith a ε -graded structure in the following way (in our notation). Let A p ε q “ A A | µ p a q E and B p ε q “ A A | µ p b q E be binary graded algebras with the multiplications µ p a q ” p¨ a q and µ p b q ” p¨ b q and the same commutation factor ε (see ( )), that is the same G -graded structure. Consider thetensor product A p ε q b B p ε q , and introduce the total ε -graded multiplication ´ A p ε q b B p ε q ¯ ‹ p ε q ´ A p ε q b B p ε q ¯ ÝÑ A p ε q b B p ε q defined by the deformation (cf. ( )) ε p b , a q p a b b q ‹ p ε q p a b b q , b , a P G, a i P A, b i P B, i “ , . ( ) Proposition 5.6.
If the ε -graded multiplication ( ) satisfies (cf. ( )) ε p b , a q p a b b q ‹ p ε q p a b b q “ p a ¨ a a q b p b ¨ b b q , ( ) then A p ε q b B p ε q is a ε -graded commutative algebra. Proposition 5.7. If A p ε q and B p ε q are associative, then A p ε q b B p ε q is also associative.Proof. This follows from ( ), ( ) and the properties of the commutation factor ε ( )–( ). (cid:3) In the matrix form ( ) becomes (with ‹ p ε q ” µ ‹p ε q ) ε p b , a q µ ‹p ε q „ µ b r a , b s µ b r a , b s “ µ b « µ p a q r a , a s µ p b q r b , b s ff , ( )where µ b is the standard binary tensor product. For numerous generalizations (including braidings),see, e.g., L ´ OPEZ P E ˜ NA ET AL . [2007], and refs. therein.Now we can extend ( ) to almost medial algebras.
Definition 5.8.
Let A p ρ q and B p ρ q be two binary medial algebras with the same medial-ity factor ρ . The total ρ -mediality graded multiplication µ ‹p ρ q : ´ A p ρ q b B p ρ q ¯ ‹ p ρ q ´ A p ρ q b B p ρ q ¯ ÝÑ A p ρ q b B p ρ q is defined by the mediality deformation (cf. ( )) ρ ˆ a b a b ˙ p a b b q ‹ p ρ q p a b b q , b , a P G, a i P A, b i P B, i “ , . ( ) Proposition 5.9.
If the ρ -graded multiplication ( ) satisfies (cf. ( )) ρ ˆ a b a b ˙ µ ‹p ρ q „ µ b r a , b s µ b r a , b s “ µ b « µ p a q r a , a s µ p b q r b , b s ff , ( ) then A p ρ q b B p ρ q is a ρ -graded binary (almost medial) algebra. Using the matrix form ( ) one can generalize the ρ -graded medial algebras to arbitrary arity.Let B p ρ q , n , . . . , B p ρ q ,nn be n ρ -graded (almost medial) n -ary algebras ( B p ρ q ,in “ A B i | µ p i q n E )with the same mediality factor ρ and the same graded structure. Consider their tensor product B p ρ q , n b . . . b B p ρ q ,nn and the ρ -graded n -ary multiplication µ ‹p ρ q n on it.- 13 -. A LMOST MEDIAL N - ARY GRADED ALGEBRAS
Higher level mediality n -ary brackets Proposition 5.10.
If the ρ -graded n -ary multiplication µ ‹p ρ q n satisfies (cf. ( )) ρ ¨˝ b . . . b n ... . . . ... b n . . . b n n ˛‚ µ ‹p ρ q n »– µ b n r b , . . . , b n s ... µ b n r b n , . . . , b nn s fifl “ µ b n »—– µ p q n r b , . . . , b n s ... µ p n q n r b n , . . . , b nn s fiffifl , ( ) b i , . . . , b in P A p ρ q ,in , b i n , . . . , b i n P G, i “ , . . . , n. ( ) then the tensor product B p ρ q , n b . . . b B p ρ q ,nn is a ρ -graded n -ary (almost medial) algebra. Symbolically, we can write this in the form, similar to the almost mediality condition ( ) ρ ´ ˆ B n q ¯ µ ‹p ρ q n ˝ µ b n ” ˆ B p n q ı “ µ b n ˝ µ p q n ˝ . . . ˝ µ p n q n ” ˆ B T p n q ı , ( )where ˆ B n q “ ¨˝ b . . . b n ... . . . ... b n . . . b n n ˛‚ , ˆ B p n q “ ¨˝ b . . . b n ... . . . ... b n . . . b nn ˛‚ ( )and ˆ B T p n q is its transpose. Example . In the lowest non-binary example, for 3 ternary ρ -graded algebras A p ρ q “ A A | µ p a q E , B p ρ q “ A B | µ p b q E , C p ρ q “ A C | µ p c q E , from ( ) we have the ternary multipli-cation µ ‹p ρ q for their ternary tensor product A p ρ q b B p ρ q b C p ρ q given by ρ ¨˝ a b c a b c a b c ˛‚ µ ‹p ρ q ¨˝ p a b b b c qp a b b b c qp a b b b c q ˛‚ “ ´ µ p a q r a , a , a s b µ p a q r b , b , b s b µ p c q r c , c , c s ¯ , ( )where a i P A, b i P B, c i P C, a i , b i , c i P G , i “ , , .5.1. Higher level mediality n -ary brackets. Binary almost mediality algebras for n “ wereconsidered in ( ), together with the tower of mediality factors ( ), ( )–( ). Here wegeneralize this construction to any arity n which can be done using the matrix polyad construction.First, we deform the almost mediality condition ( ) ρ p n q ´ ˆ A n q ¯ A p µ q n “ A T p µ q n ` M p ρ q ´ ˆ A p n q ¯ , a ij P A ; a ij P G, i, j “ , . . . , n, ( )where M p ρ q : A b n Ñ A is the higher mediality n -ary bracket of -level . Consider M p ρ q as a new n -ary (bracket) multiplication µ p ρ ,M q n ” ˆ A p n q ı : “ M p ρ q ´ ˆ A p n q ¯ . ( ) Definition 5.12. A n -ary algebra A p ρ ,M q n “ A A | µ p ρ ,M q n E ( )is called a -level mediality bracket n -ary algebra .- 14 -. A LMOST MEDIAL N - ARY GRADED ALGEBRAS
Proposition 5.13.
The n -ary algebra A p ρ ,M q n is almost medial with the mediality factor ˜ ´ ˆ ρ p n q ˙ ´ ¸ .Proof. We multiply the definition ( ) by ρ p n q ˆ´ ˆ A n q ¯ T ˙ and use ( ) to obtain ρ p n q ˆ´ ˆ A n q ¯ T ˙ M p ρ q ´ ˆ A p n q ¯ “ A p µ q n ´ ρ p n q ˆ´ ˆ A n q ¯ T ˙ A T p µ q n . ( )Taking into account that the r.h.s. here is exactly ´ M p ρ q ´ ˆ A T p n q ¯ , we have ´ ρ p n q ˆ´ ˆ A n q ¯ T ˙ M p ρ q ´ ˆ A p n q ¯ “ M p ρ q ´ ˆ A T p n q ¯ , ( )and using ( ) again, we get ´ ρ p n q ´ ˆ A n q ¯ ´ M p ρ q ´ ˆ A p n q ¯ “ M p ρ q ´ ˆ A T p n q ¯ , ( )which should be compared with ( ). (cid:3) Now we “deform” ( ) successively by defining further n -ary brackets M k and higher levelmediality factors ρ p n q k : G ˆ n Ñ k as follows. Definition 5.14.
The k -level mediality n -ary brackets and factors are defined by ρ p n q ´ ˆ A n q ¯ M p ρ q ´ ˆ A p n q ¯ “ M p ρ q ´ ˆ A T p n q ¯ ` M p ρ ,ρ q ´ ˆ A p n q ¯ , ( ) ρ p n q ´ ˆ A n q ¯ M p ρ ,ρ q ´ ˆ A p n q ¯ “ M p ρ ,ρ q ´ ˆ A T p n q ¯ ` M p ρ ,ρ ,ρ q ´ ˆ A p n q ¯ , ( )... ρ p n q k ´ ˆ A n q ¯ M p ρ ,ρ ,...,ρ k ´ q k ´ ´ ˆ A p n q ¯ “ M p ρ ,ρ ,...,ρ k ´ q k ´ ´ ˆ A T p n q ¯ ` M p ρ ,ρ ,...,ρ k q k ´ ˆ A p n q ¯ ( ) @ a ij P A ; a ij P G, i, j “ , . . . , n. Definition 5.15. k -level n -ary almost mediality is given by the vanishing of the last “deforming”medial n -ary bracket M p ρ ,ρ ,...,ρ k q k ´ ˆ A p n q ¯ “ , @ a ij P A, ( )and has the form ρ p n q k ´ ˆ A n q ¯ M p ρ ,ρ ,...,ρ k ´ q k ´ ´ ˆ A p n q ¯ “ M p ρ ,ρ ,...,ρ k ´ q k ´ ´ ˆ A T p n q ¯ . ( ) Proposition 5.16.
The higher level “deforming” functions ( n -ary brackets) M p ρ ,ρ ,...,ρ k q i ´ ˆ A p n q ¯ , i “ , . . . , k can be expressed through M p ρ q ´ ˆ A p n q ¯ from ( ) using a combination of the lowerlevel n -ary mediality factors ρ p n q k ´ ˆ A n q ¯ , i “ , . . . , k .Proof. This follows from the equations ( )–( ). (cid:3) - 15 -. T OYODA ’ S THEOREM FOR ALMOST MEDIAL ALGEBRAS
6. T
OYODA ’ S THEOREM FOR ALMOST MEDIAL ALGEBRAS
The structure of the almost medial graded algebras (binary and n -ary) can be established bysearching for possible analogs of Toyoda’s theorem ( ) (see, B RUCK [1944], M
URDOCH [1941],T
OYODA [1941]) which is the main statement for medial groupoids J
EEK AND K EPKA [1983] andquasigroups S
HCHERBACOV [2017]. As Toyoda’s theorem connects medial algebras with abelianalgebras, we can foresee that in the same way the almost medial algebras can be connected withalmost commutative algebras.First, let us consider almost medial graded binary algebras, as defined in ( )–( ). Theorem 6.1.
Let A p ρ q “ x A | µ y be an almost medial ( ρ -commutative) G -graded binary algebra,then there exists an almost commutative ( ε -commutative G -graded binary algebra ¯ A p ε q “ x A | ¯ µ y ,two grading preserving automorphisms ϕ , and a fixed element h P A , such that (cf. ( )) µ r a, b s “ ¯ µ r ¯ µ r ϕ p a q , ϕ p b qs , h s or a ¨ b “ ϕ p a q ϕ p b q h, ( ) ρ p a , b , c , d q “ ε p b , c q , @ a, b, c, d P A, a , b , c , d P G, ( ) where we denote µ ” p¨q and ¯ µ r a, b s ” ab .Proof. We use the “linear” presentation ( ) for the product in A p ρ q and insert it into the conditionof almost mediality ( ) to obtain ρ p a , b , c , d q p a ¨ b q ¨ p c ¨ d q “ p a ¨ c q ¨ p b ¨ d q ñ ρ p a , b , c , d q ϕ p ϕ p a q ϕ p b q h q ϕ pp ϕ p c q ϕ p d q h qq h “ ϕ p ϕ p a q ϕ p c q h q ϕ pp ϕ p b q ϕ p d q h qq ñ ρ p a , b , c , d q ϕ ˝ ϕ p a q ϕ ˝ ϕ p b q ϕ p h q ϕ ˝ ϕ p c q ϕ ˝ ϕ p d q ϕ p h q h “ ϕ ˝ ϕ p a q ϕ ˝ ϕ p c q ϕ p h q ϕ ˝ ϕ p b q ϕ ˝ ϕ p d q ϕ p h q h, ( )where p˝q is the composition of automorphisms. Using the cancellativity of ¯ A p ε q , we get ρ p a , b , c , d q ϕ ˝ ϕ p b q ϕ ˝ ϕ p c q “ ϕ ˝ ϕ p c q ϕ ˝ ϕ p b q . ( )Because the automorphisms ϕ , preserve grading, after implementing almost ( ε -) commutativity( ), the r.h.s. of ( ) becomes ε p b , c q ϕ ˝ ϕ p b q ϕ ˝ ϕ p c q which gives ( ) for commutingautomorphisms. (cid:3) The higher arity cases are more non-trivial, and very cumbersome. Therefore, we restrict ourselvesby the case n “ only. Theorem 6.2.
Let A p ρ q “ x A | µ , ν y be an almost medial ( ρ -commutative) G -graded ternaryalgebra over a field k . Then there exists an almost commutative ( ε -commutative G -graded binaryalgebra ¯ A p ε q “ x A | ¯ µ y , three commuting grading preserving automorphisms ϕ , , and a fixedelement h P A , such that (cf. ( )) µ r a, b, c s “ ¯ µ r ¯ µ r ¯ µ r ϕ p a q , ϕ p b qs , h s , h s” ϕ p a q ϕ p b q ϕ p c q h, @ a, b, c, h P A ( ) ρ p q ´ ˆ A q ¯ “ ε p a , a q ε p a , a q ε p a , a q ε p a , a q ε p a , a q ε p a , a q , ( ) ˆ A q “ ` a ij ˘ , @ a ij P G, i, j “ , . . . , , We use the multiplicative notation for the algebra ¯ A p ε q , because it is non-commutative. - 16 -. T OYODA ’ S THEOREM FOR ALMOST MEDIAL ALGEBRAS where we denote ¯ µ r a, b s ” ab .Proof. Using the matrix form of ternary ( n “ ) almost regularity ( ) and inserting there theternary “linear” presentation ( ) we get (in matrix form), @ a ij P A, i, j “ , . . . , , ρ p q ´ ˆ A q ¯ ¨˝ ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ˛‚ ( ) “ ¨˝ ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ˛‚ . ( )Applying the cancellativity of the binary algebra ¯ A p ε q , we have ρ p q ´ ˆ A q ¯ ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q“ ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q ϕ ˝ ϕ p a q . ( )Implementing almost ( ε -) commutativity ( ) on the r.h.s. of ( ), we arrive (for pairwise commut-ing grading preserving automorphisms ϕ i ˝ ϕ j “ ϕ j ˝ ϕ i , i, j “ , , ) at ( ). (cid:3)
7. B
INARY TENSOR CATEGORIES
We now apply the above ideas to construct a special kind of categories with multiplication
B ´
ENABOU [1963], M AC L ANE [1963] which appeared already in T
ANNAKA [1939] and later onwere called tensor categories and monoidal categories (as they “remind” us of the structure of amonoid) M AC L ANE [1971]. For reviews, see, e.g. C
ALAQUE AND E TINGOF [2008], M ¨
UGER [2010]. The monoidal categories can be considered as the categorification B AEZ AND D OLAN [1998a] of a monoid object, and can be treated as an instance of the microcosm principle : “ certainalgebraic structures can be defined in any category equipped with a categorified version of the samestructure ” B
AEZ AND D OLAN [1998b]. We start from the definitions of categories A D ´ AMEK ET AL .[1990], B
ORCEUX [1994] and binary tensor categories M AC L ANE [1971] (in our notation).Let C be a category with the class of objects Ob C and morphisms Mor C , such that the arrowfrom the source X to the target X is defined by Mor C Q f : X Ñ X , X , P Ob C , andusually Hom C p X , X q denotes all arrows which do not intersect. If Ob C and Mor C are sets,the category is small . The composition p˝q of three morphisms, their associativity and the identitymorphism ( id X ) are defined in the standard way M AC L ANE [1971].If C and C are two categories, then a mapping between them is called a covariant functor F : C Ñ C which consists of two different components: 1) the X -component is a mapping of objects F Ob : Ob C Ñ Ob C ; 2) the f -component is a mapping of morphisms F Mor : Mor C Ñ Mor C such that F “ F Ob , F Mor ( . A functor preserves the identity morphism F Mor p id X q “ id F Ob p X q andthe composition of morphisms F Mor p f ˝ f q “ F Mor p f q ˝ F Mor p f q ( “ F Mor p f q ˝ F Mor p f q for a contravariant functor ), where p˝ q is the composition in C .The (binary) product category C ˆ C consists of all pairs of objects p Ob C , Ob C q , morphisms p Mor C , Mor C q and identities p id X , id X q , while the composition p˝ q is made component-wise p f , f q ˝ p f , f q “ p f ˝ f , f ˝ f q , ( ) f ij : X i Ñ X j , @ X i P Ob C , f ij : X i Ñ X j , @ X i P Ob C , i, j “ , , , - 17 -. B INARY TENSOR CATEGORIES and by analogy this may be extended for more multipliers. A functor on a binary product category iscalled a bifunctor ( multifunctor ). A functor consists of two components F Ob , F Mor ( , and thereforea mapping between two functors F and G should also be two-component T F G “ T F G Ob , T F G
Mor ( .Without other conditions T F G is called an infra-natural transformation from F to G . A naturaltransformation (denoted by the double arrow T F G : F ñ G ) is defined by the consistency conditionof the above mappings in C T F G Ob ˝ F Mor “ G Mor ˝ T F G Ob . ( )Application to objects gives the following commutative diagram for the natural transformations(bifunctoriality) F Ob p X q ” X F T F G Ob p X q (cid:15) (cid:15) F Mor p f q” f F / / T F G
Mor p f q * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ F Ob p X q ” X F T F G Ob p X q (cid:15) (cid:15) G Ob p X q ” X G G Mor p f q” f G / / G Ob p X q ” X G ( )which is the consistency of the objects in C transformed by F and G . The the diagonal in ( ) mayalso be interpreted as the action of the natural transformation on a morphism T F G
Mor p f q : F Ob p X q Ñ G Ob p X q , f : X Ñ X , f P Mor C , X , X P Ob C , such that T F G
Mor p f q “ T F G Ob p X q ˝ F Mor p f q “ G Mor p f q ˝ T F G Ob p X q , ( )where the second equality holds valid due to the naturality ( ).In a concise form the natural transformations are described by the commutative diagram C F ) ) G C T F G (cid:11) (cid:19) ( )For a category C , the identity functor I d C “ ` I d C , Ob , I d C , Mor ˘ is defined by I d C , Ob p X q “ X , I d C , Mor p f q “ f , @ X P Ob C , @ f P Mor C . Two categories C and C are equivalent , if there existtwo functors F and G and two natural transformations T F G : I d C ñ F ˝ G and T GF : G ˝ F ñ I d C .For more details and standard properties of categories, see, e.g. M AC L ANE [1971],A D ´ AMEK ET AL . [1990], B
ORCEUX [1994] and refs therein.The categorification B
AEZ AND D OLAN [1998a], C
RANE AND Y ETTER [1994] of most algebraicstructures can be provided by endowing categories with an additional operation B ´
ENABOU [1963],M AC L ANE [1963] “reminding” us of the tensor product M AC L ANE [1971]. Usually M AC L ANE [1971], which are often denoted by the same letter, but for clarity we will distinguish them,because their action, arguments and corresponding commutative diagrams are different. - 18 -. B
INARY TENSOR CATEGORIES
A binary “ magmatic ” tensor category is ` C , M p bq ˘ , where M p bq ” b : C ˆ C Ñ C is abifunctor . In component form the bifunctor is M p bq “ ! M p bq Ob , M p bq Mor ) , where M p bq Mor is M p bq Mor r f , f s “ M p bq Ob r X , X s Ñ M p bq Ob r X , X s , ( ) f ii : X i Ñ X i , f ii P Mor C , @ X i , X i P Ob C , i “ , . The composition of the f -components is determined by the binary mediality property (cf. ( )) M p bq Mor r f , g s ˝ M p bq Mor r f , g s “ M p bq Mor r f ˝ f , g ˝ g s , ( ) f ij : X i Ñ X j , g ij : Y i Ñ Y j , f ij , g ij P Mor C , @ X i , Y i P Ob C , i “ , , . The identity of the tensor product satisfies M p bq Mor r id X , id X s “ id M p bq Ob r X ,X s . ( )We call a category C a strict ( binary ) semigroupal Y ETTER [2001], L
U ET AL . [2019] (or strictlyassociative semigroupal category B OYARCHENKO [2007], also, semi-monoidal K OCK [2008]), ifthe bifunctor M p bq satisfies only (without unit objects and unitors) the binary associativity condition p X b X qb X “ X bp X b X q and p f b f qb f “ f bp f b f q , where X i P Ob C , f i P Mor C , i “ , , (also denoted by sSGCat ). Strict associativity is the equivalence M p bq Ob ” M p bq Ob r X , X s , X ı “ M p bq Ob ” X , M p bq Ob r X , X s ı , ( ) M p bq Mor ” M p bq Mor r f , f s , f ı “ M p bq Mor ” f , M p bq Mor r f , f s ı . ( ) Remark . Usually, only the first equation for the X -components is presented in the definition ofassociativity (and other properties), while the equation for the f -components is assumed to be sat-isfied “automatically” having the same form M AC L ANE [1971], S
TASHEFF [1970]. In some cases,the diagrams for M p bq Ob and M p bq Mor can fail to coincide and have different shapes, for instance, inthe case of the dagger categories dealing with the “reverse” morphisms A
BRAMSKY AND C OECKE [2008].The associativity relations guarantee that in any product of objects or morphisms different waysof inserting parentheses lead to equivalent results (as for semigroups).In the case of a non-strict semigroupal category
SGCat (with no unit objects and unitors) Y
ETTER [2001], B
OYARCHENKO [2007] (see, also, L
U ET AL . [2019], E
LGUETA [2004], D
AVYDOV [2007])a collection of mappings can be introduced which are just the isomorphisms ( associators ) A p bq “ ! A p bq Ob , A p bq Mor ) from the left side functor to the right side functor of ( )–( ) as A p bq Ob p X , X , X q : M p bq Ob ” M p bq Ob r X , X s , X ı » Ñ M p bq Ob ” X , M p bq Ob r X , X s ı , ( )where A p bq Mor may be interpreted similar to the diagonal in ( ), because the associators arenatural transformations M AC L ANE [1971] or tri-functorial isomorphisms (in the terminology ofB
OYARCHENKO [2007]). Now different ways of inserting parentheses in a product of N objectsgive different results in the absence of conditions on the associator A p bq . However, if the associator A p bq satisfies some consistency relations, they can give isomorphic results, such that the correspond-ing diagrams commute, which is the statement of the coherence theorem M AC L ANE [1963], K
ELLY We use this notation with brackets M p bq A GUIAR AND M AHAJAN [2010], because they are convenient for furtherconsideration of the n -ary case D UPLIJ [2019]. - 19 -. B
INARY TENSOR CATEGORIES [1964]. This can also be applied to
SGCat , because it can be proved independently of existence ofunits Y
ETTER [2001], B
OYARCHENKO [2007], L
U ET AL . [2019]. It was shown M AC L ANE [1963]that it is sufficient to consider one commutative diagram using the associator (the associativity con-straint ) for two different rearrangements of parentheses for 3 tensor multiplications of 4 objects,giving the following isomorphism M p bq Ob ” M p bq Ob ” M p bq Ob r X , X s , X ı , X ı » Ñ M p bq Ob ” X , M p bq Ob ” X , M p bq Ob r X , X s ıı . ( )The associativity constraint is called a pentagon axiom M AC L ANE [1971], such that the diagram rr X , r X , X ss , X s / / A p bq , , r X , rr X , X s , X ss id X b A p bq , , $ $ ❍❍❍❍❍❍❍❍❍❍❍❍ rrr X , X s , X s , X s A p bq , , b id X : : ✉✉✉✉✉✉✉✉✉✉✉✉ » / / A p bq , , ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ r X , r X , r X , X sssrr X , X s , r X , X ss A p bq , , ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ ( )commutes.A similar condition for morphisms, but in another context (for H -spaces), was presented inS TASHEFF [1963, 1970]. Note that there exists a different (but not alternative) approach to natu-ral associativity without the use of the pentagon axiom J
OYCE [2001].The transition from the semigroupal non-strict category
SGCat to the monoidal non-strict cat-egory
MonCat can be done in a way similar to passing from a semigroup to a monoid: byadding the unit object E P Ob C and the ( right and left ) unitors U p bqp q “ ! U p bqp q Ob , U p bqp q Mor ) and U p bqp q “ ! U p bqp q Ob , U p bqp q Mor ) (“unit morphisms” which are functorial isomorphisms, natural transfor-mations) M AC L ANE [1971] U p bqp q Ob : M p bq Ob r X, E s » Ñ X, ( ) U p bqp q Ob : M p bq Ob r E, X s » Ñ X, @ X P Ob C , ( )and U p bqp , q Mor can be viewed as the diagonal in the diagram of naturality similar to ( ). The unitorsare connected with the associator A p bq , such that the diagram ( triangle axiom ) rr X , E s , X s U p bqp q Ob b id X ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ A p bq Ob / / r X , r E, X ss id X b U p bqp q Ob w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ r X , X s ( )commutes. We omit M p bq Ob in diagrams by leaving the square brackets only and use the obvious subscripts in A p bq . - 20 -. B INARY TENSOR CATEGORIES
Using the above, the definition of a binary non-strict monoidal category
MonCat can be given asthe 6-tuple ´ C , M p bq , A p bq , E, U p bq ¯ such that the pentagon axiom ( ) and the triangle axiom( ) are satisfied M AC L ANE [1963], M AC L ANE [1971] (see, also, K
ELLY [1964, 1965]).The following “normalizing” relations for the unitors of a monoidal non-strict category U p bqp q Ob p E q “ U p bqp q Ob p E q , ( )can be proven J OYAL AND S TREET [1993], as well as that the diagrams rr X , X s , E s U p bqp q Ob (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄ A p bq Ob / / r X , r X , E ss id X b U p bqp q Ob (cid:127) (cid:127) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) r X , X s rr E, X s , X s U p bqp q Ob b id X (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄ A p bq Ob / / r E, r X , X ss U p bqp q Ob (cid:127) (cid:127) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) r X , X s ( )commute.The coherence theorem B ´
ENABOU [1963], M AC L ANE [1963] proves that any diagram in a non-strict monoidal category, which can be built from an associator satisfying the pentagon axiom ( )and unitors satisfying the triangle axiom ( ), commutes. Another formulation M AC L ANE [1971]states that every monoidal non-strict category is (monoidally) equivalent to a monoidal strict one(see, also, K
ASSEL [1995]).Thus, it is important to prove analogs of the coherence theorem for various existing generaliza-tions of categories (having weak modification of units K
OCK [2008], J
OYAL AND K OCK [2013],A
NDRIANOPOULOS [2017], and from the “periodic table” of higher categories B
AEZ AND D OLAN [1995]), as well as for further generalizations (e.g., n -ary ones below).8. P OLYADIC TENSOR CATEGORIES
The arity of the additional multiplication in a category (the tensor product) was previ-ously taken to be binary. Here we introduce categories with tensor multiplication which “re-mind” n -ary semigroups, n -ary monoids and n -ary groups D ¨ ORNTE [1929], P
OST [1940] (see,also, G AL ’ MAK [2003]), i.e. we provide the categorification C
RANE AND F RENKEL [1994],C
RANE AND Y ETTER [1994] of “higher-arity” structures according to the Baez-Dolan microcosmprinciple B
AEZ AND D OLAN [1998b]. In our considerations we use the term “tensor category” ina wider context, because it can include not only binary monoid-like structures and their combina-tions, but also n -ary-like algebraic structures. It is important to note that our construction is dif-ferent from other higher generalizations of categories , such as -categories K ELLY AND S TREET [1974] and bicategories B ´
ENABOU [1967], n -categories B AEZ [1997], L
EINSTER [2002] and n -categories of n -groups A LDROVANDI AND N OOHI [2009], multicategories L
AMBEK [1969],L
EINSTER [1998], C
RUTTWELL AND S HULMAN [2010], n -tuple categories and multiple categoriesG RANDIS [2020], iterated ( n -fold) monoidal categories B ALTEANU ET AL . [2003], iterated iconsC
HENG AND G URSKI [2014], and obstructed categories D
UPLIJ AND M ARCINEK [2002, 2018b].We introduce the categorification of “higher-arity” structures along D
UPLIJ [2019] and considertheir properties, some of them are different from the binary case (as in n -ary (semi)groups and n -arymonoids). The terms “ k -ary algebraic category” and “ k -ary category” appeared in H ERRLICH [1971] and S
HULMAN [2012],respectively, but they describe different constructions. - 21 -. P
OLYADIC TENSOR CATEGORIES
Polyadic semigroupal categoriesLet C be a category M AC L ANE [1971], and introduce an additional multiplication as an n -arytensor product as in D UPLIJ [2018a, 2019].
Definition 8.1. An n -ary tensor product in a category C is an n -ary functor M p n bq : n hkkkkkkikkkkkkj C ˆ . . . ˆ C Ñ C ( )having the component form M p n bq “ ! M p n bq Ob , M p n bq Mor ) where the f -component M p n bq Mor is M p n bq Mor r f , f , . . . f nn s “ M p n bq Ob r X , X , . . . X n s Ñ M p n bq Ob r X , X , . . . X n s , ( ) f ii : X i Ñ X i , f ii P Mor C , @ X i , X i P Ob C , i “ , . . . , n. The n -ary composition of the f -components (morphism products of length n ) is determined by the n -ary mediality property (cf. ( )) M p n bq Mor “ f p , q , f p , q , . . . , f p ,n q ‰ ˝ . . . ˝ M p n bq Mor “ f p n, q , f p n, q , . . . , f p n,n q ‰ , “ M p n bq Mor “ f p , q ˝ f p , q ˝ . . . ˝ f p n, q , . . . , f p ,n q ˝ . . . ˝ f p n,n q ‰ , ( ) f p i,j q P Mor C , i, j “ , . . . , n. The identity morphism of the n -ary tensor product satisfies M p n bq Mor r id X , id X , . . . , id X n s “ id M p n bq Ob r X ,X ...,X n s . ( ) Definition 8.2. An n -ary tensor product M p n bq which can be constructed from a binary tensor prod-uct M bq by successive (iterative) repetitions is called an arity-reduced tensor product , and other-wise it is called an arity-nonreduced tensor product .Categories containing iterations of the binary tensor product were considered in B ALTEANU ET AL .[2003], C
HENG AND G URSKI [2014]. We will mostly be interested in the arity-nonreducible tensorproducts and their corresponding categories.
Definition 8.3.
A polyadic ( n -ary) “ magmatic ” tensor category is ` C , M p n bq ˘ , where M p n bq is an n -ary tensor product (functor ( )), and it is called an arity-reduced category or arity-nonreducedcategory depending on its tensor product.8.1. Polyadic semigroupal categories.
We call sequences of objects and morphisms X -polyadsand f -polyads P OST [1940], and denote them X and f , respectively (as in ( )). Definition 8.4.
The n -ary functor M p n bq is totally ( n -ary) associative , if it satisfies the following p n ´ q pairs of X equivalences M p n bq Ob ” X , M p n bq Ob r Y s , Z ı “ equivalent, ( )where X , Y , Z are X -polyads of the necessary length, and the total length of each p X , Y , Z q -polyadis n ´ , while the internal tensor products in ( ) can be on any of the n places. By analogy with the “derived n -ary group” D ¨ ORNTE [1929], P
OST [1940]. - 22 - -ary coherence 8. P
OLYADIC TENSOR CATEGORIES
Example . In the ternary case ( n “ ) the total associativity for the X -polyads of the length “ ¨ ´ gives “ ´ pairs of equivalences M p bq Ob ” M p bq Ob r X , X , X s , X , X ı “ M p bq Ob ” X , M p bq Ob r X , X , X s , X ı “ M p bq Ob ” X , X , M p bq Ob r X , X , X s ı , ( ) @ X i P Ob C , @ f i P Mor C , i “ , . . . , . Definition 8.6.
A category ` C , M p n bq ˘ is called a polyadic ( n - ary ) strict semigroupal category sSGCat n , if the bifunctor M p n bq satisfies objects and unitors) the n -ary associativity condition( ).Thus, in a polyadic strict semigroupal category for any (allowed, i.e. having the size k p n ´ q ` , @ k P N , where k is the number of n -ary tensor multiplications) product of objects (or morphisms),all different ways of inserting parentheses give equivalent results (as for n -ary semigroups).8.2. N -ary coherence. As in the binary case ( ), the transition to non-strict categories results inthe consideration of independent isomorphisms instead of the equivalence ( ). Definition 8.7.
The p n ´ q pairs of X and f isomorphisms A p n ´ qb “ ! A p n ´ qb Ob , A p n ´ qb Mor ) suchthat A p n ´ qb i, Ob : M p n bq Ob ” X , M p n bq i, Ob r Y s , Z ı » Ñ M p n bq Ob ” X , M p n bq i ` , Ob r Y s , Z ı , ( )are called n - ary associators being p n ´ q -place natural transformations, where A p n ´ qb Mor may beviewed as corresponding diagonals as in ( ). Here i “ , . . . , n ´ is the place of the internalbrackets.In the ternary case ( n “ ) we have “ ´ pairs of the ternary associators A p bq , Ob : M p bq Ob ” M p bq Ob r X , X , X s , X , X ı » Ñ M p bq Ob ” X , M p bq Ob r X , X , X s , X ı , ( )and A p bq , Ob : M p bq Ob ” X , M p bq Ob r X , X , X s , X ı » Ñ M p bq Ob ” X , X , M p bq Ob r X , X , X s ı . ( )It is now definite that different ways of inserting parentheses in a product of N objects will givedifferent results (the same will be true for morphisms as well), if we do not impose constraints onthe associators. We anticipate that we will need (as in the binary case ( )) only one more (i.e.three) tensor multiplication than appears in the associativity conditions ( ) to make a commutativediagram for the following isomorphism of ¨ p n ´ q ` “ n ´ objects M p n bq Ob ” M p n bq Ob ” M p n bq Ob r X , . . . , X n s , X n ` , . . . , X n ´ ı , X n , . . . , X n ´ ı » Ñ M p n bq Ob ” X , . . . , X n ´ , M p n bq Ob ” X n , . . . , X n ´ , M p n bq Ob r X n ´ , . . . , X n ´ s ıı . ( ) Conjecture 8.8 ( N - ary coherence ) . If the n -ary associator A p n ´ qb satisfies such n - ary coherenceconditions that the isomorphism ( ) takes place, then any diagram containing A p n ´ qb togetherwith the identities ( ) commutes.The n -ary coherence conditions are described by a “ p n ` q -gon”, which is the pentagon ( )for n “ (for classification of “ N -gons” see, e.g., W ENNINGER [1974]).- 23 -. P
OLYADIC TENSOR CATEGORIES
Polyadic monoidal categories
Definition 8.9.
A category ` C , M p n bq ˘ is called a polyadic ( n - ary ) non-strict semigroupal category sSGCat n , if the bifunctor M p n bq satisfies the n -ary coherence. Example . In the ternary case n “ we have pairs of -place associators ( )–( ) A p bq and A p bq which act on “ ¨ ´ objects ( ). We consider the diagram for objects only, then theassociativity constraint for the associators A p bq , Ob and A p bq , Ob will be a decagon axiom requiring thatthe diagram r X , r X , r X , X , X s , X s , X s A p bq , Ob 1 , , , , ' ' PPPPPPPPPPPPPPPP rr X , X , r X , X , X ss , X , X s A p bq , Ob 1 , , , , ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ r X , X , rr X , X , X s , X , X ss id X b id X b A p bq , Ob 3 , , , , (cid:15) (cid:15) rr X , r X , X , X s , X s , X , X s A p bq , Ob 1 , , , , b id X b id X O O r X , X , r X , r X , X , X s , X ss id X b id X b A p bq , Ob 3 , , , , (cid:15) (cid:15) rrr X , X , X s , X , X s , X , X s A p bq , Ob 1 , , , , b id X b id X O O A p bq , Ob 123 , , , , (cid:15) (cid:15) »p q / / r X , X , r X , X , r X , X , X sssrr X , X , X s , r X , X , X s , X s A p bq , Ob 123 , , , , ' ' PPPPPPPPPPPPPPPP r X , r X , X , X s , r X , X , X ss A p bq , Ob 1 , , , , O O rr X , X X s , X , r X , X , X ss A p bq , Ob 1 , , , , ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ ( )commutes (cf. ( ) and the pentagon axiom ( ) for binary non-strict tensor categories).9. N - ARY UNITS , UNITORS AND QUERTORS
Introducing n -ary analogs of units and unitors is non-trivial, because in n -ary structures there arevarious possibilities: one unit, many units, all elements are units or there are no units at all (see,e.g., for n -ary groups D ¨ ORNTE [1929], P
OST [1940], G AL ’ MAK [2003], and for n -ary monoidsP OP AND P OP [2004]). A similar situation is expected in category theory after proper categorifica-tion C RANE AND F RENKEL [1994], C
RANE AND Y ETTER [1994], B
AEZ AND D OLAN [1998a] of n -ary structures.9.1. Polyadic monoidal categories.
Let ´ C , M p n bq , A p n ´ qb ¯ be an n -ary non-strict semigroupalcategory SGCat n (see Definition 8.6 ) with n -ary tensor product M p n bq and the associator A p n ´ qb satisfying n -ary coherence. If a category has a unit neutral sequence of objects E p n ´ q “p E , . . . , E i q , E i P Ob C , i “ , . . . , n ´ , we call it a unital category. Note that the unit neu-tral sequence may not be unique. If all E i coincide E i “ E P Ob C , then E is called a unit object of C . The n -ary unitors U p n bqp i q , i “ , . . . , n ( n -ary “unit morphisms” being natural transformations)- 24 -olyadic nonunital groupal categories 9. N - ARY UNITS , UNITORS AND QUERTORS are defined by U p n bqp i q Ob : M p n bq Ob r E , . . . E i ´ , X, E i ` , . . . E n s » Ñ X, @ X, E i P Ob C , i “ , . . . , n ´ . ( )The n -ary unitors U p n bqp i q are compatible with the n -ary associators A p n ´ qb by the analog of thetriangle axiom ( ). In the binary case ( )–( ), we have U p bqp q “ R p bq , U p bqp q “ L p bq . Definition 9.1.
A polyadic ( n -ary ) non-strict monoidal category MonCat n is a polyadic ( n -ary)non-strict semigroupal category SGCat n endowed with a unit neutral sequence E p n ´ q and n unitors U p n bqp i q , i “ , . . . , n , that is a 5-tuple ´ C , M p n bq , A p n bq , E p n ´ q , U p n bq ¯ satisfying the “ p n ` q -gon” axiom for the p n ´ q associators A p n ´ qbp i q and the triangle axiom (the analog of ( )) for theunitors and associators compatibility condition. Example . If we consider the ternary non-strict monoidal category
MonCat with one unit object E P Ob C , then we have associators A p bq and A p bq satisfying the decagon axiom ( ) and unitors U p bqp q Ob : M p bq Ob r X, E, E s » Ñ X, ( ) U p bqp q Ob : M p bq Ob r E, X, E s » Ñ X, ( ) U p bqp q Ob : M p bq Ob r E, E, X s » Ñ X, @ X P Ob C , ( )which satisfy the “normalizing” conditions U p bqp i q Ob p E q “ E , i “ , , and the ternary analog of thetriangle axiom ( ), such that the diagram r E, r E, X, E s , E s A p bq , Ob % % ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ U p bqp q Ob E,EXE,E (cid:15) (cid:15) rr E, E, X s , E, E s A p bq , Ob rrrrrrrrrrrrrrrrrrrrr U p bqp q Ob EEX,E,E (cid:15) (cid:15) r E, E, r X, E, E ss U p bqp q Ob E,E,XEE (cid:15) (cid:15) r E, E, X s U p bqp q Ob E,E,X & & ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ r E, X, E s U p bqp q Ob E,X,E (cid:15) (cid:15) r X, E, E s U p bqp q Ob X,E,E x x rrrrrrrrrrrrrrrrrrrrrr X ( )commutes.9.2. Polyadic nonunital groupal categories.
The main result of n -ary group theory D ¨ ORNTE [1929], P
OST [1940] is connected with units and neutral polyads: if they exist, then such n -arygroup is reducible to a binary group. A similar statement can be true in some sense for categories.- 25 -. N - ARY UNITS , UNITORS AND QUERTORS
Polyadic nonunital groupal categories
Conjecture 9.3.
If a polyadic ( n -ary) tensor category has unit object and unitors, it can be arity-reducible to a binary category, such that the n -ary product can be obtained by iterations of the binarytensor product.Therefore, it would be worthwhile to introduce and study non-reducible polyadic tensor categorieswhich do not possess unit objects and unitors at all. This can be done by “categorification” of the querelement concept D ¨ ORNTE [1929]. Recall that, for instance, in a ternary group x G | µ y foran element g P G a querelement ¯ g is uniquely defined by µ r g, g, ¯ g s “ g , which can be treatedas a generalization of the inverse element concept to the n -ary case. The mapping g Ñ ¯ g can beconsidered as an additional unary operation ( queroperation ) in the ternary (and n -ary) group, whileviewing it as an abstract algebra G LEICHGEWICHT AND G ŁAZEK [1967] such that the notion ofthe identity is not used. The (binary) category of n -ary groups and corresponding functors wereconsidered in M ICHALSKI [1979, 1984], I
ANCU [1991].Let ´ C , M p n bq , A p n ´ qb ¯ be a polyadic ( n -ary) non-strict semigroupal category, where M p n bq is the n -ary tensor product, and A p n ´ qb is the associator making the “ p n ` q -gon” diagram of n -ary coherence commutative. We propose a “categorification” analog of the queroperation to be acovariant endofunctor of C . Definition 9.4. A querfunctor Q : C Ñ C is an endofunctor of C sending Q Ob p X q “ ¯ X and Q Mor p f q “ ¯f , where ¯ X and ¯f are the querobject and the quermorphism of X and f , respectively, suchthat the i diagrams ( i “ , . . . , n ) »– n hkkkkikkkkj X, . . . , X fifl P r p n bq Ob ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ i ´ hkkkkkkkkkikkkkkkkkkj id X b , . . . , b id X b Q Ob b n ´ i hkkkkkkkkkikkkkkkkkkj id X b , . . . , b id X / / »– i ´ hkkkkikkkkj X, . . . , X, ¯ X, n ´ i hkkkkikkkkj X, . . . , X fifl Q p n bqp i q Ob v v ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ X ( )commute (and analogously for morphisms), where Q p n bqp i q are quertors Q p n bqp i q Ob : M p n bq Ob »– i ´ hkkkkikkkkj X, . . . , X, ¯ X, n ´ i hkkkkikkkkj X, . . . , X fifl » Ñ X, @ X P Ob C , i “ , . . . , n, ( )and P r p n bq : C n b Ñ C is the projection. The action on morphisms Q p n bqp i q Mor can be found using thediagonal arrow in the corresponding natural transformation, as in ( ). Example . In the ternary case we have (for objects) the querfunctor Q Ob p X q “ ¯ X and 3 quertorisomorphisms Q p bqp q Ob : M p bq Ob “ ¯ X, X, X ‰ » Ñ X, ( ) Q p bqp q Ob : M p bq Ob “ X, ¯ X, X ‰ » Ñ X, ( ) Q p bqp q Ob : M p bq Ob “ X, X, ¯ X ‰ » Ñ X, @ X P Ob C . ( )- 26 -raided binary tensor categories 10. N - ARY UNITS , UNITORS AND QUERTORS
The three quertors Q p bqp i q Ob and the querfunctor Q are connected with two ternary associators A p bq , Ob , A p bq , Ob ( )–( ) such that the following diagram r X, X, X s D iag p bq b id b id v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ id b id b D iag p bq ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ id b D iag p bq b id (cid:15) (cid:15) rr X, X, X s , X, X s A p bq , Ob / / id b id b Q Ob b id b id (cid:15) (cid:15) r X, r X, X, X s , X s A p bq , Ob / / id b id b Q Ob b id b id (cid:15) (cid:15) r X, X, r X, X, X ss id b id b Q Ob b id b id (cid:15) (cid:15) ““ X, X, ¯ X ‰ , X, X ‰ Q p bqp q Ob b id X b id X ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ A p bq , Ob / / “ X, “ X, ¯ X, X ‰ , X ‰ A p bq , Ob / / id X b Q p bqp q Ob b id X (cid:15) (cid:15) “ X, X, “ ¯ X, X, X ‰‰ id X b id X b Q p bqp q Ob v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ r X, X, X s ( )commutes, where D iag p n bq : C Ñ C n b is the diagonal. Definition 9.6. A nonunital non-strict groupal category GCat n is ´ C , M p n bq , A p n ´ qb , Q , Q p n bq ¯ ,i.e. a polyadic non-strict semigroupal category SGCat n equipped with the querfunctor Q and thequertors Q p n bq satisfying ( ). Conjecture 9.7.
There exist polyadic nonunital non-strict groupal categories which are arity-non-reducible (see
Definition 8.3 ), and so their n -ary tensor product cannot be presented in the form ofbinary tensor product iterations.10. B RAIDED TENSOR CATEGORIES
The next step in the investigation of binary tensor categories is consideration of the tensor product“commutativity” property. The tensor product can be “commutative” such that for a tensor category C there exists the equivalence X b Y “ Y b X , @ X, Y, P Ob C , and such tensor categories are called symmetric M AC L ANE [1971]. By analogy with associativity, one can introduce non-strict “commu-tativity”, which leads to the notion of a braided (binary) tensor category and the corresponding coher-ence theorems J
OYAL AND S TREET [1993]. Various generalizations of braiding were considered inG
ARNER AND F RANCO [2016], D
UPLIJ AND M ARCINEK [2002, 2018a], and their higher versionsare found, e.g., in K
APRANOV AND V OEVODSKY [1994], B
ATANIN [2010], W
EBER [2005].10.1.
Braided binary tensor categories.
Let ´ C , M p bq , A p bq ¯ be a non-strict semigroupal cat-egory with the bifunctor M p bq and the associator A p bq ( ) satisfying the pentagon axiom ( )Y ETTER [2001], B
OYARCHENKO [2007]. - 27 -0. B
RAIDED TENSOR CATEGORIES
Braided binary tensor categories
Definition 10.1.
A ( binary ) braiding B p bq “ ! B p bq Ob , B p bq Mor ) of a semigroupal category SGCat isa natural transformation of the bifunctor M p bq (bifunctorial isomorphism) such that B p bq Ob : M p bq Ob r X , X s » Ñ M p bq Ob r X , X s , @ X i P Ob C , i “ , , ( )and the action on morphisms B p bq Mor may be interpreted as a diagonal, similarly to ( ). Definition 10.2.
A non-strict semigroupal category endowed with a binary braiding is called a ( bi-nary ) braided semigroupal category bSGCat ´ C , M p bq , A p bq , B p bq ¯ .The braiding B p bq is connected with the associator A p bq by the hexagon identity rr X , X s , X s A p bq Ob 1 , , ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ B p bq Ob 1 , b id X w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ rr X , X s , X s A p bq Ob 2 , , (cid:15) (cid:15) r X , r X , X ss B p bqp q Ob 1 , (cid:15) (cid:15) r X , r X , X ss id X b B p bq Ob 1 , ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ rr X , X s , X s A p bq Ob 2 , , w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ r X , r X , X ss ( )for objects, and similarly for the inverse associator. Definition 10.3. A symmetric braided semigroupal category sbSGCat has the “invertible” braid-ing B p bq Ob X ,X ˝ B p bq Ob X ,X “ id X b X or ( ) B p bq Ob X ,X “ B p bq , ´ X ,X , @ X i P Ob C ( )A von Neumann regular generalization VON N EUMANN [1936] (weakening) of ( ) leads to
Definition 10.4.
A ( von Neumann ) regular braided semigroupal category is defined by a braidingwhich satisfies D UPLIJ AND M ARCINEK [2001, 2018a] B p bq Ob X ,X ˝ B ˚p bq Ob X ,X ˝ B p bq Ob X ,X “ B p bq Ob X ,X , ( )where B ˚p bq Ob X ,X is a generalized inverse P ENROSE [1955], N
ASHED [1976] of B p bq Ob X ,X , and suchthat B ˚p bq Ob X ,X ‰ B p bq , ´ X ,X (cf. ( )). Proposition 10.5.
If the (binary) braided semigroupal category is strict (the associator becomes theequivalence ( )–( ), and we can omit internal brackets), then the diagram - 28 -raided polyadic tensor categories 10. B
RAIDED TENSOR CATEGORIES r X , X , X s id X b B p bq Ob 2 , ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ B p bq Ob 1 , b id X u u ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ B p bq Ob 1 , | | ①①①①①①①①①①①①①①①①①①①①①①①①①①① r X , X , X s id X b B p bq Ob 1 , (cid:15) (cid:15) r X , X , X s B p bq Ob 1 , b id X (cid:15) (cid:15) B p bq Ob 1 , | | ①①①①①①①①①①①①①①①①①①①①①①①①①①① r X , X , X s B p bq Ob 2 , b id X ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ r X , X , X s id X b B p bq Ob 1 , u u ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ r X , X , X s ( ) commutes S TASHEFF [1963, 1970].Proof.
The triangles commute due to the hexagon identity ( ) and the internal rectangle com-mutes, because the binary braiding B p bq is a natural transformation (bifunctorial isomorphism). (cid:3) Omitting indices ( ) becomes the
Yang-Baxter equation in terms of tensor products D
RINFELD [1989] (or the binary braid group relation—for their difference see S
TREET [1995]) ´ B p bq Ob b id ¯ ˝ ´ id b B p bq Ob ¯ ˝ ´ B p bq Ob b id ¯ “ ´ id b B p bq Ob ¯ ˝ ´ B p bq Ob b id ¯ ˝ ´ id b B p bq Ob ¯ . ( )If the braided semigroupal category bSGCat contains a unit object, then we have Definition 10.6.
A ( binary ) braided monoidal category MonCat ´ C , M p bq , A p bq , E, U p bq , B p bq ¯ is bSGCat together with a unit object E P Ob C satisfying the triangle axiom ( ) and a unitor U p bq ( )–( ) the compatibility condition with the braiding B p bq such that the diagram (forobjects) r X, E s U p bqp q Ob $ $ ■■■■■■■■■■■■■■■ B p bq Ob / / r E, X s U p bqp q Ob z z ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ X ( )commutes.For more details on binary braided monoidal categories, see F REYD AND Y ETTER [1989],J
OYAL AND S TREET [1993] and for review, see, e.g., C
HARI AND P RESSLEY [1996],E
TINGOF ET AL . [2015], B
ULACU ET AL . [2019].10.2.
Braided polyadic tensor categories.
Higher braidings for binary tensor categorieswere considered (from an n -category viewpoint) in M ANIN AND S CHECHTMAN [1989],K
APRANOV AND V OEVODSKY [1994]. We will discuss them for polyadic categories, defined abovein
Section 8 . The difference will be clearer if a polyadic category is not arity-reduced (see
Definition8.3 ) and for non-unital groupal categories (
Subsection ´ C , M p n bq , A p n ´ qb ¯ be a polyadic non-strict semigroupal category, where M p n bq is a (notarity reduced) n -ary tensor product ( n -ary functor) and A p n ´ qb is an associator, i.e. n ´ different- 29 -0. B RAIDED TENSOR CATEGORIES
Braided polyadic tensor categories p n ´ q -ary natural transformations (see Definition 8.6 ). Now the braiding becomes an n -ary natu-ral transformation, which leads to any of n permutations from the symmetry (permutation) group S n ,rather than one possibility only, as for the binary braiding ( ). Note that in the consideration ofhigher braidings M ANIN AND S CHECHTMAN [1989], K
APRANOV AND V OEVODSKY [1994] one(“order reversing”) element of S n was used σ p rev q n ” ˆ . . . nn n ´ . . . ˙ P S n . Thus, we arriveat the most general Definition 10.7. An n - ary braiding B n b “ ! B p n bq Ob , B p n bq Mor ) of a polyadic non-strict semigroupalcategory is an n -ary natural (or infra-natural) transformation B p n bq Ob : M p n bq Ob r X s » Ñ M p n bq Ob r σ n ˝ X s , ( )where X is an X -polyad (see Definition 8.4 ) of the necessary length (which is n here), and σ n P S n are permutations that may satisfy some consistency conditions. The action on morphisms B p n bq Mor maybe found from the corresponding diagonal of the natural transformation square (cf. ( )).The binary non-mixed (standard) braiding ( ) has σ “ σ p rev q “ ˆ ˙ P S . Definition 10.8.
A polyadic (non-strict) semigroupal category endowed with the n -ary braiding ´ C , M p n bq , A p n ´ qb , B p n bq ¯ is called a braided semigroupal polyadic category bSGCat n .The n -ary braiding B p n bq is connected with the associator A p n ´ qb by a polyadic analog of thehexagon identity ( ). Example . In the case n “ , the braided non-strict semigroupal ternary category bSGCat contains two associators A p bq and A p bq (see Example ) satisfying the decagon axiom ( ).Let us take for the ternary braiding B p bq its “order reversing” version B p bq Ob : M p bq Ob r X , X , X s » Ñ M p bq Ob r X , X , X s , @ X i P Ob C , i “ , , . ( )Then the ternary analog of the hexagon identity is the decagon identity such that the diagram- 30 -raided polyadic tensor categories 10. B RAIDED TENSOR CATEGORIES rr X , X , X s , X , X s A p bq , Ob 1 , , , , ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ B p bq Ob 1 , , b id X b id X w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ rr X , X , X s , X , X s A p bq , Ob 3 , , , , (cid:15) (cid:15) r X , r X , X , X s , X s A p bq , Ob 1 , , , , (cid:15) (cid:15) r X , r X , X , X s , X s id X b B p bq Ob 2 , , b id X (cid:15) (cid:15) r X , X , r X , X , X ss B p bq Ob 1 , , (cid:15) (cid:15) r X , r X , X , X s , X s A p bq , Ob 3 , , , , (cid:15) (cid:15) rr X , X , X s , X , X s A p bq , Ob 3 , , , , (cid:15) (cid:15) r X , X , r X , X , X ss id X b id X b B p bq Ob 1 , , ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ r X , r X , X , X s , X s A p bq , Ob 3 , , , , w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ r X , X , r X , X , X ss ( )commutes. Conjecture 10.10 ( Braided n - ary coherence ) . If the n -ary associator A p n ´ qb satisfies such n - arycoherence conditions that the isomorphism ( ) takes place, and the n -ary braiding B p n bq satis-fies the polyadic analog of the hexagon identity, then any diagram containing A p n ´ qb and B p n bq commutes. Proposition 10.11.
If the braided semigroupal ternary category bSGCat is strict (the associatorsbecomes equivalences, and we can omit internal brackets), then the diagram containing only theternary braidings B p bq - 31 -0. B RAIDED TENSOR CATEGORIES
Braided polyadic tensor categories r X , X , X , X , X s id X b id X b B p bq Ob 3 , , ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ B p bq Ob 1 , , b id X b id X u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ B p bq Ob 1 , , (cid:1) (cid:1) ✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄ r X , X , X , X , X s id X b id X b B p bq Ob 1 , , (cid:15) (cid:15) r X , X , X , X , X s id X b B p bq Ob 2 , , b id X (cid:15) (cid:15) B p bq Ob 1 , , (cid:1) (cid:1) ✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄ r X , X , X , X , X s id X b B p bq Ob 2 , , b id X (cid:15) (cid:15) r X , X , X , X , X s B p bq Ob 1 , , b id X b id X (cid:15) (cid:15) r X , X , X , X , X s B p bq Ob 3 , , b id X b id X ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ r X , X , X , X , X s id X b id X b B p bq Ob 1 , , u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ r X , X , X , X , X s ( ) commutes (cf. the binary braiding ( )).Proof. This is analogous to ( ). (cid:3) There follows from ( ), omitting indices, the ternary braid group relation in termsof tensor products (cf. the tetrahedron equation B
AZHANOV AND S TROGANOV [1982],K
APRANOV AND V OEVODSKY [1994], B
AEZ AND N EUCHL [1995]) ´ B p bq b id b id ¯ ˝ ´ id b B p bq b id ¯ ˝ ´ id b id b B p bq ¯ ˝ ´ B p bq b id b id ¯ “ ´ id b id b B p bq ¯ ˝ ´ B p bq b id b id ¯ ˝ ´ id b B p bq b id ¯ ˝ ´ id b id b B p bq ¯ , ( )which was obtained in D UPLIJ [2018b] using another approach: by the associative quiver techniquefrom D
UPLIJ [2018a]. For instance, the -ary braid group relation for -ary braiding B p bq has theform ´ B p bq b id b id b id ¯ ˝ ´ id b B p bq b id b id ¯ ˝ ´ id b id b B p bq b id ¯ ˝ ´ id b id b id b B p bq ¯ ˝ ´ B p bq b id b id b id ¯ “ ´ id b id b id b B p bq ¯ ( ) ˝ ´ B p bq b id b id b id ¯ ˝ ´ id b id b B p bq b id ¯ ˝ ´ id b B p bq b id b id ¯ ˝ ´ id b id b id b B p bq ¯ . For the non-mixed “order reversing” n -ary braiding (see Definition 10.7 ) we have D
UPLIJ [2018b]- 32 -raided polyadic tensor categories 10. B
RAIDED TENSOR CATEGORIES
Proposition 10.12.
The n -ary braid equation contains p n ` q multipliers, and each one acts on p n ´ q tensor products as ¨˝ B p n bq b n ´ hkkkkkkikkkkkkj id b . . . b id ˛‚ ˝ ¨˝ id b B p n bq b n ´ hkkkkkkikkkkkkj id b . . . b id ˛‚ ˝ ¨˝ id b id b B p n bq b n ´ hkkkkkkikkkkkkj id b . . . b id ˛‚ ˝ . . . ˝ ¨˝ n ´ hkkkkkkikkkkkkj id b . . . b id b B p n bq b id ˛‚ ˝ ¨˝ n ´ hkkkkkkikkkkkkj id b . . . b id b B p n bq ˛‚ ˝ ¨˝ B p n bq b n ´ hkkkkkkikkkkkkj id b . . . b id ˛‚ “ ¨˝ n ´ hkkkkkkikkkkkkj id b . . . b id b B p n bq ˛‚ ˝ ¨˝ B p n bq b n ´ hkkkkkkikkkkkkj id b . . . b id ˛‚ ˝ ¨˝ id b B p n bq b n ´ hkkkkkkikkkkkkj id b . . . b id ˛‚ ˝ . . . ˝ ¨˝ n ´ hkkkkkkikkkkkkj id b . . . b id b B p n bq b id b id ˛‚ ˝ ¨˝ n ´ hkkkkkkikkkkkkj id b . . . b id b B p n bq b id ˛‚ ˝ ¨˝ n ´ hkkkkkkikkkkkkj id b . . . b id b B p n bq ˛‚ . ( ) Remark . If a polyadic category is arity-nonreducible, then the higher n -ary braid relationscannot be “iterated”, i.e. obtained from the lower n ones.Consider a polyadic monoidal category MonCat n with one unit object E (see Definition 9.1 ).Then the n -ary braiding B p n bq satisfies the triangle identity connecting it with the unitors U p n bq . Example . In the case of the ternary monoidal category
MonCat (see Example ) the “orderreversing” braiding B p bq ( ) satisfies an additional triangle identity analogous to ( ) suchthat the diagram r X, E, E s U p bqp q Ob % % ❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑ B p bq Ob / / r E, E, X s U p bqp q Ob y y ssssssssssssssss X ( )commutes.For the polyadic non-unital groupal category GCat n (see Definition 9.6 ) the n -ary braiding B p n bq should be consistent with the quertors U p n bq and the querfunctor Q (see Definition 9.4 ). Definition 10.15. A braided polyadic groupal category bGCat n is a polyadic groupal category GCat n endowed with the n -ary braiding ´ C , M p n bq , A p n ´ qb , Q , Q p n bq , B p n bq ¯ . Example . In the ternary groupal category
GCat (see Example ) the “order reversing”braiding B p bq ( ) satisfies the additional identity of consistency with the querfunctor Q and- 33 -1. B RAIDED TENSOR CATEGORIES quertor Q p bq such that the diagram r X, X, X s Q Ob b id b id x x ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ id b id b Q Ob & & ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ “ ¯ X, X, X ‰ Q p bqp q Ob ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ B p bq Ob / / “ X, X, ¯ X ‰ Q p bqp q Ob w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ X ( )commutes.The above diagrams ensure that some version of coherence can also be proven for braided polyadiccategories. 11. M EDIALED POLYADIC TENSOR CATEGORIES
Here we consider a medial approach to braiding inspired by the first part of our paper. As op-posed to binary braiding which is defined by one unique permutation ( ), the n -ary braiding canbe defined by the enormous number of possible allowed permutations ( ). Therefore, in mostcases only one permutation, that is the “order reversing”, is usually (and artificially) used (see, e.g.,M ANIN AND S CHECHTMAN [1989]) ignoring other possible cases. On the other side, for n -arystructures it is natural to use the mediality property ( ) which is unique in the n -ary case and for bi-nary groups reduces to commutativity. So we introduce a medialing instead of braiding for the tensorproduct in categories, and (by analogy with braided categories) we call them medialed categories .Let ´ C , M p n bq , A p n ´ qb ¯ be a polyadic non-strict semigroupal category SGCat n (see Definition8.6 ). Definition 11.1. An n -ary medialing M p n b q (or “medial braiding”) is a mediality constraint whichis a natural (or infra-natural) transformation of two composed n -ary tensor product functors M p n bq (or functorial n -isomorphism) M p n b q Ob : M p n bq Ob »———– M p n bq Ob r X , X , . . . , X n s , M p n bq Ob r X , X , . . . , X n s , ... M p n bq Ob r X n , X n , . . . , X nn s fiffiffiffifl » Ñ M p n bq Ob »———– M p n bq Ob r X , X , . . . , X n s , M p n bq Ob r X , X , . . . , X n s , ... M p n bq Ob r X n , X n , . . . , X nn s fiffiffiffifl , ( )where the action on morphisms M p n bq Mor can be viewed as the corresponding diagonal in the naturaltransformation diagram as in ( ). Remark . The advantage of n -ary medialing is its uniqueness , because it does not contain a hugenumber of possible permutations σ n P S n as does the n -ary braiding ( ). Example . In the binary case n “ we have (using the standard notation M p bq ÝÑ b ) M p bq Ob : p X b X q b p X b X q » Ñ p X b X q b p X b X q , @ X i P Ob C , i “ , . . . , , ( )which is called a binary medialing by analogy with binary braiding ( ).- 34 -edialed binary and ternary categories 11. M EDIALED POLYADIC TENSOR CATEGORIES
In the compact matrix notation (see
Definition 4.1 ) instead of ( ) we have (symbolically) M p n b q Ob : ´ M p n bq Ob ¯ ” ˆ X p n q ı » Ñ ´ M p n bq Ob ¯ ” ˆ X T p n q ı , ( )where the matrix polyads of objects is (cf. ( )) ˆ X p n q “ p X ij q P p Ob C q b n , X ij P Ob C , ( )and p q T is matrix transposition. Definition 11.4. A medialed polyadic semigroupal category ´ C , M p n bq , A p n ´ qb , M p n b q ¯ mSGCat n is a polyadic non-strict semigroupal category SGCat n (see Definition 8.9 ) endowed withthe n -ary medialing M p n b q satisfying the n -ary medial coherence condition (a medial analog of thehexagon identity ( )). Definition 11.5. A medialed polyadic monoidal category ´ C , M p n bq , A p n ´ qb , E, U p n bq , M p n b q ¯ mMonCat n is a medialed polyadic semigroupal category mSGCat n with the unit object E P Ob C and the unitor U p n bq satisfying some compatibility condition.Let us consider the polyadic nonunital groupal category GCat n (see Definition 9.6 ), then the n -ary medialing M p n b q should be consistent with the quertors U p n bq and the querfunctor Q (see Definition 9.4 and also the consistency condition for the ternary braiding ( )) .
Definition 11.6. A braided polyadic groupal category ´ C , M p n bq , A p n ´ qb , Q , Q p n bq , M p n b q ¯ mGCat n is a polyadic groupal category GCat n endowed with the n -ary medialing M p n b q .11.1. Medialed binary and ternary categories.
Due to the complexity of the relevant polyadicdiagrams, it is not possible to draw them in a general case for arbitrary arity n . Therefore, it wouldbe worthwhile to consider first the binary case, and then some of the diagrams for the ternary case. Example . Let ´ C , M p bq , A p bq , M p bq ¯ be a binary medialed semigroupal category mSGCat ,and the binary medialing be in ( ). Then the medial analog of the hexagon identity ( ) is givenby the binary medial coherence condition such that the diagram- 35 -1. M EDIALED POLYADIC TENSOR CATEGORIES
Medialed binary and ternary categories rrrr X , X s , X s , X s , X s A p bq Ob 12 , , b id X (cid:15) (cid:15) A p bq Ob 1 , , b id X b id X / / rrr X , r X , X ss , X s , X s A p bq Ob 123 , , (cid:15) (cid:15) rrr X , X s , r X , X ss , X s M p bq Ob 1 , , , b id X (cid:15) (cid:15) rr X , r X , X ss , r X , X ss M p bq Ob 1 , , , (cid:15) (cid:15) rrr X , X s , r X , X ss , X s A p bq Ob 13 , , (cid:15) (cid:15) rr X , X s , rr X , X s , X ss A p bq Ob 1 , , (cid:15) (cid:15) rr X , X s , rr X , X s , X ss M p bq Ob 1 , , , (cid:15) (cid:15) r X , r X , rr X , X s , X sss id X b id X b A p bq Ob 2 , , (cid:15) (cid:15) rr X , r X , X ss , r X , X ss A p bq´ , , b id X b id X (cid:15) (cid:15) r X , r X , r X , r X , X ssssrrr X , X s , X s , r X , X ss A p bq´ , , (cid:15) (cid:15) rr X , X s , r X , r X , X sss A p bq Ob 1 , , O O rrrr X , X s , X s , X s , X s A p bq Ob 12 , , b id X (cid:15) (cid:15) rr X , X s , rr X , X s , X ss id X b id X b A p bq Ob 2 , , O O rrr X , X s , r X , X ss , X s M p bq Ob 1 , , , b id X / / rrr X , X s , r X , X ss , X s A p bq Ob 14 , , O O ( )commutes.If a medialed semigroupal category mSGCat contains a unit object and the unitor, then we have Definition 11.8. A medialed monoidal category mMonCat ´ C , M p bq , A p bq , E, U p bq , M p bq ¯ isa ( binary ) medialed semigroupal category mSGCat together with a unit object E P Ob C and aunitor U p bq ( )–( ) satisfying the triangle axiom ( ).- 36 -edialed binary and ternary categories 11. M EDIALED POLYADIC TENSOR CATEGORIES
For mMonCat the compatibility condition of the medialing M p bq with E and U p bq is given bythe commutative diagram rr X , E s , r X, X ss M p bq Ob / / A p bq Ob X ,E,XX (cid:15) (cid:15) rr X , X s , r E, X ss A p bq Ob X ,X,EX (cid:15) (cid:15) r X , r E, r X, X sss id X b A p bq´ (cid:15) (cid:15) r X , r X, r E, X sss id X b A p bq´ (cid:15) (cid:15) r X , rr E, X s , X ss id X b U p bqp q Ob b id X % % ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ r X , rr X, E s , X ss id X b U p bqp q Ob b id X y y sssssssssssssssssssssss r X , r X, X ss ( )which is an analog of the triangle diagram for braiding ( ). Example . In the ternary nonunital groupal category
GCat (see Example ) the medialing M p bq satisfies the additional identity of consistency with the querfunctor Q and quertor Q p bq suchthat the diagram - 37 -1. M EDIALED POLYADIC TENSOR CATEGORIES
Medialed binary and ternary categories rr X, X, X s , r X, X, X s , r X, X, X ss id b Q Ob b Q Ob b id b id b Q Ob b id b id b id u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ id b id b id b Q Ob b id b id b Q Ob b Q Ob b id ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ ““ X, ¯ X, ¯ X ‰ , “ X, X, ¯ X ‰ , r X, X, X s ‰ M p bq Ob / / id b id b id b Q p bqp q Ob b id b id b id (cid:15) (cid:15) “ r X, X, X s , “ ¯ X, X, X ‰ , “ ¯ X, ¯ X, X ‰‰ id b id b id b Q p bqp q Ob b id b id b id (cid:15) (cid:15) ““ X, ¯ X, ¯ X ‰ , X, r X, X, X s ‰ A p bq , Ob X, ¯ X, ¯ X,X,XXX (cid:15) (cid:15) “ r X, X, X s , X “ ¯ X, ¯ X, X ‰‰ A p bq´ , Ob XXX,X, ¯ X, ¯ X,X (cid:15) (cid:15) “ X, “ ¯ X, ¯ X, X ‰ , r X, X, X s ‰ A p bq , Ob X, ¯ X, ¯ X,X,XXX (cid:15) (cid:15) “ r X, X, X s , “ X, ¯ X, ¯ X ‰ , X ‰ A p bq´ , Ob XXX,X, ¯ X, ¯ X,X (cid:15) (cid:15) “ X, ¯ X, “ ¯ X, X, r X, X, X s ‰‰ id b id b A p bq´ , Ob (cid:15) (cid:15) ““ r X, X, X s X, ¯ X, ‰ , ¯ X, X ‰ A p bq , Ob b id b id (cid:15) (cid:15) “ X, ¯ X, “ ¯ X, r X, X, X s , X ‰‰ id b id b A p bq´ , Ob (cid:15) (cid:15) ““ X, r X, X, X s , ¯ X ‰ , ¯ X, X ‰ A p bq , Ob b id b id (cid:15) (cid:15) “ X, ¯ X, ““ ¯ X, X, X ‰ , X, X ‰‰ id b id b Q p bqp q Ob b id b id (cid:15) (cid:15) ““ X, X, “ X, X, ¯ X ‰‰ , ¯ X, X ‰ id b id b Q p bqp q Ob b id b id (cid:15) (cid:15) “ X, ¯ X, r X, X, X s ‰ A p bq´ , Ob (cid:15) (cid:15) “ r X, X, X s , ¯ X, X ‰ A p bq , Ob (cid:15) (cid:15) “ X, “ ¯ X, X, X ‰ , X ‰ id b Q p bqp q Ob b id ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ “ X, “ X, X, ¯ X ‰ , X ‰ id b Q p bqp q Ob b id u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ r X, X, X s ( )commutes. An analog of the hexagon identity in GCat can be expressed by a diagram which issimilar to ( ). - 38 -2. M EDIALED POLYADIC TENSOR CATEGORIES
12. C
ONCLUSIONS
Commutativity in polyadic algebraic structures is defined non-uniquely, if consider permutationsand their combinations. We proposed a canonical way out: to substitute the commutativity propertyby mediality. Following this “commutativity-to-mediality” ansatz we first investigated mediality forgraded linear n -ary algebras and arrived at the concept of almost mediality, which is an analog ofalmost commutativity. We constructed “deforming” medial brackets, which could be treated as amedial analog of Lie brackets. We then proved Toyoda’s theorem for almost medial n -ary algebras.Inspired by the above as examples, we proposed generalizing tensor and braided categories in asimilar way. We defined polyadic tensor categories with an additional n -ary tensor multiplication forwhich a polyadic analog of the pentagon axiom was given. Instead of braiding we introduced n -ary“medialing” which satisfies a medial analog of the hexagon identity, and constructed the “medialed”polyadic version of tensor categories. More details and examples will be presented in a forthcomingpaper. Acknowledgements . The author would like to express his deep thankfulness to Andrew JamesBruce, Grigorij Kurinnoj, Mike Hewitt, Richard Kerner, Maurice Kibler, Dimitry Leites, Yuri Manin,Thomas Nordahl, Valentin Ovsienko, Norbert Poncin, Vladimir Tkach, Raimund Vogl, AlexanderVoronov, and Wend Werner for numerous fruitful discussions and valuable support.R
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