Graded medial $n$-ary algebras and polyadic tensor categories

Algebraic structures in which the property of commutativity is substituted by the mediality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or $\varepsilon$-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 (universal) medial algebras with abelian algebras is proven for the almost medial graded algebras introduced here. 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 $n-1$ associators of the arity $2n-1$ satisfying a $\left( n^{2}+1\right) $-gon relation, which is a polyadic analog of the pentagon axiom. Polyadic monoidal categories may contain several unit objects, and it is also possible that all objects are units. A new kind of polyadic categories (called"groupal") is defined: they are close to monoidal categories, but may not contain units: instead the querfunctor and (natural) functorial isomorphisms, the quertors, are considered (by analogy with the querelements in $n$-ary groups). The arity-nonreducible $n$-ary braiding is introduced and the equation for it is derived, which for $n=2$ coincides 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.

The commutativity property and its "breaking" are quite obvious and unique for binary algebraic structures, because the permutation group S 2 has only one non-identity element. If the operation is n-ary however, then one has n!´1 non-identity permutations from S n , and the uniqueness is lost. 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 by definition, but which can give commutativity in special cases. Mediality MURDOCH [1939] (acting on n 2 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 " 2, any medial magma is a commutative monoid, and moreover for binary groups commutativity immediately follows from mediality.
The second part of the paper is devoted to a similar consideration of tensor categories MACLANE [1971], ETINGOF ET AL. [2015]. We define polyadic tensor categories by considering an n-ary tensor product (which may not be iterated from binary tensor products) and n-ary coherence conditions for the corresponding associators. The peculiarities of polyadic semigroupal and monoidal categories 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 contain a "querfunctor" and "quertors" (similar to querelements in n-ary groups DÖRNTE [1929], POST [1940]). We introduce arity-nonreducible n-ary braidings and find the equation for them that in the binary case turns into the Yang-Baxter equation in the tensor product form. Finally, we apply the "commutativity-to-mediality" ansatz to braided tensor categories JOYAL AND STREET [1993] and introduce "medialing" and corresponding "medialed" tensor categories.
The proposed "commutativity-to-mediality" ansatz can lead to medial n-ary superalgebras and Lie superalgebras, as well as to a medial analog of noncommutative geometry.

PRELIMINARIES
The standard way to generalize the commutativity is using graded vector spaces and corresponding algebras together with the commutation factor defined on some abelian grading group (see, e.g. RITTENBERG AND WYLER [1978], SCHEUNERT [1979] and BOURBAKI [1998], NASTASESCU AND VAN OYSTAEYEN [2004]). First, recall this concept from a slightly different viewpoint.
2.1. Binary gradation. Let A " A p2q " xA | µ 2 , ν 2 ; λ 1 y be an associative (binary) algebra over a field k (having unit 1 P k and zero 0 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 2 : A b A Ñ A is the (bilinear) binary multiplication (which we write as µ 2 ra, bs, a, b P A), usually in the binary case denoted by dot µ 2 " p¨q, and ν 2 : AbA Ñ A is the (binary) addition denoted by p`q, and a third (linear) operation λ 1 is the action λ 1 : K b A Ñ A (widely called a "scalar multiplication", but this is not always true, as can be seen from the polyadic case DUPLIJ [2019]).
where equality corresponds to strong gradation.
If there exist invertible elements of each degree a 1 P G, then A is called a cross product, and if all non-zero homogeneous elements are invertible, A is a graded division algebra DADE [1980]. Homogeneous (binary) morphisms ϕ : A G Ñ B G preserve the grading ϕ pA a 1 q Ă B a 1 , @a 1 P G, and the kernel of ϕ is an homogeneous ideal. The corresponding class of G-algebras and the homogeneous morphisms form a category of G-algebras G-Alg (for details, see, e.g. BOURBAKI [1998], DADE [1980]).
2.2. Almost commutativity. The graded algebras have a rich multiplicative structure, because of the possibility to deform (or twist) the algebra product µ 2 by a function depending on the gradation. Let us consider the twisting function (twist factor) τ : GˆG Ñ k.
The classes of σ form the (Schur) multiplier group SCHEUNERT [1979], and for further properties of σ and a connection with the cohomology classes H 2 pG, kq, see, e.g., COVOLO AND MICHEL [2016].
In general, the twisted product (2.2) can be any polynomial in algebra elements. Nevertheless, the special cases where µ pε 0 q 2 ra, bs 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 call the twisting function a 0-level commutation factor τ Þ Ñ ε 0 : GˆG Ñ kˆ, such that µ pε 0 q 2 ra, bs " µ 2 rb, as , or ε 0 pa 1 , b 1 q a¨b " b¨a, @a, b P A, a 1 , b 1 P G.
(2.12) 2.3. Tower of higher level commutation brackets. Let us now construct the tower of higher level commutation factors and brackets using the following informal reasoning. We "deform" the almost commutativity relation (2.4) by a function L 0 : AˆA Ñ A as where ε 0 pa 1 , b 1 q is the 0-level commuting factor satisfying (2.5)-(2.7). Consider the function (bracket) L pε 0 q 0 pa, bq as a multiplication of a new algebra called a 0-level bracket algebra. Then (2.13) can be treated as its "representation" by the associative algebra A.
Proposition 2.9. The algebra A L 0 2 is almost commutative with the commutation factor`´ε´1 0˘.
In the more symmetric form using (2.8) we have By analogy with (2.13) we successively further "deform" (2.15) then introduce "deforming" functions and higher level commutation factors in the following way.
Definition 2.10. The k-level commutation factor ε k pa 1 , b 1 q is defined by the following "differencelike" equations (2.20) Definition 2.11. k-level almost commutativity is defined by the vanishing of the last "deforming" function L pε 0 ,ε 1 ,...,ε k q k pa, bq " 0, @a, b P A, (2.21) and can be expressed in a form analogous to (2.4) pb, aq .
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 sums of 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 relation between n 2 elements in n-ary case), as a polyadic inductive generalization of commutativity. We Medial binary magmas and quasigroups 2. PRELIMINARIES then twist the multiplication by a gradation (as in the binary case above) to obtain the polyadic version of almost commutativity as almost mediality. However, let us first recall the binary and polyadic versions of the mediality property.
Let M " xM | µ 2 y be a binary magma (a closed set M with one binary operation µ 2 without any additional properties, also called a (Hausmann-Ore) groupoid 1 ).
Obviously, if a magma M contains a neutral element (identity) e P M, such that µ 2 ra, es " µ 2 re, as " a, @a P M, then M is commutative µ 2 ra, bs " µ 2 rb, as, @a, b P M. Therefore, any commutative monoid is an example of a medial magma. Numerous different kinds of magma and their classification are given in JEEK AND KEPKA [1983]. If a magma M is cancellative (µ 2 ra, bs " µ 2 ra, cs ñ b " c, µ 2 ra, cs " µ 2 rb, cs ñ a " b, @a, b, c P M), it is a binary quasigroup Q " xQ | µ 2 y for which the equations µ 2 ra, xs " b, µ 2 ry, as " b, @a, b P Q , have a unique solution HOWROYD [1973]. Moreover SHOLANDER [1949], every medial cancellative magma can be embedded in a medial quasigroup (satisfying (2.27)), and the reverse statement is also true JEŽEK AND KEPKA [1993]. For a recent comprehensive review on quasigroups(including medial and n-ary ones), see, e.g. SHCHERBACOV [2017], and references therein.
Let A " xA | µ 2 , ν 2 ; λ 1 y be a binary k-algebra, not necessarily unital, cancellative and associative. Then mediality provides the corresponding behavior which depends on the properties of the "vector multiplication" µ 2 . For instance, for unital cancellative and associative algebras, mediality implies commutativity, as for groups GŁAZEK AND GLEICHGEWICHT [1982].

ALMOST MEDIAL BINARY GRADED ALGEBRAS
Consider an associative binary algebra A over a field k. We introduce a weaker version of gradation than in (2.1).
Definition 3.1. An associative algebra A is called a binary higher graded algebra over k, if the algebra multiplication of four (" 2 2 ) elements respects the gradation where equality corresponds to strong higher gradation.
Definition 3.2. A twisted (binary) higher graded product µ pτ q 4 is defined for homogeneous elements by µ pτ q 4 ra, b, c, ds " τ pa 1 , b 1 , c 1 , d 1 q a¨b¨c¨d, a, b, c, d P A; a 1 , b 1 , c 1 , d 1 P G. (3.2) An analog of (total) associativity for the twisted binary higher graded product operation µ pτ q 4 is the following condition on seven elements (7 " 2¨2 2´1 ) for all a, b, c, d, t, u, v P A Next we propose a medial analog of almost commutativity as follows. Instead of deforming commutativity by the grading twist factor ε 0 as in (2.4), we deform the mediality (2.27) by the higher twisting function τ (3.2).
Proposition 3.6. If the algebra for which (3.6) holds is associative, the 0-level mediality factor ρ 0 satisfies the relations (3.10) Proof. As in (2.5), the relation (3.8) follows from applying (3.6) twice. The next ones follow from permutation in two ways using (3.5): for (3.9) a¨b¨pc¨d¨f¨gq¨h Þ Ñ a¨pc¨d¨f¨gq¨b¨h, a, c, d, f, g, b, h P A, (3.11) a¨b¨c¨d¨f¨g¨h Þ Ñ a¨c¨b¨d¨f¨g¨h Þ Ñ a¨c¨d¨b¨f¨g¨h Þ Ñ a¨c¨d¨f¨b¨g¨h Þ Ñ a¨c¨d¨f¨g¨b¨h, (3.12) and for (3.10) a¨pb¨c¨d¨f q¨g¨h Þ Ñ a¨g¨pb¨c¨d¨f q¨h, (3.13) a¨b¨c¨d¨f¨g¨h Þ Ñ a¨b¨c¨d¨g¨f¨h Þ Ñ a¨b¨c¨g¨d¨f¨h Þ Ñ a¨b¨g¨c¨d¨f¨h Þ Ñ a¨g¨b¨c¨d¨f¨h. (3.14) Assertion 3.7. If the 0-level almost medial algebra A pρ 0 q 2 is cancellative, then it is isomorphic to an almost commutative algebra.
Let us next introduce a 4-ary multiplication µ Definition 3.8. A 4-ary algebra is called a 0-level medial bracket algebra.

16) multiplied by a combination of the lower level mediality factors
Proof. It follows from the equations (3.19)-(3.21).

MEDIAL N-ARY ALGEBRAS
We now extend the concept of almost mediality from binary to polyadic (n-ary) algebras in the unique way which uses the construction from the previous section.
Let A pnq " xA | µ n , ν 2 y be an associative n-ary algebra (with n-ary linear multiplication A bn Ñ A) over a field k with (possible) polyadic unit e (then A pnq a unital k-algebra) defined by µ n re n´1 , as " a, @a P A (where a can be on any place) and (binary) zero z P A. We restrict ourselves (as in MICHOR AND VINOGRADOV [1996], GOZE ET AL. [2010]) by the binary addition ν 2 : A b A Ñ A which is denoted by p`q (for more general cases, see DUPLIJ [2019]). Now polyadic (total) associativity GOZE ET AL. [2010] can be defined as a kind of invariance DUPLIJ where a, c are (linear) polyads (sequences of elements from A) of the necessary length POST [1940], b pnq is a polyad of the length n, and the internal multiplication can be on any place. To describe the mediality for arbitrary arity n we need the following matrix generalization of polyads (as was implicitly used in DUPLIJ [2018a, 2019]).
Definition 4.1. A matrix (n-ary) polyadÂ pn 2 q "Â pnˆnq of size nˆn is the sequence of n 2 elementŝ A pnˆnq " pa ij q P A bn 2 , i, j " 1, . . . , n, and their product A pµq n 2 : A bn 2 Ñ A contains n`1 of n-ary multiplications µ n , which can be written as (we use hat for matrices of arguments, even informally) due to the total associativity (4.1) (by "omitting brackets").
This construction is the stack reshape of a matrix or row-major order of an array.

5)
A pn 2 q " pa ij q P A bn 2 . (4.6) Definition 4.4. A polyadic medial twist map χ pn 2 q medial is defined on the matrix polyads as DUPLIJ Definition 4.5. A n-ary algebra A pnq is called medial, if it satisfies the n-ary mediality property (4.5) for all a ij P A.
It follows from (4.3), that not all medial binary algebras are abelian.
Corollary 4.6. If a binary medial algebra A p2q is cancellative, it is abelian.
Assertion 4.7. If a n-ary medial algebra A pnq is cancellative, each matrix polyadÂ pn 2 q satisfies n 2´2 commutativity-like relations.
The gradation for associative n-ary algebras was considered in MICHOR AND VINOGRADOV [1996], GNEDBAYE [1995]. Here we introduce a weaker version of gradation, because we need to define the grading twist not for n-ary multiplication, i. e. the polyads of the length n, but only for the matrix polyads (4.6) of the length n 2 (for the binary case, see (3.1)).
Definition 5.1. An associative n-ary algebra A pnq is called a higher graded n-ary algebra over k, if the algebra multiplication of n 2 elements respects the gradation i.e.
Let us define the higher twisting function (higher twist factor) for n 2 elements τ n 2 : Gˆn 2 Ñ k by using matrix polyads (for n " 2 see (3.2)).
Definition 5.2. A n-ary higher graded twisted product µ pτ q n 2 is defined for homogeneous elements by µ whereÂ pn 2 q " pa ij q P A bn 2 is the matrix polyad of elements (4.6), andÂ 1 pn 2 q "`a 1 ij˘P G bn 2 is the matrix polyad of their gradings .
A medial analog of n-ary almost mediality can be introduced in a way analogous to the binary case (3.6).
Definition 5.3. If the higher twisted product coincides with the medially symmetric product (see (4.7)) for all a ij P A, we call the twisting function a 0-level n-ary mediality factor τ n 2 Þ Ñ ρ pn 2 q 0 : It follows from (5.4) that the normalization condition for the n-ary mediality factor is ρ pn 2 q 0¨n 2 hkkkikkkj a 1 , . . . , a 1‹ ‚" 1, @a 1 P G. Proof. It follows from (5.4) and its transpose together with the relation`B T˘T " B for any matrix over k.
Definition 5.5. An n-ary algebra for which the higher twisted product coincides with the medially symmetric product (5.4), is called a 0-level almost medial (ρ 0 -commutative) n-ary algebra A pρ 0 q n . Recall BOURBAKI [1998], that a tensor product of binary algebras can be naturally endowed with a ε 0 -graded structure in the following way (in our notation).
Using the matrix form (5.11) one can generalize the ρ 0 -graded medial algebras to arbitrary arity.
Let B pρ 0 q,1 n , . . . , B pρ 0 q,n n be n ρ 0 -graded (almost medial) n-ary algebras (B with the same mediality factor ρ 0 and the same graded structure. Consider their tensor product Proposition 5.10. If the ρ 0 -graded n-ary multiplication µ Symbolically, we can write this in the form, similar to the almost mediality condition (5.4) andB T pn 2 q is its transpose.
Example 5.11. In the lowest non-binary example, for 3 ternary ρ 0 -graded algebras A pρ 0 q , from (5.12) we have the ternary multiplication µ ‹pρ 0 q 3 for their ternary tensor product A pρ 0 q
Proof. We multiply the definition (5.17) by ρ pn 2 q 0ˆ´Â 1 pn 2 q¯T˙a nd use (5.6) to obtain Taking into account that the r.h.s. here is exactly´M and using (5.6) again, we get which should be compared with (5.4).
Now we "deform" (5.17) successively by defining further n 2 -ary brackets M k and higher level mediality factors ρ pn 2 q k : Gˆn 2 Ñ k as follows.
Definition 5.14. The k-level mediality n 2 -ary brackets and factors are defined by ρ pn 2 q Proof. This follows from the equations (5.23)-(5.25).

TOYODA'S THEOREM FOR ALMOST MEDIAL ALGEBRAS
The structure of the almost medial graded algebras (binary and n-ary) can be established by searching for possible analogs of Toyoda's theorem (2.28) (see, BRUCK [1944], MURDOCH [1941], TOYODA [1941]) which is the main statement for medial groupoids JEEK AND KEPKA [1983] and quasigroups SHCHERBACOV [2017]. As Toyoda's theorem connects medial algebras with abelian algebras, we can foresee that in the same way the almost medial algebras can be connected with almost commutative algebras.
The higher arity cases are more non-trivial, and very cumbersome. Therefore, we restrict ourselves by the case n " 3 only.

BINARY TENSOR CATEGORIES
We now apply the above ideas to construct a special kind of categories with multiplication BÉNABOU [1963], MAC LANE [1963] which appeared already in TANNAKA [1939] and later on were called tensor categories and monoidal categories (as they "remind" us of the structure of a monoid) MACLANE [1971]. For reviews, see, e.g. CALAQUE AND ETINGOF [2008], MÜGER [2010]. The monoidal categories can be considered as the categorification BAEZ AND DOLAN [1998a] of a monoid object, and can be treated as an instance of the microcosm principle: "certain algebraic structures can be defined in any category equipped with a categorified version of the same structure" BAEZ AND DOLAN [1998b]. We start from the definitions of categories ADÁMEK ET AL. [1990], BORCEUX [1994] and binary tensor categories MACLANE [1971] (in our notation).
Let C be a category with the class of objects Ob C and morphisms Mor C, such that the arrow from the source X 1 to the target X 2 is defined by Mor C Q f 12 : X 1 Ñ X 2 , X 1,2 P Ob C, and usually Hom C pX 1 , X 2 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 identity morphism (id X ) are defined in the standard way MACLANE [1971].
If C and C 1 are two categories, then a mapping between them is called a covariant functor F : C Ñ C 1 which consists of two different components: 1) the X-component is a mapping of objects F Ob : Ob C Ñ Ob C 1 ; 2) the f-component is a mapping of morphisms F Mor : Mor C Ñ Mor C 1 such that F " F Ob , F Mor ( . A functor preserves the identity morphism F Mor pid X q " id F Ob pXq and the composition of morphisms F Mor pf 23˝f12 q " F Mor pf 23 q˝1 F Mor pf 12 q (" F Mor pf 12 q˝1 F Mor pf 23 q for a contravariant functor), where p˝1q is the composition in C 1 .
The (binary) product category CˆC 1 consists of all pairs of objects pOb C, Ob C 1 q, morphisms pMor C, Mor C 1 q and identities pid X , id X 1 q, while the composition p˝2q is made component-wise pf 23 , f 1 23 q˝2 pf 12 , f 1 12 q " pf 23˝f12 , f 1 23˝1 f 1 12 q , (7.1) and by analogy this may be extended for more multipliers. A functor on a binary product category is called a bifunctor (multifunctor). A functor consists of two components 3 F Ob , F Mor ( , and therefore a mapping between two functors F and G should also be two-component Without other conditions T F G is called an infra-natural transformation from F to G. A natural transformation (denoted by the double arrow T F G : F ñ G) is defined by the consistency condition of the above mappings in C 1 Application to objects gives the following commutative diagram for the natural transformations (bifunctoriality) which is the consistency of the objects in C 1 transformed by F and G. The the diagonal in (7.3) may also be interpreted as the action of the natural transformation on a morphism T F G Mor pfq : F Ob pX 1 q Ñ G Ob pX 2 q, f : X 1 Ñ X 2 , f P Mor C, X 1 , X 2 P Ob C , such that where the second equality holds valid due to the naturality (7.2). In a concise form the natural transformations are described by the commutative diagram For a category C, the identity functor Id C "`Id C,Ob , Id C,Mor˘i s defined by Id C,Ob pXq " X, Id C,Mor pfq " f, @X P Ob C, @f P Mor C. Two categories C and C 1 are equivalent, if there exist two functors F and G and two natural transformations T F G : Id C 1 ñ F˝1G and T GF : G˝F ñ Id C .
The categorification BAEZ AND DOLAN [1998a], CRANE AND YETTER [1994] of most algebraic structures can be provided by endowing categories with an additional operation BÉNABOU [1963], MAC LANE [1963] "reminding" us of the tensor product MACLANE [1971].
A binary "magmatic" tensor category is`C, M p2bq˘, where M p2bq " b : CˆC Ñ C is a bifunctor 4
i " 1, 2, 3 (also denoted by sSGCat). Strict associativity is the equivalence Mor rf 2 , f 3 s ı . (7.10) Remark 7.1. Usually, only the first equation for the X-components is presented in the definition of associativity (and other properties), while the equation for the f-components is assumed to be satisfied "automatically" having the same form MACLANE [1971], STASHEFF [1970]. In some cases, the diagrams for M p2bq Ob and M p2bq Mor can fail to coincide and have different shapes, for instance, in the case of the dagger categories dealing with the "reverse" morphisms ABRAMSKY AND COECKE [2008].
The associativity relations guarantee that in any product of objects or morphisms different ways of 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) YETTER from the left side functor to the right side functor of (7.9)-(7.10) as Mor may be interpreted similar to the diagonal in (7.3), because the associators are natural transformations MACLANE [1971] or tri-functorial isomorphisms (in the terminology of BOYARCHENKO [2007]). Now different ways of inserting parentheses in a product of N objects give different results in the absence of conditions on the associator A p3bq . However, if the associator A p3bq satisfies some consistency relations, they can give isomorphic results, such that the corresponding diagrams commute, which is the statement of the coherence theorem MAC LANE [1963], KELLY [1964]. This can also be applied to SGCat, because it can be proved independently of existence of units YETTER [2001], BOYARCHENKO [2007], LU ET AL. [2019]. It was shown MAC LANE [1963] that it is sufficient to consider one commutative diagram using the associator (the associativity constraint) for two different rearrangements of parentheses for 3 tensor multiplications of 4 objects, giving the following isomorphism Ob rX 3 , X 4 s ıı . (7.12) The associativity constraint is called a pentagon axiom MACLANE [1971], such that the diagram 5

commutes.
A similar condition for morphisms, but in another context (for H-spaces), was presented in STASHEFF [1963,1970]. Note that there exists a different (but not alternative) approach to natural associativity without the use of the pentagon axiom JOYCE [2001].
The transition from the semigroupal non-strict category SGCat to the monoidal non-strict category MonCat can be done in a way similar to passing from a semigroup to a monoid: by adding the unit object E P Ob C and the (right and left) unitors U p2bq and commutes. 5 We omit M p2bq Ob in diagrams by leaving the square brackets only and use the obvious subscripts in A p3bq .
Using the above, the definition of a binary non-strict monoidal category MonCat can be given as the 6-tuple´C, M p2bq , A p3bq , E, U p2bq¯s uch that the pentagon axiom (7.13) and the triangle axiom (7.16) are satisfied MAC LANE [1963], MACLANE [1971] (see, also, KELLY [1964, 1965).
The following "normalizing" relations for the unitors of a monoidal non-strict category (7.17) can be proven JOYAL AND STREET [1993], as well as that the diagrams rrX 1 , X 2 s , Es The coherence theorem BÉNABOU [1963], MAC LANE [1963] proves that any diagram in a nonstrict monoidal category, which can be built from an associator satisfying the pentagon axiom (7.13) and unitors satisfying the triangle axiom (7.16), commutes. Another formulation MACLANE [1971] states that every monoidal non-strict category is (monoidally) equivalent to a monoidal strict one (see, also, KASSEL [1995]).
Thus, it is important to prove analogs of the coherence theorem for various existing generalizations of categories (having weak modification of units KOCK [2008], JOYAL AND KOCK [2013], ANDRIANOPOULOS [2017], and from the "periodic table" of higher categories BAEZ AND DOLAN [1995]), as well as for further generalizations (e.g., n-ary ones below).

POLYADIC TENSOR CATEGORIES
The arity of the additional multiplication in a category (the tensor product) was previously taken to be binary. Here we introduce categories with tensor multiplication which "remind" n-ary semigroups, n-ary monoids and n-ary groups DÖRNTE [1929], POST [1940] (see, also, GAL'MAK [2003), i.e. we provide the categorification CRANE AND FRENKEL [1994], CRANE AND YETTER [1994] of "higher-arity" structures according to the Baez-Dolan microcosm principle BAEZ AND DOLAN [1998b]. In our considerations we use the term "tensor category" in a wider context, because it can include not only binary monoid-like structures and their combinations, but also n-ary-like algebraic structures. It is important to note that our construction is different from other higher generalizations of categories 6 , such as 2-categories KELLY AND STREET [1974] GURSKI [2014], and obstructed categories DUPLIJ AND MARCINEK [2002, 2018b]. We introduce the categorification of "higher-arity" structures along DUPLIJ [2019] and consider their properties, some of them are different from the binary case (as in n-ary (semi)groups and n-ary monoids).

POLYADIC TENSOR CATEGORIES
Polyadic semigroupal categories Let C be a category MACLANE [1971], and introduce an additional multiplication as an n-ary tensor product as in DUPLIJ [2018aDUPLIJ [ , 2019.
Definition 8.1. An n-ary tensor product in a category C is an n-ary functor The n-ary composition of the f-components (morphism products of length n) is determined by the n-ary mediality property (cf. (4.5)) The identity morphism of the n-ary tensor product satisfies

(8.4)
Definition 8.2. An n-ary tensor product M pnbq which can be constructed from a binary tensor product M 1p2bq by successive (iterative) repetitions is called an arity-reduced tensor product 7 , and otherwise it is called an arity-nonreduced tensor product.
Categories containing iterations of the binary tensor product were considered in BALTEANU ET AL. [2003], CHENG AND GURSKI [2014]. We will mostly be interested in the arity-nonreducible tensor products and their corresponding categories.
Definition 8.3. A polyadic (n-ary) "magmatic" tensor category is`C, M pnbq˘, where M pnbq is an n-ary tensor product (functor (8.1)), and it is called an arity-reduced category or arity-nonreduced category depending on its tensor product. 8.1. Polyadic semigroupal categories. We call sequences of objects and morphisms X-polyads and f-polyads POST [1940], and denote them X and f, respectively (as in (4.1)). where X, Y, Z are X-polyads of the necessary length, and the total length of each pX, Y, Zq-polyad is 2n´1, while the internal tensor products in (8.5) can be on any of the n places.
Definition 8.6. A category`C, M pnbq˘i s called a polyadic (n-ary) strict semigroupal category sSGCat n , if the bifunctor M pnbq satisfies objects and unitors) the n-ary associativity condition (8.5).
Thus, in a polyadic strict semigroupal category for any (allowed, i.e. having the size k pn´1q`1, @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 (7.11), the transition to non-strict categories results in the consideration of independent isomorphisms instead of the equivalence (8.5).
Definition 8.7. The pn´1q pairs of X and f isomorphisms A p2n´1qb " are called n-ary associators being p2n´1q-place natural transformations, where A p2n´1qb Mor may be viewed as corresponding diagonals as in (7.3). Here i " 1, . . . , n´1 is the place of the internal brackets.
In the ternary case (n " 3) we have 2 " 3´1 pairs of the ternary associators Ob rX 2 , X 3 , X 4 s , X 5 Ob rX 3 , X 4 , X 5 s ı . (8.9) It is now definite that different ways of inserting parentheses in a product of N objects will give different results (the same will be true for morphisms as well), if we do not impose constraints on the associators. We anticipate that we will need (as in the binary case (7.12)) only one more (i.e. three) tensor multiplication than appears in the associativity conditions (8.5) to make a commutative diagram for the following isomorphism of 3¨pn´1q`1 " 3n´2 objects Conjecture 8.8 (N-ary coherence). If the n-ary associator A p2n´1qb satisfies such n-ary coherence conditions that the isomorphism (8.10) takes place, then any diagram containing A p2n´1qb together with the identities (8.4) commutes.

N -ARY UNITS, UNITORS AND QUERTORS
Introducing n-ary analogs of units and unitors is non-trivial, because in n-ary structures there are various possibilities: one unit, many units, all elements are units or there are no units at all (see, e.g., for n-ary groups DÖRNTE [1929], POST [1940], GAL'MAK [2003], and for n-ary monoids POP AND POP [2004]). A similar situation is expected in category theory after proper categorification CRANE AND FRENKEL [1994], CRANE AND YETTER [1994], BAEZ AND DOLAN [1998a] of n-ary structures.
9.1. Polyadic monoidal categories. Let´C, M pnbq , A p2n´1qb¯b e an n-ary non-strict semigroupal category SGCat n (see Definition 8.6) with n-ary tensor product M pnbq and the associator A p2n´1qb satisfying n-ary coherence. If a category has a unit neutral sequence of objects E pn´1q " pE 1 , . . . , E i q, E i P Ob C, i " 1, . . . , n´1, we call it a unital category. Note that the unit neutral 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 pnbq piq , i " 1, . . . , n (n-ary "unit morphisms" being natural transformations) are defined by U pnbq piq Ob : M pnbq Ob rE 1 , . . . E i´1 , X, E i`1 , . . . E n s » Ñ X, @X, E i P Ob C, i " 1, . . . , n´1. (9.1) The n-ary unitors U pnbq piq are compatible with the n-ary associators A p2n´1qb by the analog of the triangle axiom (7.16). In the binary case (7.14)-(7.15), we have U p2bq p1q " R p2bq , U p2bq p2q " L p2bq . 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 pn´1q and n unitors U pnbq piq , i " 1, . . . , n, that is a 5-tuple´C, M pnbq , A pnbq , E pn´1q , U pnbq¯s atisfying the "pn 2`1 qgon" axiom for the pn´1q associators A p2n´1qb piq and the triangle axiom (the analog of (7.16)) for the unitors and associators compatibility condition.
p1q Ob X,E,E x x r r r r r r r r r r r r r r r r r r r r r r X (9.5) commutes.
9.2. Polyadic nonunital groupal categories. The main result of n-ary group theory DÖRNTE [1929], POST [1940] is connected with units and neutral polyads: if they exist, then such n-ary group is reducible to a binary group. A similar statement can be true in some sense for categories. Conjecture 9.3. If a polyadic (n-ary) tensor category has unit object and unitors, it can be arityreducible to a binary category, such that the n-ary product can be obtained by iterations of the binary tensor product.
Therefore, it would be worthwhile to introduce and study non-reducible polyadic tensor categories which do not possess unit objects and unitors at all. This can be done by "categorification" of the querelement concept DÖRNTE [1929]. Recall that, for instance, in a ternary group xG | µ 3 y for an element g P G a querelementḡ is uniquely defined by µ 3 rg, g,ḡs " g, which can be treated as a generalization of the inverse element concept to the n-ary case. The mapping g Ñḡ can be considered as an additional unary operation (queroperation) in the ternary (and n-ary) group, while viewing it as an abstract algebra GLEICHGEWICHT AND GŁAZEK [1967] such that the notion of the identity is not used. The (binary) category of n-ary groups and corresponding functors were considered in MICHALSKI [1979,1984], IANCU [1991].
Let´C, M pnbq , A p2n´1qb¯b e a polyadic (n-ary) non-strict semigroupal category, where M pnbq is the n-ary tensor product, and A p2n´1qb is the associator making the "pn 2`1 q-gon" diagram of n-ary coherence commutative. We propose a "categorification" analog of the queroperation to be a covariant endofunctor of C.
Definition 9.4. A querfunctor Q : C Ñ C is an endofunctor of C sending Q Ob pXq "X and Q Mor pfq "f, whereX andf are the querobject and the quermorphism of X and f, respectively, such that the i diagrams (i " 1, . . . , n) » -n hkkkkikkkkj X, . . . , X fi fl Pr pnbq and Pr pnbq : C nb Ñ C is the projection. The action on morphisms Q pnbq piq Mor can be found using the diagonal arrow in the corresponding natural transformation, as in (7.3).
Conjecture 9.7. There exist polyadic nonunital non-strict groupal categories which are arity-nonreducible (see Definition 8.3), and so their n-ary tensor product cannot be presented in the form of binary tensor product iterations.

BRAIDED 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 bY " Y bX, @X, Y, P Ob C, and such tensor categories are called symmetric MACLANE [1971]. By analogy with associativity, one can introduce non-strict "commutativity", which leads to the notion of a braided (binary) tensor category and the corresponding coherence theorems JOYAL AND STREET [1993]. Various generalizations of braiding were considered in GARNER AND FRANCO [2016], DUPLIJ AND MARCINEK [2002, 2018a], and their higher versions are found, e.g., in KAPRANOV AND VOEVODSKY [1994], BATANIN [2010] The braiding B p2bq is connected with the associator A p3bq by the hexagon identity Ob 1,2,3 Ob 2,1,3 p3q Ob 1,23 Ob 2,3,1 w w ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ rX 2 , rX 3 , X 1 ss for objects, and similarly for the inverse associator.
Proposition 10.5. If the (binary) braided semigroupal category is strict (the associator becomes the equivalence (7.9)-(7.10), and we can omit internal brackets), then the diagram p2n´1q-ary natural transformations (see Definition 8.6). Now the braiding becomes an n-ary natural 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 (10.1). Note that in the consideration of higher braidings MANIN AND SCHECHTMAN [1989], KAPRANOV AND VOEVODSKY [1994] one ("order reversing") element of S n was used σ 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 pnbq Mor may be found from the corresponding diagonal of the natural transformation square (cf. (7.3)).
The n-ary braiding B pnbq is connected with the associator A p2n´1qb by a polyadic analog of the hexagon identity (10.2).
Conjecture 10.10 (Braided n-ary coherence). If the n-ary associator A p2n´1qb satisfies such n-ary coherence conditions that the isomorphism (8.10) takes place, and the n-ary braiding B pnbq satisfies the polyadic analog of the hexagon identity, then any diagram containing A p2n´1qb and B pnbq commutes.
There follows from (10.12), omitting indices, the ternary braid group relation in terms of tensor products (cf. the tetrahedron equation BAZHANOV AND STROGANOV [1982], KAPRANOV AND VOEVODSKY [1994], BAEZ AND NEUCHL [1995]) For the non-mixed "order reversing" n-ary braiding (see Definition 10.7) we have DUPLIJ [2018b] Proposition 10.12. The n-ary braid equation contains pn`1q multipliers, and each one acts on p2n´1q tensor products as (10.15) Remark 10.13. If a polyadic category is arity-nonreducible, then the higher n-ary braid relations cannot 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 pnbq satisfies the triangle identity connecting it with the unitors U pnbq .
Example 10.14. In the case of the ternary monoidal category MonCat 3 (see Example 9.2) the "order reversing" braiding B p3bq (10.10) satisfies an additional triangle identity analogous to (10.8) such that the diagram rX, E, Es For the polyadic non-unital groupal category GCat n (see Definition 9.6) the n-ary braiding B pnbq should be consistent with the quertors U pnbq and the querfunctor Q (see Definition 9.4).
Example 10.16. In the ternary groupal category GCat 3 (see Example 9.5) the "order reversing" braiding B p3bq (10.10) satisfies the additional identity of consistency with the querfunctor Q and In the compact matrix notation (see Definition 4.1) instead of (11.1) we have (symbolically) where the matrix polyads of objects is (cf. (4.5)) X pn 2 q " pX ij q P pOb Cq bn 2 , X ij P Ob C, (11.4) and p q T is matrix transposition.
Definition 11.4. A medialed polyadic semigroupal category´C, M pnbq , A p2n´1qb , M pn 2 bqm SGCat n is a polyadic non-strict semigroupal category SGCat n (see Definition 8.9) endowed with the n-ary medialing M p n 2 bq satisfying the n-ary medial coherence condition (a medial analog of the hexagon identity (10.2)).
Definition 11.5. A medialed polyadic monoidal category´C, M pnbq , A p2n´1qb , E, U pnbq , M p n 2 bqm MonCat n is a medialed polyadic semigroupal category mSGCat n with the unit object E P Ob C and the unitor U pnbq satisfying some compatibility condition.
Let us consider the polyadic nonunital groupal category GCat n (see Definition 9.6), then the n-ary medialing M pn 2 bq should be consistent with the quertors U pnbq and the querfunctor Q (see Definition 9.4 and also the consistency condition for the ternary braiding (10.17)) .
Definition 11.6. A braided polyadic groupal category´C, M pnbq , A p2n´1qb , Q, Q pnbq , M pn 2 bqm GCat n is a polyadic groupal category GCat n endowed with the n-ary medialing M p n 2 bq . 11.1. Medialed binary and ternary categories. Due to the complexity of the relevant polyadic diagrams, it is not possible to draw them in a general case for arbitrary arity n. Therefore, it would be worthwhile to consider first the binary case, and then some of the diagrams for the ternary case.
If a medialed semigroupal category mSGCat 2 contains a unit object and the unitor, then we have Definition 11.8. A medialed monoidal category mMonCat 2´C , M p2bq , A p3bq , E, U p2bq , M p4bq¯i s a (binary) medialed semigroupal category mSGCat 2 together with a unit object E P Ob C and a unitor U p2bq (7.14)-(7.15) satisfying the triangle axiom (7.16).
For mMonCat 2 the compatibility condition of the medialing M p4bq with E and U p2bq is given by the commutative diagram rrX 1 , Es , rX, X 2 ss M which is an analog of the triangle diagram for braiding (10.8).
Example 11.9. In the ternary nonunital groupal category GCat 3 (see Example 9.5) the medialing M p9bq satisfies the additional identity of consistency with the querfunctor Q and quertor Q p3bq such that the diagram Commutativity in polyadic algebraic structures is defined non-uniquely, if consider permutations and their combinations. We proposed a canonical way out: to substitute the commutativity property by mediality. Following this "commutativity-to-mediality" ansatz we first investigated mediality for graded linear n-ary algebras and arrived at the concept of almost mediality, which is an analog of almost commutativity. We constructed "deforming" medial brackets, which could be treated as a medial 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 a similar way. We defined polyadic tensor categories with an additional n-ary tensor multiplication for which 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 forthcoming paper.