Weighted Tensor Products of Joyal Species, Graphs, and Charades

Motivated by the weighted Hurwitz product on sequences in an algebra, we produce a family of monoidal structures on the category of Joyal species. We suggest a family of tensor products for charades. We begin by seeing weighted derivational algebras and weighted Rota-Baxter algebras as special monoids and special semigroups, respectively, for the same monoidal structure on the category of graphs in a monoidal additive category. Weighted derivations are lifted to the categorical level.


Introduction
There are monoidal structures on the category GphV of graphs in a monoidal additive category V for which weighted derivational monoids and weighted Rota-Baxter monoids (see [10], for example) can be seen as monoids (also called "algebras") and semigroups (or "non-unital associative algebras"). We also produce a family of monoidal structures on the category of Joyal species. In particular, this defines an interesting family of tensor products for linear representations of the symmetric groups. We also suggest a family of tensor products for charades (see [8,21]) which generalizes, in particular, the essentially classical tensor product of representations of the general linear groups over a finite field, proved braided in [16].
Weighted derivations are defined on monoidal categories with finite coproducts over which the tensor product distributes.

Review of λ-weighted derivations and Rota-Baxter operators
The inspiration for the family of tensor products on species came from the λweighted product of Hurwitz series as discussed in [10,11] and their references. They begin by defining a derivation of weight λ on an algebra A over a commutative ring k, with given λ ∈ k, to be a k-module morphism d : However, I prefer to write this in the form d n (ab) = n=r+s+t n r, s, t λ t d r+t (a)d s+t (b) (2.1) to emphasise the relationship to the trinomial expansion rule for (x + y + λxy) n . Here n r, s, t = n! r!s!t! .
The λ-Hurwitz product on A N can be defined by the clearly related equation (f · λ g)(n) = n=r+s+t n r, s, t λ t f (r + t)g(s + t) .
is a λ-weighted derivation.
Example 3. Define d : A N → A N by d(s)(n) = s(n + 1) − s(n). This d is a 1-weighted derivation when A N is equipped with the pointwise addition and multiplication.
Example 4. Define d : A N → A N by d(f )(n) = f (n + 1). This d is a λ-weighted derivation when A N is equipped with the λ-Hurwitz product for any λ. Notice that we have an algebra morphism d * : A N → (A N ) N defined by d * (f )(m)(n) = f (m+n). This may motivate the next definition.
Define d * : A → A N by d * (a)(n) = d n (a). We see that the Leibnitz rule (2.1) amounts to: on the category Alg k of k-algebras whose Eilenberg-Moore-coalgebras are k-algebras A equipped with a λ-derivation, so-called λ-derivation algebras; write DA λ for the category of these. The morphism d * : A → A N is the coaction of the comonad.
Where there is differentiation, there should also be integration. A Rota-Baxter operator of weight λ on a k-algebra A is a k-linear morphism P : A → A satisfying P (a)P (b) = P (P (a)b + P (a)b + λab) . (2.4) The pair (A, P ) is called a λ-weighted Rota-Baxter algebra. Write RBA λ for the category of these.
Example 6. For λ = 0, k = R and A the algebra of continuous functions f : R → R under pointwise addition and multiplication, the integration function P : , is a 0-weighted Rota-Baxter operator by the classical integration-by-parts rule.
Example 7. For λ = 1 and any k-algebra A, define P : A N → A N to take a sequence u in A to its sequence P (u) of partial sums: Then P is a 1-weighted Rota-Baxter operator on A N with pointwise addition and multiplication. See [1,20]. For d the consecutive difference operator as defined in Example 3, notice that d • P = 1 A N . Example 8. If Q is a 1-weighted Rota-Baxter operator on A then P (a) = λQ(a) defines a λ-weighted Rota-Baxter operator P on A.
A λ-weighted derivation RB-algebra is a k-algebra A equipped with a λ-weighted derivation d and a λ-weighted Rota-Baxter operator P such that d • P = 1 A . Write DRB λ for the category of these.

Proposition 9.
[See [10]] Let P be a RB-operator of weight λ on A. Then A N equipped with the λ-Hurwitz product, the derivation d of Example 4, and P defined by is a λ-weighted derivation RB-algebra. Moreover, the following square commutes.
With a little more work following Proposition 9, we see that the comonad G (2.3) lifts to RBA λ . In particular, with V denoting the forgetful functor, we have a comonadḠ and a commutative square Write RBA λ− for the category of λ-weighted Rota-Baxter algebras where we do not insist on the algebras having a unit.
The monoids in this monoidal category are precisely λ-derivation algebras. Moreover, U (2.5) and its right adjoint form a monoidal adjunction which therefore defines an adjunction between the categories of monoids. This adjunction generates the comonad G (2.3) on the category MonV of monoids in V .
2. There is a bialgebra structure on the polynomial algebra k[x] with comultiplication the algebra morphism δ : Then the convolution product on the left-hand side of the canonical isomorphism transports to the λ-Hurwitz product on A N .
3. It feels like there should be a multicategory/promonoidal/substitude structure on [ΣN, V ] for dealing with RB-algebras.

Graphs in monoidal additive categories
Let V be a monoidal additive category. We act as if the monoidal structure were strict. Let GphV be the category of directed graphs in V . So an object has the form of a pair of parallel morphisms s, t : E −→ A in V ; we use s and t for source and target morphisms in all graphs. A morphism (f, φ) : (A, E) −→ (B, F ) in GphV consists of morphisms f and φ making the following diagram commute.
Write ver : GphV −→ V for the forgetful functor taking (A, E) to A and write edg : GphV −→ V for the forgetful functor taking (A, E) to E. We will use the notation n = {1, 2, . . . , n}. For R ⊆ n , write for the characteristic function of R defined by Choose an endomorphism λ : I → I of the tensor unit I in V . For any f : Given a list (A 1 , E 1 ), . . . , (A n , E n ) of objects of GphV , we define an n-fold tensor product For n = 2 this gives a binary tensor product The unit for this tensor is the graph (I, I) with s = 0 : I → I and t = 1 I : I → I.

Proposition 12.
A monoidal structure on GphV is defined by (3.6) for any given λ ∈ V (I, I). Both ver and edg : GphV −→ V are strict monoidal.
Proof. Easy calculations of the source morphisms for show they agree with that of the triple tensor product. The target morphisms obviously agree. What this really means is that the associativity constraints for V lift through ver and edg to GphV and are therefore coherent.
where φ is forced to be f • t : F → A. Clearly J is fully faithful and the monoidal structure of Proposition 12 restricts to a monoidal structure on [ΣN, V ] yielding (3.8) as a strict monoidal functor.
equipped with the monoidal structure obtained as the restriction through (3.8) of that of Proposition 12 on GphV .
More explicitly, a λ-weighted derivational monoid is a monoid A in V equipped with an endomorphism d : A → A satisfying the λ-weighted equation: There is an isomorphism of categories taking (A, E) to (A, E) op for which A and E are unchanged but s and t have been interchanged. Put Like J, this composite J op is fully faithful. However, the image of J op is not closed under the monoidal structure of Proposition 12. All we obtain on [ΣN, V ] is a structure of multicategory (sometimes called a "coloured operad"). The sets of multimorphisms are defined by To be more explicit, for R ⊆ n and i ∈ n , put Then, an element of the set (3.13), a multimorphism, is a morphism satisfying the equation This definition should make the calculation of free weighted Rota-Baxter monoids possible; compare [20,2] for λ = 1, −1 for the case of commutative algebras.
To make Definition 14 a little more explicit, as expected, a λ-weighted Rota- (3.14) Derivations and Rota-Baxter operators are not the only sources of semigroups and monoids for the monoidal structure of Proposition 12. The forgetful functor taking the graph (A, E) to the pair (A, E), is strict monoidal and has a right adjoint R defined by with s = pr 1 (the first projection) and t = pr 2 (the second projection). It follows that R is monoidal and hence takes monoids to monoids.
Example 15. Take V = Mod k , the category of modules over a commutative ring k. For a graph (A, E) in this V , we can write e : a → b to mean a, b ∈ A, e ∈ E with s(e) = a, t(e) = b. For k-algebras A and B, we obtain a monoid R(A, B) in GphV : the graph is pr 1 , pr 2 : A ⊕ A ⊕ B → A and the multiplication is defined by: Of course the V -functor J (3.8) has both adjoints if V is complete and cocomplete enough. In particular, the right adjoint is defined by taking K(A, E) to be the equalizer of the two morphisms Here succ : N → N is the successor function n → n + 1. Since J (3.8) is strong monoidal for the monoidal structures under discussion, the adjunction J ⊣ K is monoidal. So K takes semigroups to semigroups and monoids to monoids. In particular, if (A, p) is a λ-weighted Rota-Baxter monoid in V , then K takes the graph (A, A) with s = 1 A and t = p to a λ-weighted derivational monoid in V . The underlying monoid is the limit of the diagram with d(a) n = a n+1 . Moreover, K(J(A, p) op ) supports a λ-weighted Rota-Baxter operator p defined by p(a) n = p(a n ). Notice too that d • p = 1.
We conclude this section by describing the promonoidal structure in the sense of Day [4] with respect to which the monoidal structure of Proposition 12 is convolution.
Let G denote the category whose only objects are 0 and 1, with the only non-identity morphisms σ, τ : 1 → 0. Write I * G for the free V -category on V .
where the first set of square brackets means the ordinary functor category while the second means the V -enriched functor category. The promonoidal structure in question is technically on I * G in the V -enriched sense. However, we can look at it as consisting of an ordinary a functor and an object J ∈ GphV . Of course J is just the graph 0, 1 : I → I which is the tensor unit. We can regard P as a "cograph of cographs of graphs" (although a cograph looks just like a graph): The L-tensor product of species Let S denote the groupoid whose objects are finite sets and whose morphisms are bijective functions. We write U + V for the disjoint union of sets U and V ; this is the binary coproduct as objects of the category Set of sets and all functions. It is not the coproduct in S; yet it does provide the symmetric monoidal structure on S of interest here. When we write X = A + B for A and B subsets of a set X, we mean X = A ∪ B and ∅ = A ∩ B.
Let V denote a monoidal category with finite coproducts which are preserved by tensoring on either side by an object. The tensor product of V, W ∈ V is denoted by V ⊗ W and the unit object by I. Justified by coherence theorems (see [15] for example), we write as if the monoidal structure on V were strictly associative and strictly unital. For any set S, write S · V for the coproduct of S copies of V ∈ V , when it exists (as it does for S finite).
The category of V -valued Joyal species, after [12,13], is the functor category [S, V ]. The objects will simply be called species when V is understood.
Suppose L : S → ZV is a braided strong monoidal functor into the monoidal centre (in the sense of [14]) of V . We have natural isomorphisms If V itself is braided (a fortiori symmetric), we can take a braided strong monoidal functor L : S → V since then there is a canonical braided strong monoidal functor V → ZV . By way of example, we could have any finite set Λ and LX = Λ X · I with Lσ = Λ σ −1 · I for any bijective function σ.
Define the L-tensor product F ⊗ L G of species F and G on objects X ∈ S by The definition of F ⊗ L G on morphisms is clear since any bijective function σ : X → Y restricts to bijections Let J : S → V be the species whose value at X is the unit I for tensor in V when X is empty and is initial in V otherwise. Clearly J is a unit for the L-tensor product in the sense that we have canonical isomorphisms λ G : J ⊗ L G → G and ρ F : F → F ⊗ L J .

Associativity isomorphisms
are obtained using the following fact easily proved by Venn diagrams.
In the case L = J, we recover from (4.19) the usual convolution (Cauchy) product of species appearing in [12]. For a general L, the term L(U ∩ V ) can be considered a measure of the failure of U and V to be disjoint.

A combinatorial interpretation
We consider the case where V = Set so that [S, Set] is the category of species as studied in [12]. Fix any set Λ. Define the species L by and (Lσ)S = (σS λ ) λ∈Λ . In other words, a structure of the species L on the set X is a partition of X into a Λ-indexed family of disjoint (possibly empty) subsets. A structure of the species F ⊗ L G on the set consists of a quintuplet (U, V, S, φ, γ) where U, V are subsets of X such that X = U ∪V , and S,φ,γ are L-,F -,G-structures on U ∩ V , U , V , respectively.
We write #S for the cardinality of the set S. We assume Λ is finite and put λ = #Λ.
The cardinality sequence of a species F is the sequence #F : N → Z defined by (#F )(n) = #F n .
We consider the λ-Hurwitz product (2.2) on Z N .
The iterated tensor and coherence Proposition 19. An alternative definition of F ⊗ L G is Proof. Given X = A + B + C, put U = A + C and V = B + C. Given X = U ∪ V , put A = U \V , B = V \U , and C = U ∩ V .
The n-fold version of this tensor product is This yields the formula in Proposition 19 for n = 2 by taking A = A 1 , B = A 2 , C = A {1,2} . Note that (6.22) is unchanged if we replace n by any set of cardinality n. Let us consider the effect of inserting one pair of parentheses in a multiple tensor (6.22). We look at Using (6.22) twice, once with n = p + 1 + r and once with n = q, we obtain the expression providing a partition X = T B T of X, together with all families providing a partition ⋆∈T B T = R C R of ⋆∈T . Using the fact that L lands in ZV and that L is strong monoidal, we obtain the isomorphic expression partitioning X by defining Then we can recover the B and C families via We have the following equations: This shows that the sum of the expressions (6.25) over the pairs (B, C) is equal to (6.22) with n = p + q + r. Remember however that the tensor product + on S is not strict symmetric; the symmetry on S provides canonical bijections between the left-and right-hand sides of (i) and (ii). Since L is braided, we have constructed a natural isomorphism a p,q,r : ⊗ L n (F 1 , . . . , F p+q+r ) (6.26) ∼ = ⊗ L p+1+r (F 1 , . . . , F p , ⊗ L q (F p+1 , . . . , F p+q ), F p+q+1 , . . . , F p+q+r ) .
Now consider the Mac Lane-Stasheff pentagon for 2-fold bracketings of F 1 ⊗ L F 2 ⊗ L F 3 ⊗ L F 4 as the vertices. Let a : H → K denote one of the edges of the pentagon obtained using the associativity isomorphisms (4.20). There is a composite b of two isomorphisms, each using one instance of an isomorphism (6.26), which goes from ⊗ L 4 (F 1 , F 2 , F 3 , F 4 ) to H, and another one c : ⊗ L 4 (F 1 , F 2 , F 3 , F 4 ) → H. By coherence of the braided strong monoidal functor L, it follows that a • b = c. Commutativity of the pentagon is a consequence of commutativity of all these triangular sides of the so-formed pentagonal cone.

Promonoidal structures on S
For finite sets A, B and X, let Cov(A, B; X) denote the set of jointly surjective pairs (µ, ν) of injective functions We write A × X B for the pullback of µ and ν.
Define a functor P :

The weighted bimonoidale structure on famS
Consider the 2-category Cat + of (small) categories admitting finite coproducts, and finite-coproduct-preserving functors. This becomes a symmetric closed monoidal bicategory (see [5]) with tensor product A ⊠ B representing functors H : A × B → X for which each H(A, −) and each H(−, B) is finite coproduct preserving. Clearly the monoidal category V of Section 4 is a monoidale (= pseudomonoid) in Cat + .
For any category C , we write famC for the free finite coproduct completion of C . That is, fam provides the left biadjoint to the forgetful 2-functor Cat + → Cat. Indeed, fam is a strong monoidal pseudofunctor; in particular, there is a canonical equivalence famC ⊠ famD ≃ fam (C × D) .
Every monoidal category C determines a monoidale famC in Cat + . Explicitly, the objects of famC can be written formally as s∈S C s where S is a finite set and C s ∈ C . Then, if C is monoidal, the monoidale structure on famC is defined by We are interested in famS. By what we have just said, this is a monoidale in Fix a finite set Λ and define L : S → Set by LX = Λ X and Lσ = Λ σ −1 . Define a coproduct-preserving functor Proposition 23. The functor ∆ of (8.28) is strong monoidal.
Proof. In ∆(X + Y ) = X+Y =A+B+C L(C) · (A + C, B + C) we can put as required.

Weighted categorical derivations
Harking back to Remark 11, we are prompted to consider the 2-category Here ΣS denotes the bicategory with one object (denoted ⋆) whose homcategory is the symmetric groupoid S; composition is provided by the monoidal structure + on S. Also V -Cat L,+ denotes the 2-category of V -categories admitting finite coproducts and tensoring with the object L(X) of V ; the morphisms are V -functors preserving these colimits; the 2-cells are V -natural transformations. The objects of (9.29) are pseudofunctors T : ΣS → V -Cat L,+ , the morphisms are pseudonatural transformations, and the 2-cells are modifications (in terminology of [18]). Such In this way, E (9.29) becomes a monoidal bicategory. An L-weighted derivation D * on a monoidale M in V -Cat L,+ is a lifting of the monoidale structure on M to a monoidale structure on (M , D * ) in E (9.29).
Example 24. An L-weighted derivation D * : The main point is the canonical isomorphism below.
Remark 25. The first item of Remark 11 has a categorical version. The forgetful 2-functor U : E → V -Cat L,+ has a right biadjoint JS taking the V -category A to the object of E determined be the V -category JSA = [S, A ] of species in A , equipped with L-weighted derivation the D * just as in Example 24 with the codomain V replaced by A . Since U is strong monoidal, the biadjunction U ⊣ bi JS is monoidal. Consequently the biadjunction lifts to one between the 2-categories of monoidales in E and V -Cat L,+ . Indeed U is pseudocomonadic.

The iterated tensor product again
Observe the following simple reindexing of (4.19).
Proposition 26. An alternative definition of F ⊗ L G is This leads us to another formula for the n-fold L-weighted tensor product. Define the modified n-filtration set on any finite set X by: Proposition 27. An alternative definition of the n-fold tensor product (6.22) is Proof. The formula follows by repeated application of the formula of Proposition 26 in evaluating the left bracketing Let us relate the formulas (6.22) and (10.33) in the case n = 3. A modified 3-filtration (U, V ) ∈ mFil 3 X of X amounts to subsets U 1 ⊆ U 2 ⊆ X and V 1 ⊆ U 1 , V 2 ⊆ U 2 . With this we can define and verify that X = A 1 + A 2 + A 3 + A 12 + A 13 + A 23 + A 123 . Conversely, given the partition A of X, we can define

Tensor products for charades
Motivated by Proposition 26, we consider generalizing the tensor product of [16]. Let G q be the groupoid of finite vector spaces over the field F q of cardinality q; the morphisms are linear bijections. We write V ≤ U to mean V is an F q -linear subspace of U , and we write U/V for the quotient space.
To be specific, take V = Vect C to be the category of complex vector spaces with all linear functions.
Let L : G q → V be a suitable functor: we will consider conditions on it later. For functors F, G : G q → V , define F ⊗ L G : G q → V by This leads us to an n-fold tensor product in a manner analogous to (10.33). Define the modified n-flag set on any finite F q -vector space X by: mFlg n X = (U, V ) | U = (0 = U 0 ≤ U 1 ≤ · · · ≤ U n−1 ≤ U n = X) , V = (V 0 , V 1 , . . . , V n−1 ) with V i ≤ U i for 0 ≤ i < n . (11.36) Now we put ⊗ L n (F 1 , . . . , F n )X = (U,V )∈mFlg n X L(U 1 \V 1 ) ⊗ · · · ⊗ L(U n−1 \V n−1 ) ⊗ F 1 (U 1 \V 0 ) ⊗ · · · ⊗ F n (U n \V n−1 ) . (11.37) Should this be the case, the tensor ⊗ L on [G q , V ] would be obtained from quite an interesting promonoidal structure on G q . A short sequence 11.50) in Vect Fq might be called short pre-exact when f is a monomorphism, g is an epimorphism and kerf ≤ img. Write Spes(A, B; X) for the set of such (f, g). Put This P : G op q × G op q × G q −→ V , defined on morphisms in the obvious way, would give the promonoidal structure in question. The term L (im(g • f )) measures the failure of the sequence (11.50) to be exact.

The dimension sequence
Following on from Section 11, we take F ∈ [G q , Vect C ] and define its dimension sequence dimF ∈ Z N by (dimF )n = dim F (F n q ) . (12.52) This inspires an algebra structure on A N for any k-algebra A. We assume we have λ ∈ k as before, but also some integer q (not necessarily a prime power). As in [16], we use φ n (q) = (q n − 1)(q n−1 − 1) . . . (q − 1) .
For f, g ∈ A N , put f · λ q g = r+s+t=n n r, s, t q λ t f (r + t)g(s + t) . (12.53) The calculations of Section 11 show that this is associative at least when A = Z, q is a prime power and λ = dimL(F). More generally, I claim A N is an associative k-algebra.