Theory and Applications of Categories, Vol. 18, No. 16, 2007, pp. 440–472.

NUCLEI OF CATEGORIES WITH TENSOR PRODUCTS

ALEXEI DAVYDOV

Abstract. Following the analogy between algebras (monoids) and monoidal categories the construction of nucleus for non-associative algebras is simulated on the categorical level. Nuclei of categories of modules are considered as an example.

1. Introduction

The property

xJ(y, z, w) − J(xy, z, w) + J(x, yz, w) − J(x, y, zw) + J(x, y, z)w = 0  (1)

of the associator J(x, y, z) = x(yz) − (xy)z guarantees that the subspaces

N_{l}(A) = {a ∈ A : J(a, x, y) = 0 ∀x, y ∈ A},  (2)
N_{m}(A) = {a ∈ A : J(x, a, y) = 0 ∀x, y ∈ A},  (3)
N_{r}(A) = {a ∈ A : J(x, y, a) = 0 ∀x, y ∈ A}  (4)

of an algebra A are associative subalgebras (see for example [4]). We call these subalgebras left, middle and right nucleus respectively.

In section 2.1 we categorify these constructions by defining left, middle and right nuclei N_{l}(G), N_{m}(G), N_{r}(G) for a category G with a tensor product (magmoidal category). The tensor product transports to the nuclei. Moreover there is a natural choice of associativity constraint making them semigroupal categories (the difference between semigroupal and monoidal is the lack of unit object).

The construction resembles the well-know categorification of the centre of an associative algebra (the so called monoidal centre [3]). In both cases the categorifications are categories of pairs. The first component of a pair is an object of the original category when the second should be seen as a specialisation of the appropriate constraint (braiding in the case of the monoidal centre, associativity in the case of nucleus). Coherence then used to define tensor product of pairs and to check the desired properties of resulting monoidal categories.

The work on the paper was supported by Australian Research Council grant DP00663514. The author thanks members of the Australian Category Seminar for stimulating discussions and the referees for numerous corrections and suggestions. Special thanks are due to R. Street for invaluable support during the work on the paper.

Received by the editors 2007-03-12 and, in revised form, 2007-10-08.

Transmitted by RJ Wood. Published on 2007-10-11.

2000 Mathematics Subject Classification: 18D10.

Key words and phrases: monoidal categories, monoidal functors, quasi-bialgebras.

\begin{matrix}c \\ ◯ \end{matrix} Alexei Davydov, 2007. Permission to copy for private use granted.

Categorical analogues of the inclusions N_{l}(A), N_{m}(A), N_{r}(A) ⊂ A are tensor product preserving (forgetful) functors N_{l}(G), N_{m}(G), N_{r}(G) [figure] G.

If an algebra A is already associative, its nuclei coincide with it. For a semigroupal category G we only get semigroupal functors G [figure] N_{l}(G), N_{m}(G), N_{r}(G), which split the corresponding forgetful functor. Semigroupal structures on G are in 1-to-1 correspondence with splittings of the forgetful functor.

If a (non-associative) algebra A has a unit (an element e ∈ A such that ex = xe = x for any x ∈ A) then the unit belongs to all nuclei N_{ϵ}(A) making them associative unital. In section 2.5 we modify the constructions of nuclei for the case of a magmoidal category with unit. The modified constructions give monoidal categories N_{l}^{1}(G), N_{m}^{1}(G), N_{r}^{1}(G).

Constructions of self-symmetries of categorical nuclei which we describe in section 2.9 do not have any analogues in algebra. Here we associate, to any natural transformation of the tensor product functor on G, monoidal autoequivalences of the nuclei of G.

In section 2.13 we define certain subcategories of nuclei and show how they are acted upon by the self-symmetries of section 2.9.

Note that in the classical algebraic construction the multiplication map of an algebra A makes it a left N_{l}(A)-module, a right N_{r}(A)-module, and a left- and right- N_{m}(A)-module. Moreover, left multiplications by elements of N_{l}(A) commute with right multiplications by elements of N_{m}(A) and N_{r}(A); while right multiplications by elements of N_{r}(A) commute with left multiplications by elements of N_{m}(A) and N_{l}(A). Finally, the multiplication map A ⊗ A [figure] A factors through A ⊗_{N_{m}(A)} A and is a morphism of N_{l}(A)-N_{r}(A)-bimodules. In section 2.16 we categorify some of these constructions by defining actions of nuclei N (G) on the category G.

This could give an impression that coherence laws of monoidal categories guarantee that categorifications of algebraic constructions are nicely coherent themselves. Unfortunately it is not always true, especially if commutativity enters the picture. For example, if A is a commutative (non-associative) algebra, then different nuclei are related as follows

N_{l}(A) = N_{r}(A) ⊂ N_{m}(A).

Indeed, the following chain of equalities proves that an element a of N_{l}(A) belongs to N_{m}(A):

x(ay) = (ay)x = a(yx) = a(xy) = (ax)y = (xa)y.

To see that a belongs to N_{r}(A) one can use:

x(ya) = (ya)x = (ay)x = a(yx) = (yx)a = (xy)a.

On the level of categories we have similar relations between nuclei of a category G with commutative tensor product:

N_{l}(G) ≃ N_{r}(G) [figure] N_{m}(G).

But the functors between different nuclei are not monoidal.

In the second part of the paper we deal with a similar construction making maps between associative algebras multiplicative. Let f : A [figure] B be a linear map of associative algebras. Define m_{f}(x, y) = f(xy) − f(x)f(y). The property

f(x)m_{f}(y, z) − m_{f}(xy, z) + m_{f}(x, yz) − m_{f}(x, y)f(z) = 0

implies that the subspaces

M_{l}(f) = {a ∈ A : m_{f}(a, x) = 0 ∀x ∈ A}, M_{r}(f) = {a ∈ A : m_{f}(x, a) = 0 ∀x ∈ A}

of A are subalgebras (we call them the left and right multiplicants of the map f). It is straightforward to see that the restrictions of f to M_{l}(f) and M_{r}(f) are homomorphisms of algebras.

Again we face problems when we try to bring in commutativity. For example, the left and right multiplicants M_{l}(f), M_{r}(f) of a linear map f : A [figure] B between commutative algebras coincide. The following chain of equalities shows that any a ∈ M_{l}(f) belongs to M_{r}(f):

f(xa) = f(ax) = f(a)f(x) = f(x)f(a).
But the equivalence M_{l}(F)  [figure] M_{r}(F) between multiplicants of a functor F : G  [figure] H

between symmetric monoidal categories fails to be monoidal.

2. Nuclei of a category with a tensor product

2.1. Nuclei as semigroupal categories. Let G be a category with a tensor product functor ⊗ : G × G [figure] G without any associativity condition (magmoidal category). Following (2) define the left nucleus N_{l}(G) of G as the category of pairs (A, a), where A ∈ G and a denotes a natural family of isomorphisms

a_{X,Y} : A ⊗ (X ⊗ Y) [figure] (A ⊗ X) ⊗ Y

indexed by all pairs of objects X, Y ∈ G. A morphism (A, a) [figure] (B, b) in N_{l}(G) is a morphism f : A [figure] B in G such that the diagram

\xymatrix{A ⊗ (X ⊗ Y) \ar[d]^{f⊗(X⊗Y)} \ar[r]^{a_{X,Y}} & (A ⊗ X) ⊗ Y \ar[d]^{(f⊗X)⊗Y} \\ B ⊗ (X ⊗ Y) \ar[r]^{b_{X,Y}} & (B ⊗ X) ⊗ Y}

is commutative for any X, Y ∈ G. Define the tensor product on the category N_{l}(G) by (A, a) ⊗ (B, b) = (A ⊗ B, a|b), where a|b is defined by the pentagon diagram:

[figure: A ⊗ (B ⊗ (X ⊗ Y)); (A ⊗ B) ⊗ (X ⊗ Y); a_{B,X⊗Y}; ((A ⊗ B) ⊗ X) ⊗ Y; (a|b)_{X,Y}; A ⊗ ((B ⊗ X) ⊗ Y); A⊗b_{X,Y}; (A ⊗ (B ⊗ X)) ⊗ Y; a_{B⊗X,Y}; a_{B,X}⊗Y]

The following diagram implies that for two morphisms f : (A, a) [figure](B, b), g : (A′, a′) [figure] (B′, b′) the tensor product f ⊗ g is a morphism between tensor products of pairs (A, a) ⊗ (A′, a′) [figure] (B, b) ⊗ (B′, b′).

[figure: A(B(XY)); (AB)(XY); a_{B,XY}; (A′B′)(XY); (fg)(XY); ((AB)X)Y; (a|b)_{X,Y}; ((A′B′)X)Y; ((fg)X)Y; A((BX)Y); Ab_{X,Y}; A′((B′X)Y); f((gX)Y); (A(BX))Y; a_{BX,Y}; (A′(B′X))Y; (f(gX))Y; a_{B,X}Y; A′(B′(XY)); f(g(XY)); a′_{B′,XY}; (a′|b′)_{X,Y}; A′b′_{X,Y}; a′_{B′X,Y}; a′_{B′,X}Y]  (5)

In particular, the forgetful functor N_{l}(G) [figure] G sending (A, a) to A is magmoidal. It follows from the diagram below that for (A, a), (B, b), (C, c) ∈ N_{l}(G) the isomor-

phism a_{B,C} : A ⊗ (B ⊗ C) [figure] (A ⊗ B) ⊗ C belongs to the nucleus N_{l}(G). Indeed, it is equivalent to the commutativity of the square

\xymatrix{(A ⊗ (B ⊗ C)) ⊗ (X ⊗ Y) \ar[d]^{a_{B,C}⊗(X⊗Y)} \ar[r]^{(a|(b|c))_{X,Y}} & ((A ⊗ (B ⊗ C)) ⊗ X) ⊗ Y \ar[d]^{(a_{B,C}⊗X)⊗Y} \\ ((A ⊗ B) ⊗ C) ⊗ (X ⊗ Y) \ar[r]^{((a|b)|c)_{X,Y}} & (((A ⊗ B) ⊗ C) ⊗ X) ⊗ Y}

which is one of the faces of the diagram (K_{5} in Stasheff’s notation [5]):

[figure: A(B(C(XY))); A((BC)(XY)); Ab_{C,XY}; (A(BC))(XY); a_{BC,XY}; ((AB)C)(XY); a_{B,C}(XY); (((AB)C)X)Y); ((a|b)|c)_{X,Y}; ((A(BC))X)Y; (a|(b|c))_{X,Y}; (a_{B,C}X)Y; A(((BC)X)Y); A(b|c)_{X,Y}; (A((BC)X))Y; a_{(BC)X,Y}; (a_{BC}X)Y; (AB)(C(XY)); a_{B,C(XY )}; (a|b)_{C,XY}; (AB)((CX)Y)); (AB)c_{X,Y}; ((AB)(CX))Y; (a|b)_{CX,Y}; (a|b)_{C,X}Y; A(B((CX)Y)); A(Bc_{X,Y} ); a_{B,(CX)Y}; A((B(CX))Y); Ab_{CX,Y}; A(b_{C,X}Y); (A(B(CX)))Y; a_{B(CX),Y}; (Ab_{C,X})Y; a_{B,CX}Y]

Two other square faces of this diagram are commutative by naturality while the pentagon faces commute by the definition of the tensor product in N_{l}(G). Thus

ϕ_{(A,a),(B,b),(C,c)} : (A, a) ⊗ ((B, b) ⊗ (C, c)) [figure] ((A, a) ⊗ (B, b)) ⊗ (C, c),

defined by ϕ_{(A,a),(B,b),(C,c)} = a_{B,C}, is a morphism in the category N_{l}(G).

2.2. Theorem. The category N_{l}(G), with tensor product and associativity constraint defined above, is semigroupal.

Proof. All we need to verify is that the associativity constraint satisfies the pentagon axiom, which says that the diagram

[figure: A ⊗ (B ⊗ (C ⊗ D)); (A ⊗ B) ⊗ (C ⊗ D); a_{B,C⊗D}; ((A ⊗ B) ⊗ C) ⊗ D; (a|b)_{C,D}; A ⊗ ((B ⊗ C) ⊗ D); A⊗b_{C,D}; (A ⊗ (B ⊗ C)) ⊗ D; a_{B⊗C,D}; a_{B,C}⊗D]

is commutative for any (A, a), (B, b), (C, c), (D, d) ∈ N_{l}(G). This obviously follows from the definition of a|b.

Semigroupal structures on the magmoidal category G correspond to functors G [figure] N_{l}(G) which split the forgetful functor N_{l}(G) [figure] G. Note that this is close parallel with the corresponding result for monoidal centres (Proposition 4 from [2]).

2.3. Theorem. If G is a semigroupal category with the associativity constraint ϕ, the assignment A ↦ (A, ϕ_{A,−,−}) defines a semigroupal functor G [figure] N_{l}(G) whose composition

G  [figure] N_{l}(G)  [figure] G with the forgetful functor is the identity.
Conversely, a functor G  [figure] N_{l}(G), whose composition G  [figure] N_{l}(G)  [figure] G with the

forgetful functor is the identity, defines an associativity constraint on G thus turning it into a semigroupal category.

Moreover these constructions are mutually inverse.

Proof. The pentagon axiom for ϕ implies that ϕ_{A,−,−}|ϕ_{B,−,−} = ϕ_{A⊗B,−,−}, so that A ↦ (A, ϕ_{A,−,−}) is a strict semigroupal functor.

A splitting G [figure] N_{l}(G) of the forgetful functor N_{l}(G) [figure] G must have the form: A ↦ (A, a). It is straightforward to see that ϕ_{A,X,Y} = a_{X,Y} is an associativity constraint on G.

Diagrams of this section (as well as their faces, edges and vertices) can be parameterized by certain partially parenthesised words of objects (morphisms) of G. For example, the tensor product of two objects X ⊗ Y corresponds to the word XY, tensor products A ⊗ (X ⊗ Y) and (A ⊗ X) ⊗ Y correspond to the words A(XY) and (AX)Y respectively. The word AXY corresponds to the associativity morphism a_{X,Y} and the word ABXY to the pentagon diagram which was used to define the tensor product in N_{l}(G). Five edges of this pentagon correspond to the five ways of placing a pair of brackets ( ) in the word ABXY:

A(BXY), (AB)XY, A(BX)Y, AB(XY), (ABX)Y

which represent

A ⊗ b_{X,Y} , (a|b)_{X,Y} , a_{B⊗X,Y} , a_{B,X⊗Y} , a_{B,X} ⊗ Y

respectively. The diagram (5) for tensor product of morphisms corresponds to the word fgXY. Finally, Stasheff’s polytope corresponds to the word ABCXY. Again its faces are parameterized by placements of a single pair of brackets ( ) in the word ABCXY. Six pentagon faces of the Stasheff polytope correspond to the words:

A(BCXY), (AB)CXY, A(BC)XY, AB(CX)Y, ABC(XY), (ABCX)Y,

while three square faces are labeled by:

(ABC)XY, A(BCX)Y, AB(CXY).

The fact that the associativity morphism a_{B,C} is a morphism in the nucleus N_{l}(G) follows from the commutativity of the face (ABC)XY. Finally, the pentagon axiom for its associativity is encoded by ABCD.

Using the word presentation for diagrams we can sketch the proof of the analog of theorem 2.2 for N_{m}(G). Define N_{m}(G) to be the category of pairs (A, a), where A ∈ G and a is a natural collection of isomorphisms

a_{X,Y} : X ⊗ (A ⊗ Y) [figure] (X ⊗ A) ⊗ Y

which we label by the word XAY. As before morphisms in N_{m}(G) are morphisms in G compatible with natural isomorphisms. We can define the tensor product of pairs (A, a) ⊗ (B, a) = (A ⊗ B, a|b) using the pentagon with label XABY. The diagram of shape XfgY implies that the tensor product of morphisms f ⊗ g is a morphism of tensor products. It follows from commutativity properties of the shape XABCY that the formula ϕ_{(A,a),(B,b),(C,c)} = b_{A,C} defines a natural collection of isomorphisms in N_{m}(G):

ϕ_{(A,a),(B,b),(C,c)} : (A, a) ⊗ ((B, b) ⊗ (C, c)) [figure] ((A, a) ⊗ (B, b)) ⊗ (C, c).

The word ABCD gives the pentagon axiom for this constraint.

Similarly, define N_{r}(G) to be the category of pairs (A, a), with

a_{X,Y} : X ⊗ (Y ⊗ A) [figure] (X ⊗ Y) ⊗ A

which we label by the word XY A. Again, morphisms in N_{r}(G) are morphisms in G compatible with the natural isomorphisms. The tensor product of pairs (A, a) ⊗ (B, a) = (A ⊗ B, a|b) is defined by the pentagon with the label XY AB. The diagram of shape XY fg implies that the tensor product of morphisms f ⊗ g is a morphism of tensor products. The shape XY ABC implies that the formula ϕ_{(A,a),(B,b),(C,c)} = c_{A,B} defines a natural collection of isomorphisms in N_{r}(G):

ϕ_{(A,a),(B,b),(C,c)} : (A, a) ⊗ ((B, b) ⊗ (C, c)) [figure] ((A, a) ⊗ (B, b)) ⊗ (C, c).

Finally, the word ABCD gives the pentagon axiom for this constraint.

2.5. Units. In this section we slightly modify the construction to incorporate units. First we need to modify the setup.

Now let G be a magmoidal category with unit (an object 1 ∈ G with natural isomorphisms l_{X} : 1⊗X [figure] X, r_{X} : X⊗1 [figure] X such that l_{1} = r_{1}). Define the left unital nucleus N_{l}^{1}(G) as the category of pairs (A, a) where as before a_{X,Y} : A⊗(X ⊗Y) [figure] (A⊗X)⊗Y is natural collection of isomorphism but now satisfying normalisation conditions: the diagrams

[figure: A ⊗ (1 ⊗ X); A ⊗ X; A⊗l_{X}; A ⊗ (1 ⊗ X); (A ⊗ 1) ⊗ X; a_{1,X}; (A ⊗ 1) ⊗ X; A ⊗ X; r_{A}⊗X; A ⊗ (X ⊗ 1); A ⊗ X; A⊗r_{X}; A ⊗ (X ⊗ 1); (A ⊗ X) ⊗ 1; a_{X,1}; (A ⊗ X) ⊗ 1; A ⊗ X; A⊗X]

commute for any X ∈ G.

Define a natural collection of isomorphisms i_{X,Y} : 1 ⊗ (X ⊗ Y) [figure] (1 ⊗ X) ⊗ Y as

the composition \xymatrix{1 ⊗ (X ⊗ Y) \ar[r]^{l_{X⊗Y}} & X ⊗ Y \ar[r]^{(l_{X}⊗Y)^{−1}} & (1 ⊗ X) ⊗ Y.}

2.6. Lemma. The pair (1, i) is an object of N_{l}^{1}(G).

Proof. The normalisation conditions for the collection i follow from the commutative diagrams:

\xymatrix{1 ⊗ (1 ⊗ Y) \ar[rr]^{i_{1,Y}} \ar[dr]^{l_{1⊗Y}} \ar[dr]_{1⊗l_{Y}} &  & (1 ⊗ 1) ⊗ Y \ar[dl]^{r_{1}⊗Y} \ar[dl]_{l_{1}⊗Y} \\  & 1 ⊗ Y}

\xymatrix{1 ⊗ (X ⊗ 1) \ar[rr]^{i_{,1}} \ar[dr]_{1⊗r_{X}} \ar[dddr]_{l_{X⊗1}} &  & (1 ⊗ X) ⊗ 1 \ar[dl]^{r_{1⊗X}} \ar[dddl]^{l_{X}⊗1} \\  & 1 ⊗ X \ar[d]^{l_{X}} \\  & X \\  & X ⊗ 1 \ar[u]_{r_{X}}}

Here the equation l_{1⊗Y} = 1 ⊗ l_{Y} (used in the first diagram) follows from the naturality of l:

\xymatrix{1 ⊗ (1 ⊗ Y) \ar[r]^{l_{1⊗Y}} \ar[d]_{1⊗l_{Y}} & 1 ⊗ Y \ar[d]^{l_{Y}} \\ 1 ⊗ Y \ar[r]^{l_{Y}} & Y}

In the second diagram we use the naturality of l with respect to r_{X} and the naturality of r with respect to l_{X}.

2.7. Theorem. The category N_{l}^{1}(G) is monoidal with unit object (1, i) and unit natural isomorphisms

l_{(A,a)} : (1, i) ⊗ (A, a)  [figure] (A, a), r_{(A,a)} : (A, a) ⊗ (1, i)  [figure] (A, a)  (6)

defined by l_{(A,a)} = l_{A}, r_{(A,a)} = r_{A}.

Proof. First we need to check that the tensor product of pairs as it was defined for N_{l}(G) preserves the normalisation conditions. This follow from the commutative diagrams:

[figure: A(B(1Y)); A(BY); A(Bl_{Y} ); (AB)Y; a_{B,Y}; (AB)(1Y); a_{B,1Y}; (AB)l_{Y}; ((AB)1)Y; (a|b)_{1,Y}; r_{AB}Y; A((B1)Y); Ab_{1,Y}; A(r_{b}Y); (A(B1))Y; a_{B1,Y}; a_{B,1}Y; (Ar_{B})Y]

[figure: A(B(X1)); A(BX); A(Br_{X}); (AB)X; a_{B,X}; (AB)(X1); a_{B,X1}; (ABr_{X}; ((AB)X)1; (a|b)_{X,1}; r_{(AB)X}; A((BX)1); Ab_{X,1}; Ar_{BX}; (A(BX))1; a_{BX,1}; a_{B,X}1; r_{A(BX)}]

Thus the subcategory N_{l}^{1}(G) is closed under tensor product in N_{l}(G) and is a semigroupal category. By lemma 2.6, the pair (1, i) is an object of N_{l}^{1}(G). We need to check that the unit isomorphisms (6) are morphisms in the category N_{l}^{1}(G). It is equivalent to the commutativity of the squares:

\xymatrix{(1A)(XY) \ar[r]^{(i|a)_{X,Y}} \ar[d]_{l_{A}XY} & ((1A)X)Y \ar[d]^{(l_{A}X)Y} & (A1)(XY) \ar[r]^{(a|i)_{X,Y}} \ar[d]_{r_{A}XY} & ((A1)X)Y \ar[d]^{(r_{A}X)Y} \\ A(XY) \ar[r]^{a_{X,Y}} & (AX)Y & A(XY) \ar[r]^{a_{X,Y}} & (AX)Y}

which follows from the commutative diagrams:

[figure: 1(A(XY)); A(XY); l_{A(XY )}; (AX)Y; a_{X,Y}; (1A)(XY); i_{A,XY}; l_{A}(XY); ((1A)X)Y; (i|a)_{X,Y}; (l_{A}X)Y; 1((AX)Y); 1a_{X,Y}; l_{(AX)Y}; (1(AX))Y; i_{AX,Y}; i_{A,X}Y; l_{AX}Y]

Finally, the coherence for the unit isomorphisms

\xymatrix{(A, a) ⊗ ((1, i) ⊗ (B, b)) \ar[dr]_{(A,a)⊗l_{(B,a)}} \ar[rr]^{ϕ_{(A,a),(1,i),(B,b)}} &  & ((A, a) ⊗ (1, i)) ⊗ (B, b) \ar[dl]^{r_{(A,a)}⊗(B,b)} \\  & (A, a) ⊗ (B, b)}

[figure: A(1(XY)); A(XY); Al_{XY}; (AX)Y; a_{X,Y}; (A1)(XY); a_{1,XY}; r_{A}(XY); ((A1)X)Y; (a|i)_{X,Y}; (r_{A}X)Y; A((1X)Y); Ai_{X,Y}; A(l_{X}Y); (A(1X))Y; a_{1X,Y}; a_{1,X}Y; (Al_{X})Y]

boils down to

\xymatrix{A ⊗ (1 ⊗ B) \ar[rr]^{a_{1,B}} \ar[dr]_{l_{b}} &  & (A ⊗ 1) ⊗ B \ar[dl]^{r_{A}} \\  & A ⊗ B}

which is commutative by the definition of N_{l}^{1}(G).

The constructions of the following sections can be made unital (monoidal instead of just semi-groupal). For the sake of brevity we will not be doing it.

2.9. Functoriality. Here we assign, to an invertible natural self-transformation (an automorphism) c of the tensor product functor ⊗ : G × G [figure] G, a strict semigroupal autoequivalence N_{l}(c) : N_{l}(G) [figure] N_{l}(G) of the nucleus. For an object (A, a) of N_{l}(G) set N_{l}(c)(A, a) to be (A, a^{c}) where a^{c} is defined by the diagram:

[figure: (A ⊗ X) ⊗ Y; (A ⊗ X) ⊗ Y; c_{A⊗X,Y}; A ⊗ (X ⊗ Y); (A ⊗ X) ⊗ Y; α_{x,y}; A ⊗ (X ⊗ Y); A ⊗ (X ⊗ Y); c_{A,X⊗Y}; A ⊗ (X ⊗ Y); (A ⊗ X) ⊗ Y; (A ⊗ X) ⊗ Y; (A ⊗ X) ⊗ Y; c_{A,X}⊗Y; A ⊗ (X ⊗ Y); (A ⊗ X) ⊗ Y; A ⊗ (X ⊗ Y); A ⊗ (X ⊗ Y); A⊗c_{X,Y}; A ⊗ (X ⊗ Y); (A ⊗ X) ⊗ Y; α_{X,Y}^{c}]

2.10. Proposition. The functor N_{l}(c) is strict semigroupal.

Proof. The following commutative diagram shows that (a|b)^{c} = (a^{c}|b^{c}) for (A, a), (B, b) from N_{l}(G):

[figure: A(B(XY)); (AB)(XY); a_{B,XY}; ((AB)X)Y; (a|b)_{X,Y}; ((AB)X)Y; c_{(AB)X,Y}; ((AB)X)Y; c_{AB,X}Y; ((AB)X)Y; (c_{A,B}X)Y; (AB)(XY); c_{AB,XY}; (AB)(XY); c_{A,B}(XY); (AB)(XY); (AB)c_{X,Y}; (a^{c}|b^{c})_{X,Y}; (AB)(CX); (AB)c_{X,Y}; c_{A,B}(XY); (a|b)_{X,Y}^{c}; A((BX)Y); Ab_{X,Y}; (A(BX))Y; a_{BX,Y}; a_{B,X}Y; A((BX)Y); c_{A(BX),Y}; a_{B,X}Y; (A(BX))Y; c_{A,BX}Y; (A(BX))Y; (Ac_{B,X})Y; a_{B,X}^{c}Y; A((BX)Y); c_{A,(BX)Y}; A((BX)Y); Ac_{BX,Y}; A((BX)Y); A(c_{B,X}Y); a_{BX,Y}^{c}; a_{BX,Y}^{c}; A(B(XY)); c_{A,B(XY )}; Ab_{X,Y}; A(B(XY)); Ac_{B,XY}; a_{B,XY}^{c}; A(B(XY)); A(Bc_{X,Y} ); a_{B,XY}^{c}; Ab_{X,Y}^{c}]

Thus N_{l}(c)((A, a) ⊗ (B, b)) coincides with N_{l}(c)(A, a) ⊗ N_{l}(c)(B, b).

2.9. Functoriality. A diagram of that shaped appeared in [5] in connection with A_{∞}- maps.

By naturality, c_{A,X⊗Y} commutes with A ⊗ d_{X,Y} and c_{A⊗X,Y} commutes with d_{A,X} ⊗ Y for any c, d ∈ Aut(⊗). Thus we have that (a^{c})^{d} = a^{cd} which means that the composition N_{l}(c)N_{l}(d) coincides with N_{l}(cd). In other words, the assignment c ↦ N_{l}(c) defines a group homomorphism Aut(⊗) [figure] Aut^{⊗}(N_{l}(G)) into the group of monoidal autoequivalences.

Now we establish the link between natural transformations of the identity functor on G and monoidal natural transformations of functors of the form N_{l}(c). First we define an action of the group Aut(id_{G}) of automorphisms of the identity functor on the group Aut(⊗) of automorphisms of the tensor product functor. For c ∈ Aut(⊗) and f ∈ Aut(id_{G}), define c^{f} ∈ Aut(⊗) by

c_{X,Y}^{f} = (f_{X} ⊗ f_{Y})c_{X,Y} f_{X⊗Y}^{−1} .

Note that by naturality of c, c_{X,Y} commutes with f_{X} ⊗ f_{Y} and by naturality of f, c_{X,Y} commutes with f_{X⊗Y} so c_{X,Y}^{f} coincides with

(f_{X} ⊗ f_{Y})f_{X⊗Y}^{−1} c_{X,Y} = c_{X,Y} (f_{X} ⊗ f_{Y})f_{X⊗Y}^{−1} .

This in particular implies that Aut(id_{G}) acts on the group Aut(⊗):

c_{X,Y}^{fg} = c_{X,Y} (fg_{X} ⊗ fg_{Y})fg_{X⊗Y}^{−1} = c_{X,Y} (f_{X} ⊗ f_{Y})f_{X⊗Y}^{−1} (g_{X} ⊗ g_{Y})g_{X⊗Y}^{−1} = (c^{f})_{X,Y}^{g} .

2.12. Proposition. A natural transformation f ∈ Aut(id_{G}) of the identity functor defines a natural transformation N_{l}(c) [figure] N_{l}(c^{f}).

Proof. We need to show that for any (A, a) ∈ N_{l}(G) the map f_{A} : A [figure] A is a morphism (A, a) [figure] (A, a^{c^{f}}) in N_{l}(G) or that the square

\xymatrix{A ⊗ (X ⊗ Y) \ar[d]_{f_{A}⊗(X⊗Y)} \ar[r]^{a_{X,Y}^{c}} & (A ⊗ X) ⊗ Y \ar[d]^{(f_{A}⊗X)⊗Y} \\ A ⊗ (X ⊗ Y) \ar[r]^{a_{X,Y}^{c^{f}}} & (A ⊗ X) ⊗ Y}

is commutative. The commutativity of this square is equivalent to the commutativity of the bottom square of the following diagram:

[figure: A ⊗ (X ⊗ Y); c_{A,X⊗Y}^{f}; a_{X,Y}; (A ⊗ X) ⊗ Y; c_{A⊗X,Y}^{f}; A ⊗ (X ⊗ Y); f_{A⊗(X⊗Y )}; c_{A,X⊗Y}; a_{X,Y}; (A ⊗ X) ⊗ Y; f_{A⊗X)⊗Y}; c_{A⊗X,Y}; A ⊗ (X ⊗ Y); A⊗c_{X,Y}^{f}; (A ⊗ X) ⊗ Y; c_{A,X}^{f}⊗Y; A ⊗ (X ⊗ Y); f_{A}⊗f_{X⊗Y}; A⊗c_{X,Y}; (A ⊗ X) ⊗ Y; f_{A,X}⊗f_{Y}; c_{A,X}⊗Y; A ⊗ (X ⊗ Y); a_{X,Y}^{c^{f}}; (A ⊗ X) ⊗ Y; A ⊗ (X ⊗ Y); a_{X,Y}^{c}; f_{A}⊗(f_{X}⊗f_{Y} ); (A ⊗ X) ⊗ Y; (f_{A}⊗f_{X})⊗f_{Y}]

It is clear that the natural transformation N_{l}(f) : N_{l}(c) [figure] N_{l}(c^{f}) is monoidal iff f ∈ Aut(id_{G}) is monoidal, i.e. iff f_{X⊗Y} = f_{X} ⊗ f_{Y} for any X, Y ∈ G.

2.13. Subcategories of nuclei. Here we define semigroupal (monoidal) subcategories in nuclei anchored at quasi-monoidal functors. Let G be a magmoidal category, let H be a monoidal (semi-groupal) category and let F : G [figure] H be a functor equipped with a natural collection of isomorphisms F_{X,Y} : F (X ⊗ Y) [figure] F (X) ⊗ F (Y) (a quasi-monoidal functor). Define a full subcategory N_{l}(F) of N_{l}(G) consisting of pairs (A, a) such that the following diagram commutes:

\xymatrix{F (A ⊗ (X ⊗ Y)) \ar[d]_{F (a_{X,Y} )} \ar[r]^{F_{X,Y}} & F (A) ⊗ F (X ⊗ Y) \ar[r]^{F (A)⊗F_{X,Y}} & F (A) ⊗ (F (X) ⊗ F (Y)) \ar[d]_{ψ_{F (A),F (X),F (Y )}} \\ F ((A ⊗ X) ⊗ Y) \ar[r]^{F_{A⊗X,Y}} & F (A ⊗ X) ⊗ F (X) \ar[r]^{F_{X,A}⊗F (Y)} & (F (A) ⊗ F (X)) ⊗ F (Y)} (7)

here ψ is the associativity constraint of H.

2.14. Proposition. The subcategory N_{l}(F) is closed under tensor product in N_{l}(G). The quasi-monoidal functor F lifts to a monoidal (semi-groupal) functor \overline{F} : N_{l}(F) [figure] H.

Proof. We need to verify that the tensor product (A ⊗ B, a|b) of two pairs (A, a), (B, b) from N_{l}(F) is in N_{l}(F). It follows from the commutativity of the diagram:

[figure: F (A(B(XY))); F ((AB)(XY)); F (a_{B,XY} ); F (((AB)X)Y); F ((a|b)_{X,Y} ); F ((AB)X)F (Y); F_{(AB)X,Y}; (F (AB)F (X))F (Y); F_{AB,X}F (Y); ((F (A)F (B))F (X))F (Y); (F_{A,B}F (X))F (Y); F (AB)F (XY); F_{AB,XY}; (F (A)F (B))F (XY); F_{A,B}F (XY); (F (A)F (B))(F (X)F (Y)); (F (A)F (B))F_{X,Y}; ψ; F (AB)(F (X)F (Y)); F (AB)F_{,Y} X; F_{A,B}(F (C)F (X)); ψ; F (A((BX)Y)); F (Ab_{X,Y} ); F ((A(BX))Y); F (a_{BX,Y} ); F (a_{B,X}Y); F (A(BX))F (Y); F_{A(BX),Y}; F (a_{B,X})F (Y); (F (A)F (BX))F (Y); F_{A,BX}F (Y); (F (A)(F (B)F (X)))F (Y); (F (A)F_{B,X})F (Y); ψF (X); F (A)F ((BX)Y); F_{A,(BX)Y}; F (A)(F (BX)F (Y)); F (A)F_{BX,Y}; F (A)((F (B)F (X))F (Y)); F (A)(F_{B,X}F (Y)); ψ; ψ; F (A)F (B(XY)); F_{A,B(XY )}; F (A)F (b_{X,Y} ); F (A)(F (B)F (XY)); F (A)F_{B,XY}; ψ; F (A)(F (B)(F (X)F (Y))); F (A)(F (B)F_{X,Y} ); ψ; F (A)ψ]

Now define the functor \overline{F} : N_{l}(F) [figure] H by \overline{F} (A, a) = F (A). The coherence axiom for the compatibility isomorphism F_{X,Y} : F (X ⊗ Y) [figure] F (X) ⊗ F (Y) follows from the definition of N_{l}(F).

In the next proposition we explain how monoidal autoequivalences of nuclei (defined in section 2.9) permute the subcategories of nuclei. Let c_{X,Y} : X ⊗Y [figure] X ⊗Y be a natural collection of isomorphisms defining a semi-groupal functor N_{l}(c) : N_{l}(G) [figure] N_{l}(G). Let F : G [figure] H be a quasi-monoidal functor defining a semi-groupal subcategory N_{l}(F).

Define a new quasi-monoidal structure F^{c} on F by

\xymatrix{F_{X,Y}^{c} : F (X ⊗ Y) \ar[r]^{F (c_{X,Y} )} & F (X ⊗ Y) \ar[r]^{F_{X,Y}} & F (X) ⊗ F (Y)}

2.15. Proposition. The semigroupal functor N_{l}(c) defines an equivalence between the semi-groupal subcategories N_{l}(F) and N_{l}(F^{c}).

Proof. We need to show that the coherence diagram (7) commutes if we replace F_{X,Y} by F_{X,Y}^{c} and a_{X,Y} by a_{X,Y}^{c}. This follows from the commutative diagram:

[figure: F (A(XY)); F (A)F (XY); F_{X,Y}^{c}; F (A)(F (X)F (Y)); F (A)F_{X,Y}^{c}; (F (A)F (X))F (Y); ψ_{F (A),F (X),F (Y )}; F (A)F (XY); F (A)F (c_{X,Y} ); F (A)F_{X,Y}; F (A(XY)); F (c_{A,XY} ); F_{A,XY}; F (A(XY)); F (Ac_{X,Y} ); F_{A,XY}; F ((AX)Y); F (a_{X,Y} ); F (AX)F (Y); F_{AX,Y}; F_{A,X}F (Y); F ((AX)Y); F (a_{X,Y}^{c} ); F (AX)F (Y); F_{X,Y}^{c}; F_{A,X}^{c}F (Y); F (c_{A,X})F (Y); F ((AX)Y); F (c_{AX,Y} ); F_{AX,Y}; F (c_{A,X}Y)]

2.16. Actions of nuclei. Recall that a monoidal (semigroupal) category C acts on a category M if there is given a monoidal (semigroupal) functor C [figure] End(M) into the category of endofunctors on M. In that case M is called a (left) module category over C or a (left) C-category.

Let L : C [figure] End(M), L′ : C [figure] End(M′) be two actions. A functor F : M [figure] M′ between C-categories is a C-functor if it is equipped with a collection of isomorphisms F_{A,X} : F (L(A)(X)) [figure] L′(A)(F (X)) natural in A ∈ C, X ∈ M such that the diagram

[figure: F (L(AB)(X)); F_{AB,X}; F (L_{A,B}(X)); L′(AB)(F (X)); L′_{A,B}(F (X)); L′(A)(L′(B)(F (X))); F (L(A)(L(B)(X))); F_{A,L(B)(X)}; L′(A)(F (L(B)(X))); L′(A)(F_{B,X}); L′(A)(L′(B)(F (X)))]  (8)

commutes.

For a magmoidal category G define a functor L : N_{l}(G) [figure] End(G) by L(A, a)(X) = A ⊗ X. Define a monoidal structure L_{(A,a),(B,b)} : L((A, a) ⊗ (B, b)) [figure] L(A, a) ∘ L(B, b)

(a natural collection of isomorphisms of functors) by

[figure: L((A, a) ⊗ (B, b))(X); L_{(A,a),(B,b)}(X); L(A, a)(L(B, b)(X)); (A ⊗ B) ⊗ X; A ⊗ (B ⊗ X); a_{B,X}]

2.17. Proposition. The functor L : N_{l}(G) [figure] End(G) is semigroupal.

Proof. All we need to check is the coherence axiom for the monoidal structure, which follows from the definition of the tensor product in N_{l}(G):

[figure: L((A, a)((B, b)(C, c)))(X); L(ϕ_{(A,a),(B,b),(C,c)})(X); L_{(A,a),(B,b)(C,c)}(X); L(((A, a)(B, b))(C, c))(X); L_{(A,a)(B,b),(C,c)}(X); (A(BC))X; a_{B,C}X; ((AB)C)X; L(A, a)(L((B, b)(C, c))(X)); L(A,a)L_{(B,b),(C,c)}(X); L((A, a)(B, b))(L(C)(X)); L_{(A,a),(B.b)}(L_{(C,c)}(X)); A((BC)X); a_{BC,X}; (AB)(CX); (a|b)_{C,X}; L(A, a)(L(B, b)(L(C, c)(X))); (L(A, a)L(B, b))(L(C, c)(X)); A(B(CX)); Ab_{C,X}; A(B(CX)); a_{B,CX}]

Now consider the cartesian square G × G of a magmoidal category as a left N_{l}(G)- category by making the semigroupal category N_{l}(G) act on the first factor of G ×G. Define a natural collection of isomorphism ⊗_{(A,a),X,Y} : L(A, a)(X) ⊗ Y [figure] L(A, a)(X ⊗ Y) by

[figure: L(A, a)(X) ⊗ Y; ⊗_{(A,a),X,Y}; L(A, a)(X ⊗ Y); (A ⊗ X) ⊗ Y; A ⊗ (X ⊗ Y); a_{X,Y}]  (9)

2.18. Proposition. The functor ⊗ : G × G [figure] G between N_{l}(G)-categories is a N_{l}(G)- functor.

Proof. We need to check that the natural collection (9) satisfies the coherence condition (8). It follows from the diagram:

[figure: L((A, a)(B, b))(XY); L_{(A,a),(B,b)}(X)Y; ⊗_{(A,a)(B,b),X,Y}; L(A, a)(L(B, b)(X))Y; ⊗_{(A,a),L(B,b)(X),Y}; ((AB)X)Y; a_{B,X}Y; (A(BX))Y; L((A, a)(B, b))(XY); L_{(A,a),(B.b)}(XY); L(A, a)(L(B, b)(X)Y); L(A,a)(⊗_{(B,b),X,Y} ); (AB)(XY); (a|b)_{X,Y}; A((BX)Y); a_{BX,Y}; L(A, a)(L(B, b)(XY)); (L(A, a)L(B, b))(XY)); A(B(XY)); a_{B,XY}; A(B(XY)); Ab_{X,Y}]

In the next proposition we characterise left nucleus N_{l}(G) as a category of certain endofunctors on G. For a magmoidal category G a functor F : G [figure] G will be called (right) G-invariant if it is equipped with a natural collection of isomorphisms f_{X,Y} : F (X ⊗ Y) [figure] F (X) ⊗ Y. A morphism c : F [figure] G of G-invariant functors is a natural transformation such that the diagram

\xymatrix{F (X ⊗ Y) \ar[r]^{f_{X,Y}} \ar[d]^{c_{X⊗Y}} & F (X) ⊗ Y \ar[d]^{c_{X}⊗Y} \\ G(X ⊗ Y) \ar[r]^{g_{X,Y}} & G(X) ⊗ Y}

commutes. Define End_{G}(G) to be the category of G-invariant functors. The composite of two G-invariant functors (F, f), (G, g) has a structure of G-invariant functor:

[figure: (F ∘ G)(X ⊗ Y); (f∘g)_{X,Y}; (F ∘ G)(X) ⊗ Y; F (G(X ⊗ Y)); F (g_{X,Y} ); F (G(X) ⊗ Y); f_{G(X),Y}; F (G(X)) ⊗ Y]

2.19. Lemma. The category End_{G}(G) is strict monoidal.

Proof. All we need to verify is that the horizontal composition of morphisms of Ginvariant functors preserves the G-invariant structure of the composition. This follows from the commutative diagram:

[figure: (F ∘ G)(XY); F (G(XY)); F (G(X)Y); F (g_{X,Y} ); F (G(X))Y; f_{G(X),Y}; F′(G(X))Y; c_{G(X)}Y; F′(G′(X))Y; F (d_{X})Y; F′(G(X)Y); c_{G(X)Y}; f′_{G(X),Y}; F′(G′(X)Y); F′(d_{X}Y); f′_{G′(X),Y}; F (G′(X)Y); F (d_{X}Y); c_{G′(X)Y}; F (G′(XY)); F (d_{XY} ); F (g′_{X,Y ”)}; F′(G′(XY)); c_{G′(XY )}; F′(g′_{X,Y} ); (F ∘ G)(X)Y; (f∘g)_{X,Y}; (F′ ∘ G′)(X)Y; (F′ ∘ G′)(XY); (c∘d)_{XY}; (f′∘g′)_{X,Y}]

Now we extend the (left) action L : N_{l}(G) [figure] End(G) to a functor L : N_{l}(G) [figure] End_{G}(G) by the assignment (A, a) ↦ (L(A), L(a)), where the structure L(a) of G-invariant functor on L(A) is defined as follows:

[figure: L(A)(X ⊗ Y); L(a)_{X,Y}; L(A)(X) ⊗ Y; A ⊗ (X ⊗ Y); a_{X,Y}; (A ⊗ X) ⊗ Y]

2.20. Proposition. Let G be a unital magmoidal category. Then the functor L : N_{l}^{1}(G) [figure] End_{G}(G) is a monoidal equivalence.

Proof. First we note that the monoidal structure L_{X,Y} of the functor L : N_{l}^{1}(G) [figure] End(G) is compatible with the G-invariant structures (a morphism in End_{G}(G)) thus making the extended functor L : N_{l}^{1}(G) [figure] End_{G}(G) monoidal. This follows from the commutative diagram:

[figure: L(AB)(XY); L(AB)(X)Y; L(a|b)_{X,Y}; L(A)(L(B)(X))Y; L_{A,B}(X)Y; A((BX)Y); ((AB)X)Y; a_{B,X}Y; (AB)(XY); (a|b)_{X,Y}; L(A)(L(B)(XY)); L_{A,B}(XY); L(A)(L(B)(X)Y); L(A)((L(b)_{X,Y} ); L(a)_{L(B)(X),Y}; A(B(XY)); a_{B,XY}; A((BX)Y); Ab_{X,Y}; a_{BX,Y}]

To see that L is an equivalence it is enough to note that the assignment (F, f) ↦ (F (1), \overline{f}) defines a functor quasi-inverse to L. Here

\overline{f}_{X,Y} : F (1) ⊗ (X ⊗ Y) [figure] (F (1) ⊗ X) ⊗ Y

is defined by

\xymatrix{F (1) ⊗ (X ⊗ Y) \ar[d]^{_{\overline{f}_{X,Y}}} & F (1 ⊗ (X ⊗ Y)) \ar[l]_{F_{1,X⊗Y}} \ar[r]^{F (l_{X⊗Y} )} & F (X ⊗ Y) \ar[d]^{F_{X,Y}} \\ (F (1) ⊗ X) ⊗ Y & F (1 ⊗ X) ⊗ Y \ar[l]_{F_{1,X}⊗Y} \ar[r]^{F (l_{X})⊗Y} & F (X) ⊗ Y}

3. Multiplicants of functors

Now let F : G [figure] H be a functor between semigroupal (monoidal) categories. Define M_{l}(F) to be the category of pairs (A, a), where A ∈ G and a is a natural family of isomorphisms a_{X} : F (A ⊗ X) [figure] F (A) ⊗ F (X). A morphism from (A, a) to (B, b) is a morphism f : A [figure] B in G such that the following diagram commutes for each X ∈ G

\xymatrix{F (A ⊗ X) \ar[d]_{F (f⊗X)} \ar[r]^{a_{X}} & F (A) ⊗ F (X) \ar[d]^{F (f)⊗F (X)} \\ F (B ⊗ X) \ar[r]^{b_{X}} & F (A) ⊗ F (X)}

Note that the assignment (A, a) ↦ A defines a functor M_{l}(F) [figure] G (the forgetful functor).

We can define a tensor product on M_{l}(F) by (A, a) ⊗ (B, b) = (A ⊗ B, a|b), where the natural family a|b is given by the diagram

\xymatrix{F (A ⊗ (B ⊗ X)) \ar[d]_{F (ϕ_{A,B,X})} \ar[r]^{a_{B⊗X}} & F (A) ⊗ F (B ⊗ X) \ar[r]^{F (A)⊗b_{X}} & F (A) ⊗ (F (B) ⊗ F (X)) \ar[d]_{ψ_{F (A),F (B),F (X)}} \\ F ((A ⊗ B) ⊗ X) \ar[r]^{(a|b)_{X}} & F (A ⊗ B) ⊗ F (X) \ar[r]^{a_{B}⊗F (X)} & (F (A) ⊗ F (B)) ⊗ F (X)}

3.1. Theorem. The category M_{l}(F) is semigroupal. The forgetful functor M_{l}(F) [figure] G is a strict semigroupal functor. The composition of the forgetful functor with F is a semigroupal functor \overline{F} : M_{l}(F) [figure] H with semigroupal structure:

\overline{F}_{(A,a),(B,b)} = a_{B} : F (A ⊗ B) [figure] F (A) ⊗ F (B).

Proof. The following diagram proves that the tensor product f ⊗ g of morphisms f :

(A, a)  [figure] (C, c), g : (B, b)  [figure] (D, d) is a morphism from (A, a)⊗(B, b) to (C, c)⊗(D, d):
[figure: F (C ⊗ (D ⊗ X)); c_{D⊗X}; F (ϕ_{C,D,X}); F ((C ⊗ D) ⊗ X); (c|d)_{X}; F (A ⊗ (B ⊗ X)); F (f⊗(g⊗I)); a_{B⊗X}; F (ϕ_{A,B,X}); F ((A ⊗ B) ⊗ X); F ((f⊗g)⊗I); (a|b)_{X}; F (C) ⊗ F (D ⊗ X); I⊗d_{X}; F (C ⊗ D) ⊗ F (X); c_{D}⊗I; F (A) ⊗ F (B ⊗ X); F (f)⊗F (g⊗I); I⊗b_{X}; F (A ⊗ B) ⊗ F (X); F (f⊗g)⊗I; a_{B}⊗I; F (C) ⊗ (F (D) ⊗ F (X)); ϕ_{F (C),F (D),F (X)}; (F (C) ⊗ F (D)) ⊗ F (X); F (A) ⊗ (F (B) ⊗ F (X)); ϕ_{F (A),F (B),F (X)}; F (f)⊗(F (g)⊗I); (F (A) ⊗ F (B)) ⊗ F (X); (F (f)⊗F (g))⊗I]

It follows from the diagram

[figure: F (A(B(CX))); F ((AB)(CX)); F (ϕ); F (((AB)C)X); F (ϕ); F ((AB)C)F (X); ((a|b)|c)_{X}; (F (AB)F (C))F (X); (a|b)_{C}F (X); ((F (A)F (B))F (C))F (X); (a_{B}F (C))F (X); F (AB)F (CX); (a|b)_{CX}; (F (A)F (B))F (CX); a_{B}F (CX); (F (A)F (B))(F (C)F (X)); (F (A)F (B))c_{X}; ψ; F (AB)(F (C)F (X)); F (AB)c_{X}; a_{B}(F (C)F (X)); ψ; F (A((BC)X)); F (Aϕ); F ((A(BC))X); F (ϕ); F (ϕX); F (A(BC))F (X); (a|(b|c))_{X}; F (ϕ)F (X); (F (A)F (BC))F (X); a_{BC}F (X); (F (A)(F (B)F (C)))F (X); (F (A)b_{C})F (X); ψF (X); F (A)F ((BC)X); a_{(BC)X}; F (A)(F (BC)F (X)); F (A)(b|c)_{X}; F (A)((F (B)F (C))F (X)); F (A)(b_{C}F (X)); ψ; ψ; F (A)F (B(CX)); a_{B(CX)}; F (A)F (ϕ); F (A)(F (B)F (CX)); F (A)b_{CX}; ψ; F (A)(F (B)(F (C)F (X))); F (A)(F (B)c_{X}); ψ; F (A)ψ]

that the square

\xymatrix{F ((A(BC))X) \ar[r]^{(a|(b|c))_{X}} \ar[d]_{F (ϕX)} & F (A(BC))F (X) \ar[d]^{F (ϕ)F (X)} \\ F (((AB)C)X) \ar[r]^{((a|b)|c)_{X}} & F ((AB)C)F (X)}

commutes, which means that the associativity constraint ϕ_{A,B,C} : A ⊗ (B ⊗ C) [figure] (A ⊗ B) ⊗ C of G is a morphisms of M_{l}(F):

ϕ_{(A,a),(B,b),(C,c)} = ϕ_{A,B,C}.

Thus M_{l}(F) is semigroupal and the forgetful functor M_{l}(F) [figure] G is strict semigroupal. It follows from the definition of M_{l}(F) that \overline{F} is semigroupal.

The semigroupal category M_{l}(F) becomes monoidal if we assume that the functor F preserves monoidal unit, i.e. there is given an isomorphism ϵ : F (1) [figure] 1. It can be verified that the object (1, ι) ∈ M_{l}(F) is a monoidal unit. Here ι_{X} is the composition:

\xymatrix{F (1 ⊗ X) \ar[r]^{F (l_{X})} & F (X) & 1 ⊗ F (X) \ar[l]_{l_{F (X)}} & F (1) ⊗ F (X) \ar[l]_{ϵ⊗I}}

The semigroupal functors M_{l}(F) [figure]G, H allow us to define M_{l}(F)-actions on the categories G and H. In the next proposition we show that the functor F preserves these actions.

3.3. Proposition. The functor F : G [figure] H has a natural structure of a M_{l}(F)-functor.

Proof. The M_{l}(F)-action on G is given by (X, x)(Y) = X ⊗ Y for Y ∈ G, while the M_{l}(F)-action on H is simply (X, x)(Z) = F (X) ⊗ Z for Z ∈ H. The composition

[figure: F ((X, x)(Y)); F (X ⊗ Y); x_{Y}; F (X)F (Y); (X, x)F (Y)]

defines the structure of a M_{l}(F)-functor on F. The coherence for this structure follows from the definition of tensor product in M_{l}(F).

3.4. Functoriality. Here we define a correspondence between isomorphisms c : F [figure] G of functors F, G : G [figure] H between monoidal categories and monoidal functors M_{l}(c) : M_{l}(F) [figure] M_{l}(G) between their multiplicators. For an object (A, a) of M_{l}(F), define M_{l}(c)(A, a) to be (A, a^{c}) where a^{c} is defined by the diagram:

\xymatrix{F (A ⊗ X) \ar[r]^{a_{X}} \ar[d]_{c_{A⊗X}} & F (A) ⊗ F (X) \ar[d]^{c_{A}⊗c_{X}} \\ G(A ⊗ X) \ar[r]^{a_{X}^{c}} & G(A) ⊗ G(X)}

3.5. Proposition. The functor M_{l}(c) : M_{l}(F) [figure]M_{l}(G) defined above is strict monoidal and fits into a commutative diagram of monoidal functors:

\xymatrix{M_{l}(F) \ar[r]^{M_{l}(c)} \ar[d] \ar[dr] & M_{l}(G) \ar[d] \ar[dl] \\ G & H}  (10)

Proof. The following commutative diagram shows that (a|b)^{c} = (a^{c}|b^{c}) for (A, a), (B, b) ∈ M_{l}(F):

[figure: G(A ⊗ (B ⊗ X)); a_{B⊗X}^{c}; G(ϕ_{A,B,X}); G((A ⊗ B) ⊗ X); (a|b)^{c}; (a^{c}|b^{c})_{X}; F (A ⊗ (B ⊗ X)); c_{A⊗(B⊗X)}; a_{B⊗X}; F (ϕ_{A,B,X}); F ((A ⊗ B) ⊗ X); c_{(A⊗B)⊗X}; (a|b)_{X}; G(A) ⊗ G(B ⊗ X); I⊗b_{X}^{c}; G(A ⊗ B) ⊗ G(X); a_{B}^{c}⊗I; F (A) ⊗ F (B ⊗ X); c_{A}⊗c_{B⊗X}; I⊗b_{X}; F (A ⊗ B) ⊗ F (X); c_{A⊗B}⊗c_{X}; a_{B}⊗I; G(A) ⊗ (G(B) ⊗ G(X)); ψ_{G(A),G(B),G(X)}; (G(A) ⊗ G(B)) ⊗ G(X); F (A) ⊗ (F (B) ⊗ F (X)); ψ_{F (A),F (B),F (X)}; c_{A}⊗(c_{B}⊗c_{X}); (F (A) ⊗ F (B)) ⊗ F (X); (c_{A}⊗c_{B})⊗c_{X}]

This proves that the functor M_{l}(c) is strict monoidal:

M_{l}(c)((A, a) ⊗ (B, b)) = M_{l}(c)(A ⊗ B, a|b) = (A ⊗ B, (a|b)^{c})

coincides with

M_{l}(c)(A, a) ⊗ M_{l}(c)(B, b) = (A, a^{c}) ⊗ (B, b^{c}) = (A ⊗ B, a^{c}|b^{c}).

Commutativity of the diagram (10) follows from the definition of M_{l}(c).

Note that for successive isomorphisms of functors F [figure: c]G [figure: d]H the composition of monoidal functors M_{l}(c) and M_{l}(d) coincides with M_{l}(cd). This follows from the fact that a^{cd} = (a^{c})^{d} for (A, a) ∈ M_{l}(F).

3.6. Composition properties. For a successive (set-theoretic) maps of monoids

\xymatrix{A \ar[r]^{f} & B \ar[r]^{g} & C}

(left) multiplicants form a successive pair of spans (in the category of monoids):

\xymatrix{ & M_{l}(f) \ar[dl] \ar[dr] &  & M_{l}(g) \ar[dl] \ar[dr] \\ A &  & B &  & C}

The composition of spans is given by the pullback (fibered product) of the two middle maps:

M_{l}(f) ×_{B} M_{l}(g) = {(a, f(a))| a ∈ M_{l}(f), f(a) ∈ M_{l}(g)}.

Note that the “projection” (a, f(a)) ↦ a defines a homomorphism M_{l}(f)×_{B}M_{l}(g) [figure] M_{l}(gf). Indeed, for such a and any x ∈ A we have that

gf(ax) = g(f(a)f(x)) = gf(a)gf(x).

Moreover, this homomorphism fits into a commutative diagram (morphism of spans):

[figure: M_{l}(gf); M_{l}(f) ×_{B} M_{l}(g); M_{l}(f); M_{l}(g); A; B; C]

Thus multiplicants define an oplax functor from the category of monoids and set-theoretic maps into the bicategory of spans of monoids.

Here we categorify this construction by defining a monoidal functor M_{l}(F)×_{H} M_{l}(G) [figure] M_{l}(GF) for a pair of composable functors G [figure: F]H [figure: G]J between monoidal categories. Here by M_{l}(F) ×_{H} M_{l}(G) we mean the pseudo-pullback in the 2-category of monoidal categories: with objects (A, a, B, b, x), where (A, a) ∈ M_{l}(F), (B, b) ∈ M_{l}(G), and x : F (A) [figure] B being an isomorphism in H. A morphism (A, a, B, b, x) [figure] (A′, a′, B′, b′, x′) is a pair (f, g) of morphisms f : (A, a) [figure] (A′, a′), g : (B, b) [figure] (B′, b′) in M_{l}(F), M_{l}(G) respectively, such that the square:

\xymatrix{F (A) \ar[r]^{x} \ar[d]_{F (f)} & B \ar[d]^{g} \\ F (A′) \ar[r]^{x′} & B′}

commutes. The tensor product in M_{l}(F) ×_{H} M_{l}(G) has the form:

(A, a, B, b, x) ⊗ (A′, a′, B′, b′, x′) = (A ⊗ A′, a|a′, B ⊗ B′, b|b′, x|x′),

where x|x′ is the composition

\xymatrix{F (A ⊗ A′) \ar[r]^{a_{A′}} & F (A) ⊗ F (A′) \ar[r]^{x⊗x′} & B ⊗ B′.}

Define a functor M_{l}(F, G) : M_{l}(F) ×_{H} M_{l}(G) [figure] M_{l}(GF) by (A, a, B, b, x) ↦ (A, ã), where ã is the composite:

\begin{aligned}GF (A ⊗ X) & [figure: G(a_{X})] G(F (A) ⊗ F (X)) [figure: G(x⊗F (X))] G(B ⊗ F (X)) \\ & [figure: b_{F (X)}] G(B) ⊗ GF (X) [figure: G(x)^{−1}⊗GF (X)] GF (A) ⊗ GF (X) \end{aligned}

3.7. Proposition. The functor M_{l}(F, G) is strict monoidal.

Proof. We need to show that

M_{l}(F, G)((A, a, B, b, x) ⊗ (A′, a′, B′, b′, x′)) =

= M_{l}(F, G)(A ⊗ A′, a|a′, B ⊗ B′, b|b′, x|x′) = (A ⊗ A′, ã|a′)

coincides with

M_{l}(F, G)(A, a, B, b, x) ⊗ M_{l}(F, G)(A′, a′, B′, b′, x′) =

= (A, ã) ⊗ (A′, ã′) = (A ⊗ A′, ã|ã′).

466  ALEXEI DAVYDOV

This follows from the diagram:

[figure: GF (A(A′X)); GF ((AA′)X); GF (ϕ); G(F (AA′)F (X)); G((a|a′)_{X}); G((BB′)F (X)); (3); G(BB′)GF (X); (b|b′)_{F (X)}; (G(B)G(B′))GF (X); b_{B′} GF (X); G((F (A)F (A′))F (X)); (1); (2); GF (AA′)GF (X); (ã|ã′)_{X}; (ã|a′)_{X}; (12); (GF (A)GF (A′))GF (X); ã_{A′} GF (X); (G(B)GF (A′))GF (X); (19); (20); G(F (A)F (A′))GF (X); (11); G(BF (A′))GF (X); (13); (16); (14); GF (A)GF (A′X); ã_{A′X}; GF (A)(GF (A′)GF (X)); GF (A)ã′_{X}; ψ; GF (A)(G(B′)GF (X)); (18); G(B)(G(B′)GF (X)); (15); ψ; GF (A)G(F (A′)F (X)); (5); GF (A)G(B′F (X)); (6); (17); G(B)G(B′F (X)); (9); Bb′_{F (X)}; G(B)G(F (A′)F (X)); (7); (10); G(B)GF (A′X); (4); (8); G(F (A)F (A′X)); G(a_{A′X}); G(BF (A′X)); G(xF (A′X)); b_{F (A′X)}; G(B(F (A′)F (X))); G(Ba′_{X}); b_{F (A′)F (X)}; G(B(B′F (X))); G(B(x′F (X))); G(ϕ); b_{B′F (X)}; G(F (A)(F (A′)F (X))); G(F (A)a′_{X}); G(ϕ); G(x(F (A′)F (X)))]

where some of the arrow labels are displayed below:

\begin{matrix}(1) = G(a_{A′}F (X)), & (11) = G(a_{A′})GF (X), \\ (2) = G((xx′)F (X)), & (12) = G(x|x′)GF (X), \\ (3) = G((x|x′)F (X)), & (14) = G(Bx′)GF (X), \\ (4) = G(x)GF (A′X), & (13) = G(xF (A′))GF (X), \\ (5) = GF (A)G(a′_{X}), & (15) = G(x)(G(B′)GF (X)), \\ (6) = GF (A)G(x′F (X)), & (16) = b_{F (A′)}GF (X), \\ (7) = G(x)G(F (A′)F (X)), & (17) = GF (A)b′_{F (X)}, \\ (8) = G(B)G(a′_{X}), & (18) = GF (A)(G(x′)GF (X)), \\ (9) = G(x)G(B′F (X)), & (19) = (G(x)GF (A′))GF (X), \\ (10) = G(B)G(x′F (X)), & (20) = (G(B)G(x′))GF (X). \end{matrix}

3.8. Subcategories of multiplicants. Let G, H, C be monoidal categories equipped with monoidal functors F′ : G [figure] C, F′′ : H [figure] C. Let F : G [figure] H be a functor and let γ : F′′ ∘ F [figure] F′ be an isomorphism of functors. Define the full subcategory M_{l}(γ) = M_{l}(F, F′, F′′, γ) in M_{l}(F) of objects (A, a) such that the following diagram is commutative

\xymatrix{F′(A ⊗ X) \ar[dd]_{F′_{A,X}} & F′′F (A ⊗ X) \ar[l]_{γ_{A⊗X}} \ar[d]^{F′′(a_{X})} \\  & F′′(F (A) ⊗ F (X)) \ar[d]^{F′′_{A,X}} \\ F′(A) ⊗ F′(X) & F′′F (A) ⊗ F′′F (X) \ar[l]_{γ_{A}⊗γ_{X}}}

3.9. Theorem. The subcategory M_{l}(γ) is closed under the tensor product in M_{l}(F). The natural isomorphism γ lifts to a monoidal natural isomorphism \overline{γ} which fills the square

\xymatrix{M_{l}(γ) \ar[r] \ar[d] & H \ar[d]^{F′′} \\ G \ar[r]^{F′} & C}

Proof. The fact that the tensor product (A ⊗ B, a|b) of two objects (A, a), (B, b) of M_{l}(γ) belongs to M_{l}(γ) follows from the commutative diagram:

[figure: F′′F (A(B(X)); F′(A(BX)); γ_{A(BX)}; F′((AB)X); F′(ϕ); F′(AB)F′(X); F′_{AB,X}; (F′(A)F′(B))F′(X); F′_{A,B}F′(X); F′(A)F′(BX); F′_{A,BX}; F′(A)(F′(B)F′(X)); F′(A)F′_{B,X}; ψ; F′′F ((AB)X); F′′F (ϕ); γ_{(AB)X}; F′′(F (AB)F (X)); F′′((a|b)_{X}); F′′((F (A)F (B))F (X)); F′′(a_{B}F (X)); F′′(F (A)F (B))F′′F (X); F′′_{F (A)F (B),F (X)}; (F′′F (A)F′′F (B))F′′F (X); F′′_{F (A),F (B)}F (X); (γ_{A}γ_{B})γ_{X}; F′′(F (AB))F′′F (X); F′′_{F (AB),F (X)}; F′′(a_{B})F′′F (X); γ_{AB}γ_{X}; F′′(F (A)F (BX)); F′′(a_{BX}); F′′(F (A)(F (B)F (X)); F′′(F (X)b_{X}; F′′(ϕ′); F′′F (A)F′′(F (B)F (X)); F′′_{F (A),F (B)F (X)}; F′′F (A)(F′′F (B)F′′F (X)); F′′F (A)F′′_{F (B),F (X)}; ψ; γ_{A}(γ_{B}γ_{X}); F′′F (A)F′′F (BX); F′′_{F (A),F (BX)}; F′′F (A)F′′(b_{X}); γ_{A}γ_{BX}]

It follows from the definition of M_{l}(γ) that the natural transformation \overline{γ} given by \overline{γ}_{(}A, a) = γ_{A} : F′′F (A) [figure] F′(A) is monoidal.

4. Examples, categories of modules

Here k denotes the ground field.

4.1. Nuclei and multiplicants of categories of modules. We start with the following technical statement.

4.2. Lemma. Let M be a left module over an associative algebra R and m_{X_{1},...,X_{n}} ∈ End(M ⊗ X_{1} ⊗ ⋯ ⊗ X_{n}) be a family of linear operators, which is natural in left Rmodules X_{i}. Then m_{X_{1},...,X_{n}} is given by multiplication by an element m ∈ End(M) ⊗ R ⊗ ⋯ ⊗ R. Families of invertible operators correspond to invertible elements in the algebra End(M) ⊗ R ⊗ ⋯ ⊗ R.

Proof. The R-module R is a generator in R-Mod. Thus any transformation m_{X_{1},...,X_{n}} natural in X_{i} ∈ R-Mod is defined by its specialisation m_{R,...,R} ∈ End(M ⊗ R ⊗ ⋯ ⊗ R). Naturality of m implies that m_{R,...,R} commutes with 1 ⊗ End_{R}(R) ⊗ ⋯ ⊗ End_{R}(R). Thus m_{R,...,R} is given by multiplication by an element m ∈ End(M) ⊗ R ⊗ ⋯ ⊗ R.

4.3. Example. Nucleus of a category of modules.

Let R be an algebra and Δ : R [figure] R ⊗ R a homomorphism of algebras defining a magmoidal structure on the category R-Mod of left R-modules. By lemma 4.2 the nucleus N_{l}(R-Mod) is the category of pairs (M, m) where M is a left R-module and m is an invertible element of the tensor product End(M) ⊗ R ⊗ R satisfying

(Δ ⊗ I)Δ(r)m = m(I ⊗ Δ)Δ(r)  (11)

for all r ∈ R. Here End(M) is the endomorphism ring of the vector space M. The tensor product on the category N_{l}(R-Mod) is defined by (M, m)⊗(N, n) = (M ⊗N, m|n) where m|n ∈ End(M) ⊗ End(N) ⊗ R ⊗ R is given by

(m|n) = (m ⊗ 1)(I ⊗ Δ ⊗ I)(m)(1 ⊗ n)(I ⊗ I ⊗ Δ)(m)^{−1}.  (12)

The tensor product is semigroupal with the associativity constraint:

ϕ_{(M,m),(N,n),(L,l)} = (I ⊗ ρ_{N} ⊗ ρ_{L})(m).
Here ρ_{N} : R  [figure] End(N), ρ_{L} : R  [figure] End(L) are the corresponding R-module structures.

If ε : R [figure] k is a homomorphism of algebras (counit) such that

(ε ⊗ I)Δ = I = (I ⊗ ε)Δ,

the magmoidal category R-Mod has a unit object (k, 1). The monoidal subcategory N_{l}^{1}(R-Mod) consists of those pairs (M, m), where m satisfies the normalisation condition:

(I ⊗ ε ⊗ I)(m) = 1 = (I ⊗ I ⊗ ε)(m).

In the case when Δ is coassociative (when (R, Δ) is a bialgebra) the equation (11) means that m is R-invariant with respect to the diagonal inclusion of R into End(M) ⊗ R ⊗ R. The monoidal functor R-Mod [figure] N_{l}(R-Mod) corresponding to the canonical (trivial) associativity constraint on R-Mod, sends an R-module M into a pair (M, 1).

4.4. Example. Multiplicant of a homomorphism.

Let H_{1}, H_{2} be bialgebras and let f : H_{1} [figure] H_{2} be a homomorphism of algebras. It defines a functor f^{∗} : H_{2}-Mod [figure] H_{1}-Mod between monoidal categories. Here we describe its (left) multiplicant M_{l}(f^{∗}). Objects of M_{l}(f^{∗}) are pairs (M, m) where M is H_{2}-module and m is an invertible element of End(M) ⊗ H_{2} satisfying

mΔ(f(h)) = (f ⊗ f)Δ(h)m  (13)

for all h ∈ H_{1}. The tensor product of pairs is given by the formula (M, m) ⊗ (N, n) = (M ⊗ N, m|n) where m|n ∈ End(M) ⊗ End(N) ⊗ H_{2} is defined by

(m|n) = (m ⊗ 1)^{−1}(1 ⊗ n)(I ⊗ Δ)(m).

The forgetful functor M_{l}(f^{∗}) [figure] H_{2}-Mod sends a pair (M, m) to M. The functor f^{∗} : M_{l}(f^{∗}) [figure] H_{1}-Mod, which sends (M, m) into f^{∗}(M), is semigroupal with respect to the constraint:

f_{(M,m),(N,n)}^{∗} = (I ⊗ ρ_{N})(m) ∈ End(M) ⊗ End(N).
Let F_{i} : H_{i} −Mod  [figure] Vect be the forgetful functors. An isomorphism γ : F_{1}  [figure] F_{2}f^{∗}

is given by multiplication with an invertible x ∈ H_{2}. The subcategory M_{l}(γ) of the multiplicant M_{l}(f^{∗}) consists of the pairs (M, m) such that

m = (ρ_{M} ⊗ I)(Δ(x)^{−1}(x ⊗ x)),

where ρ_{M} : H_{2} [figure] End(M) is the H_{2}-action on M. Note that this together with (13) is equivalent to the condition:

(ρ_{M} ⊗ I)(Δ(g(h))) = (ρ_{M} ⊗ I)((g ⊗ g)Δ(h)), h ∈ H_{1},

where g : H_{1} [figure] H_{2} is given by g(h) = xf(h)x^{−1}.

4.5. Algebraic constructions. For an associative algebra R consider the category E(R) of pairs (V, v) where V is a vector space and v is an element of End(V) ⊗ R with morphisms being vector space maps compatible with second components: a morphism

(V, v)  [figure] (U, u) is a map f : V  [figure] U such that the equality
u(f ⊗ 1) = (f ⊗ 1)v  (14)

is valid in Hom(V, U) ⊗ R. Denote by A(R) the full subcategory of E(R) of those pairs (V, v) for which v is invertible in End(V) ⊗ R.

4.6. Lemma. Let R be a finite dimensional associative algebra. Then the category E(R) is equivalent to the category of modules over the free algebra T (R^{∨}) generated by the dual space of R. The category A(R) is equivalent to the category of modules over the quotient of the free algebra T (R^{∨} ⊕ R^{∨}) by the ideal generated by {δ′(l) − l(1), δ”(l) − l(1); l ∈ R^{∨}} where δ′, δ” are the composites of δ : R^{∨} [figure] R^{∨} ⊗ R^{∨}, which is the dual map to the multiplication R ⊗ R [figure] R, with two inclusions of R^{∨} ⊗ R^{∨} into (R^{∨} ⊕ R^{∨})^{⊗2}.

Proof. An element v of End(V) ⊗ R corresponds to a map f_{v} : R^{∨} [figure] End(V) sending l ∈ R^{∨} into (I ⊗ l)(v), and thus a T (R^{∨})-module structure on V. Clearly compatibility condition (14) guarantees that a morphisms of pairs is T (R^{∨})-linear. In the case when v is invertible the maps f_{v}, f_{v^{−1}} satisfy the conditions:

(f_{v} ⊗ f_{v^{−1}})δ(l) = (I ⊗ δ(l)(v_{12}v_{13}^{−1}) = (I ⊗ l)(vv^{−1}) = l(1),
(f_{v^{−1}} ⊗ f_{v})δ(l) = (I ⊗ δ(l)(v_{12}^{−1}v_{13}) = (I ⊗ l)(v^{−1}v) = l(1).

Let R = k(G) be an algebra of functions on a finite group G with standard multiplication Δ given by Δ(l)(f ⊗ g) = l(fg) for l ∈ k(G) and f, g ∈ G. By p_{g} ∈ k(G) we denote the δ-function concentrated at g ∈ G. Note that δ-functions p_{g}, g ∈ G form a basis in k(G) so we can write m ∈ End(M)⊗k(G)⊗k(G) as ∑_{f,g∈G} m(f, g)⊗p_{f} ⊗p_{g}. The element m is invertible if m(f, g) ∈ End(M) are invertible for any f, g. The k(G)-invariance of m (11) means that m(f, g) commutes with the image of k(G) in End(M). Thus we can identify the nucleus N_{l}(k(G)-Mod) with the category N_{l}(k(G))-Mod of left modules over the algebra N_{l}(k(G)) = k(G) ⊗ k[F (G × G)] which is a tensor product of k(G) and the group algebra of the free group F (G × G) generated by the cartesian product G × G. Generators of F (G × G) will be denoted by u(f, g). The pair (M, m) corresponds to the N_{l}(k(G)) action on M sending u(f, g) into m(f, g). The formula (12) for the tensor product of two pairs gives

(m|n)(f, g) = ∑_{h∈G} m(h, f)m(hf, g)m(h, fg)^{−1} ⊗ p_{h}n(f, g)

which corresponds to the following comultiplication on N_{l}(k(G))

Δ(p_{h}) = ∑_{fg=h} p_{f} ⊗ p_{g},

Δ(u(f, g)) = ∑_{h∈G} u(h, f)u(hf, g)u(h, fg)^{−1} ⊗ p_{h}u(f, g).

Note that k(G) is a sub-bialgebra of N_{l}(k(G)). The coproduct on N_{l}(k(G)) is not coassociative but rather quasi-associative in the sense of [1]:

(I ⊗ Δ)(x) = Φ(Δ ⊗ I)Δ(x)Φ^{−1}

where Φ = ∑_{f,g∈G} u(f, g) ⊗ p_{f} ⊗ p_{g} is an invertible element of N_{l}(k(G))^{⊗3} (associator).

It is well known that associativity constraints on the category k(G)-Mod are in one-to-one correspondence with 3-cocycles Z^{3}(G, k^{∗}) of the group G with coefficients in invertible elements of the ground field k. For any 3-cocycle α ∈ Z^{3}(G, k^{∗}) there is defined a homomorphism of quasi-bialgebras N_{l}(k(G)) [figure] k(G) splitting the inclusion k(G) [figure] N_{l}(k(G)) which sends u(f, g) into ∑_{h∈G} α(h, f, g)p_{h}. This homomorphism maps Φ into the associator Φ_{α} = ∑_{h∈G} α(h, f, g)p_{h} ⊗ p_{f} ⊗ p_{g} in k(G) corresponding to α.

References

[1] V. Drinfeld, Quasi-Hopf algebras. (Russian) Algebra i Analiz 1 (1989), no. 6, 114–148; translation in Leningrad Math. J. 1 (1990), no. 6, 1419–1457.

[2] A. Joyal, R. Street, Tortile Yang-Baxter operators in tensor categories. Journal of Pure and Applied Algebra, 71 (1991), 43-51.

[3] A. Joyal, R. Street, Braided tensor categories. Adv. Math. 102 (1993), no. 1, 20–78.

[4] R. D. Schafer, An Introduction to Nonassociative Algebras, 1995 Courier Dover Publications, 176 pp.

[5] J. Stasheff, H-spaces from a homotopy point of view. Lecture Notes in Mathematics, Vol. 161 Springer-Verlag, Berlin-New York 1970, 95 pp.

Department of Mathematics, Division of Information and Communication Sciences Macquarie University, Sydney, NSW 2109, Australia

Email: davydov@math.mq.edu.au

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TEXnical editor. Michael Barr, McGill University: mbarr@barrs.org

Transmitting editors.

Richard Blute, Université d’ Ottawa: rblute@uottawa.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: ronnie.profbrown (at) btinternet.com Aurelio Carboni, Università dell Insubria: aurelio.carboni@uninsubria.it Valeria de Paiva, Xerox Palo Alto Research Center: paiva@parc.xerox.com Ezra Getzler, Northwestern University: getzler(at)northwestern(dot)edu Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk G. Max Kelly, University of Sydney: maxk@maths.usyd.edu.au Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, University of Western Sydney: s.lack@uws.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@acsu.buffalo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz Brooke Shipley, University of Illinois at Chicago: bshipley@math.uic.edu James Stasheff, University of North Carolina: jds@math.unc.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca