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Nuclei of categories with tensor products

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.


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}, N m (A) = {a ∈ A : J(x, a, y) = 0 ∀x, y ∈ A}, N r (A) = {a ∈ A : J(x, y, a) = 0 ∀x, y ∈ A} (4) of the algebra A are associative subalgebras. We call these subalgebras left, middle and right nucleus respectively. In section 2.1 we categorify these construction 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 categorical analogs of the inclusions N l (A), N m (A), N r (A) ⊂ A are the tensor product preserving (forgetful) functors N l (G), N m (G), N r (G) → G.
If an algebra A was already associative, its nuclei will coincide with it. For a semigroupal category G we only get semigroupal functors G → 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 ǫ (A) making them associative unital. In section 2.2 we modify the constructions of nuclei for the case of a magmoidal category with unit. The modified constructions give monodal categories N 1 l (G), N 1 m (G), N 1 r (G). Constructions of self-symmetries of categorical nuclei which we describe in section 2.3 do not have any analogs 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 2.4 we define certain subcategories of nuclei and show how they are acted upon by the self-symmetries of section 2.3.
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- This could give the 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 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.
On the level of categories we have the following relations between nuclei of a category G with commutative tensor product: 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. 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 → B between commutative algebras coincide. The following chain of equalities shows that any a ∈ M l (f ) belongs to M r (f ): But the equivalence M l (F ), → M r (F ) between multiplicants of a functor F : G → H between symmetric monoidal categories fails to be monoidal.

Acknowledgment
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. Special thanks are due to R. Street for invaluable support during the work on the paper.
2 Nuclei of a category with a tensor product

Nuclei as semigroupal categories
Let G be a category with a tensor product functor ⊗ : G × G → 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 family of isomorphisms is commutative for any X, Y ∈ G. Introduce 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: The following diagram implies that for two morphisms f : (A, a) → (B, b), g : In particular, the forgetful functor N l (G) → G sending (A, a) to A is magmoidal.
It follows from the diagram below that for (A, a), Indeed, it is equivalent to the commutativity of the square e e t t t t t t t t t t t t t t t t t t t t t x x q q q q q q q q q q q q q q q q q q q q q q q q q Two other square faces of this diagram are commutative by the naturality while the pentagon faces commute by the definition of the tensor product in N l (G). Thus defined by φ (A,a),(B,b),(C,c) = a B,C , is a morphism in the category N l (G). Theorem 2.1.1. 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 axion, which says that the diagram Semigroupal structures on the category G correspond to functors G → N l (G) which split the forgetful functor N l (G) → G.
Theorem 2.1.2. If G is a semigroupal category with the associativity constraint φ, the assignment A → (A, φ A,−,− ) defines a semigroupal functor G → N l (G) whose composition G → N l (G) → G with the forgetful functor is the identity.
Conversely, a functor G → N l (G), whose composition G → N l (G) → G with the forgetful functor is the identity, defines an associativity constraint on G thus turning it into a semigroupal category.
A splitting G → N l (G) of the forgetful functor N l (G) → 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 respectfully. 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 five ways of placing a pair of brackets ( ) in the word ABXY : 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 diagram we can sketch the proof of the analog of the theorem 2.1.1 for N m (G). Define N m (G) as the category of pairs (A, a), where A ∈ G and a is a natural collection of isomorphisms 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 the label XABY . The diagram of the shape Xf gY 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 ( The word ABCD gives the pentagon axiom for this constraint. Similarly, define N r (G) as the category of pairs (A, a), with Finally, the word ABCD gives the pentagon axiom for this constraint.

Units
In this section we slightly modify the construction to incorporate units. First we need to modify the setup. Let now G be a magmoidal category with unit (an object 1 ∈ G with natural isomorphisms l X : 1 ⊗ X → X, r X : X ⊗ 1 → X such that l 1 = r 1 ). Define the left unital nucleus N 1 l (G) as the category of pairs (A, a) where as before a X,Y : The normalisation conditions for the collection i follow from the commutative diagrams: Here the equation l 1⊗Y = 1 ⊗ l Y (used in the first diagram) follows from the naturality of l: 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 .
is monoidal with the unit object (1, i) and unit natural isomorphisms 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:

t t t t t t t t t t t t t t t t t t t t t t t t t
Thus the subcategory N 1 l (G) is closed under tensor product in N l (G) and is a semigroupal category. By lemma 2.2.1, the pair (1, i) is an object of N 1 l (G). We need to check that the unit isomorphisms (6) are morphisms in he category N 1 l (G). It is equivalent to the commutativity of the squares: aX,Y G G (AX)Y which follows from the commutative diagrams: The constructions of the following sections can be made unital (monoidal instead of just semi-groupal). We will not be doing it for the sake of simplicity.

Functoriality
Here we assign, to an invertible natural self-transformation (an automorphism) c of the tensor product functor ⊗ : G × G → G, a strict semigroupal autoequiv- where a c is defined by the diagram:

Remark 2.3.2.
A diagram of that shaped appeared in [2] 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(⊗) → 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.
Note that by naturality of c, c X,Y commutes with f X ⊗ f Y and by naturality of This in particular implies that Aut(id G ) acts on the group Aut(⊗): Proof. We need to show that for any (A, a) ∈ N l (G) the map is commutative. The commutativity of this square is equivalent to the commutativity of the bottom square of the following diagram:

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 → H be a functor equipped with a natural collection of isomorphisms F X,Y : F (X ⊗ Y ) → 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: here ψ is the associativity constraint of 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: follows from the definition of N l (F ).
In the next proposition we explain how monoidal autoequivalences of nuclei (defined in section 2.3) permute the subcategories of nuclei. Let c X,Y : X ⊗Y → X ⊗ Y be a natural collection of isomorphisms defining a semi-groupal functor N l (c) : N l (G) → N l (G). Let F : G → 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 The semigroupal functor N l (c) defines an equivalence between the semi-groupal subcategories N l (F ), N l (F c ).
Proof. We need to show that the coherence diagram (7) commutes if we replace F X,Y by F c X,Y and a X,Y by a c X,Y . This follows from the commutative diagram: Proof. We need to check that the natural collection (9) satisfies the coherence condition (8). It follows from the diagram: 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 → G will be called (right) G-invariant if it is equipped with a natural collection of isomorphisms f 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: Proof. All we need to verify is that the horizontal composition of morphisms of G-invariant functors preserves G-invariant structure of the composition. This follows from the commutative diagram:  Proof. First we note that the monoidal structure L X,Y of the functor L : is compatible with the G-invariant structures (a morphism in End G (G)) thus making the extended functor L : N 1 l (G) → End G (G) monoidal. This follows from the commutative diagram: To see that L is an equivalence it is enough to note that the assignment (F, f ) → (F (1), f ) defines a functor quisi-inverse to L. Here f X,Y :

Multiplicants of functors
Now let F : G → 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 : Note that the assignment (A, a) → A defines a functor M l (F ) → 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 It follows from the diagram commutes, which means that the associativity constraint of G φ A,B,C : Thus M l (F ) is semigroupal and the forgetful functor M l (F ) → G is strict semigroupal. It follows from the definition of M l (F ) that 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 ǫ : F (1) → 1. It can be verified that the object (1, ι) ∈ M l (F ) is a monoidal unit. Here ι X is the composition:  Proof. The M l (F )-action on G is given by ( 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 ).

Functoriality
Here we define a correspondence between isomorphisms c : F → G of functors 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) → M l (GF ) for a pair of composable functors G  morphism (A, a, B, The tensor product in M l (F ) × H M l (G) has the form: where x|x ′ is the composition by (A, a, B, b, x) → (A,ã), whereã is the composition: Proof. We need to show that It follows from the diagram: G(a A ′ )GF (X) 5 5 5 5 5 5 5 5 5 5 5 5 5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) GF (A)G(a ′ X ) 5 5 5 5 5 5 5 5 5 5 5 5 5 5 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: It follows from the definition of M l (γ) that the natural transformation γ given by γ ( A, a)

Examples, categories of modules
Here k denote the ground field.

Nuclei and multiplicants of categories of modules
We start with the following technical statement.
The tensor product is semigroupal with the associativity constraint:

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) → (U, u) is a map f : V → U such that the equality 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.
Lemma 4.2.1. 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 compositions of δ : R ∨ → R ∨ ⊗ R ∨ , which is the dual map to the multiplication R ⊗ R → 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 ∨ → 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 to the conditions: Let R = k(G) be an algebra of functions on a finite group G with standard multiplication ∆ given by ∆(l)(f ⊗ g) = l(f g) for l ∈ k(G) and f, g ∈ G. By p g ∈ k(G) we denote the δ-function concentrated at g ∈ G. Note that δfunctions 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. 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)-M od) with the category N l (k(G))-M od 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, f g) −1 ⊗ p h n(f, g) which corresponds to the following comultiplication on N l (k(G)) ∆(u(f, g)) = h∈G u(h, f )u(hf, g)u(h, f g) −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]: 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) − M od 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)) → k(G) splitting the inclusion k(G) → N l (k(G)) which sends u(f, g) into h∈G α(h, f, g)p h .