TENSOR PRODUCT OF CORRESPONDENCE FUNCTORS

SERGE BOUC AND JACQUES THÉVENAZ

Abstract. As part of the study of correspondence functors, the present paper investigates their tensor product and proves some of its main properties. In particular, the correspondence functor associated to a finite lattice has the structure of a commutative algebra in the tensor category of all correspondence functors and a converse also holds.

1. Introduction

In recent papers [BT2, BT3, BT4], we developed the theory of correspondence functors, namely functors from the category C of finite sets and correspondences to the category k-Mod of all kmodules, where k is a commutative ring. The main motivation for this work lies in its fundamental nature and in the fact that the whole theory turns out to be very rich. We describe here another piece of structure. We introduce the tensor product of any two correspondence functors (Section 3) and analyse some of its main properties, such as projectivity, finite generation, and behaviour under induction (Section 5).

Whenever M and M′ are correspondence functors, we not only define their tensor product M ⊗ M′, but also their internal hom H(M, M′), which satisfies the usual adjointness property (Section 6). Instead of proving this by some standard arguments of the representation theory of categories, we show more. We prove that the internal hom can be constructed in a very simple way by making use of the symmetric monoidal structure on C given by the disjoint union of finite sets. By means of this monoidal structure, we also define bilinear pairings of functors and show that they are related in a standard way to the tensor product. This is an instance of the general construction known as Day convolution [Da].

A main example of correspondence functor is the functor F_{T} associated to a finite lattice T, as defined in [BT3]. We prove in Section 4 that F_{T} ⊗ F_{T′} ≅ F_{T ×T′}. Finally, in Section 7, we show that F_{T} carries the additional structure of a commutative algebra in the tensor category F_{k} of all correspondence functors. Conversely, we prove the rather remarkable fact that, over a field k (or more generally if Spec(k) is connected), any commutative algebra in the tensor category F_{k} is isomorphic to F_{T} for some finite lattice T, provided it satisfies two additional conditions, one of them being of an exponential nature (see Theorem 7.4). This result establishes a very strong link between finite lattices and correspondence functors having a commutative algebra structure. Finally, a few small examples are discussed in Section 8.

Throughout this paper, k denotes a fixed commutative ring and all modules are left k-modules.

2. Correspondence functors

In this section, we recall the definitions and basic properties of correspondence functors. We refer to Sections 2–4 of [BT2] and Section 2 of [BT3] for more details. We denote by C the category of finite sets and correspondences. Its objects are the finite sets and the set C(Y, X) of morphisms from X to Y (using a reverse notation which is convenient for left actions) is the set of all correspondences from X to Y, namely all subsets of Y ×X. Given two correspondences V ⊆ Z ×Y and U ⊆ Y ×X, their composition V U is defined by

2010 Mathematics Subject Classification. 06B05, 06B15, 16B50, 18B05, 18B10, 18B35, 18E05.

Key words and phrases. finite set, correspondence, relation, functor category, tensor product.

V U := { (z, x) ∈ Z × X | ∃ y ∈ Y such that (z, y) ∈ V and (y, x) ∈ U } .

The correspondence Δ_{X} = {(x, x) | x ∈ X} is a unit element for this composition. A correspondence from X to X is also called a relation on X. In particular,

R_{X} := kC(X, X)

is a k-algebra, the algebra of the monoid of all relations on X.

For our fixed commutative ring k, we let kC be the k-linearization of C. The objects are again the finite sets and kC(Y, X) is the free k-module with basis C(Y, X). A correspondence functor over k is a k-linear functor from kC to the category k-Mod of all k-modules. We let F_{k} be the category of all correspondence functors over k (for some fixed commutative ring k).

If M is a correspondence functor and U ∈ C(Y, X) is a correspondence, then U acts as a linear map M(U) : M(X) → M(Y) and we simply write U for this left action. In other words, for any m ∈ M(X), we define

Um := M(U)(m) ∈ M(Y) .

In particular, we have (V U)m = V (Um) for any correspondence V ∈ C(Z, Y).

The following examples have been introduced in [BT2] and [BT3] respectively.

2.1. Example. For any finite set E, the representable functor kC(−, E) is a projective correspondence functor by Yoneda’s lemma. More generally, for any left R_{E}-module W, there is a correspondence functor L_{E,W} defined by

L_{E,W} (X) := kC(X, E) ⊗_{R_{E}} W .

This is left adjoint to the evaluation at E, in the sense that it satisfies the adjointness property

Hom_{F_{k}}(L_{E,W} , M) ≅ Hom_{R_{E}}(W, M(E)) .

This example is used in [BT2], but it is in fact a general construction for representations of categories which goes back to [Bo]. In particular, it is used for the construction of simple functors.

2.2. Example. The constant functor \underline{k} is defined by \underline{k}(X) = k for any finite set X and Uλ = λ for any λ ∈ \underline{k}(X) and any correspondence U ∈ C(Y, X). Actually, it is elementary to check that \underline{k} ≅ kC(−, ∅), so in particular \underline{k} is projective.

2.3. Example. For any finite lattice T, let F_{T} (X) = kT^{X} for any finite set X, where T^{X} is the set of all maps X → T and kT^{X} denotes the free k-module with basis T^{X}. The action of a correspondence U ∈ C(Y, X) on a function φ ∈ T^{X} is a function Uφ ∈ T^{Y} defined by the join

(Uφ)(y) := ⋁_{\begin{matrix}x∈X \\ (y,x)∈U \end{matrix}} φ(x) ,

with the usual comment that the join over an empty set yields the least element ˆ0 of the lattice. It is easy to check that this defines a correspondence functor F_{T} (see [BT3]).

Recall that a join-morphism f : T → T′ of finite lattices is a map such that f(⋁_{t∈A} t) = ⋁_{t∈A} f(t) for any subset A of T. In particular, f must be order-preserving. Moreover, the case when A is empty shows that f(ˆ0) = ˆ0, where ˆ0 denotes the least element of any lattice. Any joinmorphism f : T → T′ induces a morphism of correspondence functors F_{T} → F_{T′} by composition with f.

3. Tensor product

The tensor product of functors always exists in the representation theory of small categories. In this section, we recall the definition and discuss the basic properties of the tensor product in the special case of correspondence functors.

3.1. Definition. Let M and M′ be correspondence functors over k. The tensor product of M and M′ is the correspondence functor M ⊗ M′ defined as follows. For any finite set X,

(M ⊗ M′)(X) = M(X) ⊗_{k} M′(X) .

If Y is a finite set and U ∈ C(Y, X), then U acts as a k-linear map

U : (M ⊗ M′)(X) ⟶ (M ⊗ M′)(Y)

defined by

U(m ⊗ m′) = Um ⊗ Um′, ∀m ∈ M(X), ∀m′ ∈ M′(X) .

This action of the generators C(Y, X) is extended to kC(Y, X) by k-linearity.

3.2. Proposition. Let M, M′ and M′′ be correspondence functors over k.

(a) The assignment (M, M′) ↦ M ⊗ M′ is a k-linear bifunctor F_{k} × F_{k} → F_{k} and is right exact in M and M′.

(b) There are natural isomorphisms of correspondence functors

M ⊗ (M′ ⊗ M′′) ≅ (M ⊗ M′) ⊗ M′′
M ⊗ M′ ≅ M′ ⊗ M
M ⊗ (M′ ⊕ M′′) ≅ (M ⊗ M′) ⊕ (M ⊗ M′)
\underline{k} ⊗ M ≅ M ,

where \underline{k} is the constant functor introduced in Example 2.2.

Proof : (a) This is a straightforward consequence of the usual properties of tensor products.

(b) For any finite set X, the standard k-linear isomorphisms

M(X) ⊗_{k} (M′(X) ⊗_{k} M′′(X)) ≅ (M(X) ⊗_{k} M′(X)) ⊗_{k} M′′(X)
M(X) ⊗_{k} M′(X) ≅ M′(X) ⊗_{k} M(X)
M(X) ⊗ (M′(X) ⊕ M′′(X)) ≅ (M(X) ⊗ M′(X)) ⊕ (M(X) ⊗ M′′(X))
\underline{k}(X) ⊗_{k} M(X) = k ⊗_{k} M(X) ≅ M(X) ,

are clearly compatible with the action of correspondences.

An additional property of the tensor product uses the symmetric monoidal structure on C given by the disjoint union of finite sets, which we write ⊔ throughout this paper. More precisely, if X, X′, Y, Y′ are finite sets, and if U ∈ C(X′, X) and V ∈ C(Y′, Y), then the disjoint union U ⊔ V can be viewed as a subset of (X′ ⊔ Y′) × (X ⊔ Y). We will represent this correspondence in the matrix form

( \begin{matrix}U & ∅ \\ ∅ & V \end{matrix} ) .

This yields a bifunctor C × C → C, still denoted by a disjoint union symbol.

We also use the following matrix notation. If U ∈ C(X′, X) and V ∈ C(X′′, X), then

\binom{U}{V} ∈ C(X′ ⊔ X′′, X)

denotes the obvious subset of (X′ ⊔ X′′) × X. Similarly, if U ∈ C(X, X′) and V ∈ C(X, X′′), then

(U, V) ∈ C(X, X′ ⊔ X′′)

denotes the obvious subset of X × (X′ ⊔ X′′).

The monoidal structure of C is used to define bilinear pairings. If M, M′, and M′′ are correspondence functors over k, the k-module of bilinear pairings M′ × M → M′′ is the k-module of natural transformations from the bifunctor

C × C ⟶ k-Mod , (X, Y) ↦ M′(X) ⊗_{k} M(Y)

to the bifunctor

C × C ⟶ k-Mod , (X, Y) ↦ M′′(X ⊔ Y) .

The connection with the tensor product of functors is established by the following standard result.

3.3. Theorem. Let M, M′ and M′′ be correspondence functors over k. The k-module of all bilinear pairings M′ × M → M′′ is isomorphic to the k-module Hom_{F_{k}}(M′ ⊗ M, M′′).

Proof : Given a morphism of correspondence functors ψ : M′ ⊗ M → M′′, we need to construct a bilinear pairing ψ̂ : M′ × M → M′′. For any finite set X, there is a k-linear map ψ_{X} : M′(X) ⊗_{k} M(X) → M′′(X) with the property that, for any finite set Z and any correspondence U ∈ C(Z, X), the diagram

\xymatrix{M′(X) ⊗_{k} M(X) \ar[r]^{ψ_{X}} \ar[d]_{U⊗U} & M′′(X) \ar[d]^{U} \\ M′(Z) ⊗_{k} M(Z) \ar[r]^{ψ_{Z}} & M′′(Z)}

is commutative. If X and Y are finite sets, we define a map

ψ̂_{X,Y} : M′(X) ⊗_{k} M(Y) ⟶ M′′(X ⊔ Y)

as the following composition

\xymatrix{M′(X) ⊗_{k} M(Y) \ar[r]^{\binom{Δ_{X}}{∅}⊗\binom{∅}{Δ_{Y}}} & M′(X ⊔ Y) ⊗_{k} M(X ⊔ Y) \ar[r]^{ψ_{X⊔Y}} & M′′(X ⊔ Y) .}

If X′ and Y′ are finite sets, if U ∈ C(X′, X) and V ∈ C(Y′, Y), then we claim that the diagram

\xymatrix{M′(X) ⊗_{k} M(Y) \ar[d]_{U}^{V} \ar[r]^{\binom{Δ_{X}}{∅}⊗\binom{∅}{Δ_{Y}}} & M′(X ⊔ Y) ⊗_{k} M(X ⊔ Y) \ar[d]^{_{( \begin{matrix}U & ∅ \\ ∅ & V \end{matrix} )}}_{_{( \begin{matrix}U & ∅ \\ ∅ & V \end{matrix} )}} \ar[r]^{ψ_{X⊔Y}} & M′′(X ⊔ Y) \ar[d]_{_{( \begin{matrix}U & ∅ \\ ∅ & V \end{matrix} )}} \\ M′(X′) ⊗_{k} M(Y′) \ar[r]_{\binom{Δ_{X′}}{∅}⊗\binom{∅}{Δ_{Y ′}}} & M′(X′ ⊔ Y′) ⊗_{k} M(X′ ⊔ Y′) \ar[r]_{ψ_{X′⊔Y ′}} & M′′(X′ ⊔ Y′)}

is commutative. The left hand side square is commutative because

( \begin{matrix}U & ∅ \\ ∅ & V \end{matrix} ) \binom{Δ_{X}}{∅} = \binom{U}{∅} = \binom{Δ_{X′}}{∅}U ,

and similarly

( \begin{matrix}U & ∅ \\ ∅ & V \end{matrix} ) \binom{∅}{Δ_{Y}} = \binom{∅}{V} = \binom{∅}{Δ_{Y′}}V .

The right hand side square is commutative by the defining property of the morphism ψ : M′⊗M → M′′. It follows that

( \begin{matrix}U & ∅ \\ ∅ & V \end{matrix} ) ψ̂_{X,Y} = ψ̂_{X′,Y′}(U ⊗ V) ,

so that the family of maps ψ̂_{X,Y} define a natural transformation from the bifunctor (X, Y) ↦ M′(X) ⊗_{k} M(Y) to the bifunctor (X, Y) ↦ M′′(X ⊔ Y), in other words a bilinear pairing ψ̂: M′ × M → M′′.

Conversely, given a bilinear pairing η : M′ × M → M′′, we have to construct a morphism of correspondence functors η̃: M′ ⊗ M → M′′. For any finite sets X, Y, there is a k-linear map

η_{X,Y} : M′(X) ⊗_{k} M(Y) ⟶ M′′(X ⊔ Y)

such that, for any finite set X′ and any correspondences U ∈ C(X′, X) and V ∈ C(Y′, Y), the diagram

(3.4)  \xymatrix{M′(X) ⊗_{k} M(Y) \ar[d]_{U⊗V} \ar[r]^{η_{X,Y}} & M′′(X ⊔ Y) \ar[d]^{_{( \begin{matrix}U & ∅ \\ ∅ & V \end{matrix} )}} \\ M′(X′) ⊗_{k} M(Y′) \ar[r]^{η_{X′,Y ′}} & M′′(X′ ⊔ Y′)}

is commutative.

In particular, for X = Y, we have a map η_{X,X} : M′(X) ⊗_{k} M(X) → M′′(X ⊔ X) which we can compose with the map M′′(X ⊔ X) → M′′(X) given by the action of the correspondence (Δ_{X}, Δ_{X}) ∈ C(X, X ⊔ X), to get a map

η̃_{X} := (Δ_{X}, Δ_{X}) η_{X,X} : M′(X) ⊗_{k} M(X) ⟶ M′′(X) .

If Z is a finite set and U ∈ C(Z, X), we claim that the diagram

\xymatrix{M′(X) ⊗_{k} M(X) \ar[d]_{U⊗U} \ar[r]^{η_{X,X}} & M′′(X ⊔ X) \ar[d]^{_{( \begin{matrix}U & ∅ \\ ∅ & U \end{matrix} )}} \ar[r]^{(Δ_{X},Δ_{X})} & M′′(X) \ar[d]^{U} \\ M′(Z) ⊗_{k} M(Z) \ar[r]_{η_{Z,Z}} & M′′(Z ⊔ Z) \ar[r]_{(Δ_{Z},Δ_{Z})} & M′′(Z)}

is commutative. The commutativity of the left hand side square is a special case of the commutativity of the diagram (3.4). The right hand side square is commutative because

U(Δ_{X}, Δ_{X}) = (U, U) = (Δ_{Z}, Δ_{Z}) ( \begin{matrix}U & ∅ \\ ∅ & U \end{matrix} ) .

It follows that

Uη̃_{X} = η̃_{Z}(U ⊗ U)

and therefore the family of maps η̃_{X} define a morphism of correspondence functors η̃ from M′ ⊗ M to M′′.

The constructions ψ ↦ ψ̂ and η ↦ η̃ are k-linear and we need to prove that they are inverse to each other. First let ψ : M′ ⊗ M → M′′ be a morphism of correspondence functors. For any finite set X, we have

˜ψ̂_{X} = (Δ_{X}, Δ_{X}) ψ̂_{X,X} = (Δ_{X}, Δ_{X}) ψ_{X⊔X}(\binom{Δ_{X}}{∅} ⊗ \binom{∅}{Δ_{X}})
= ψ_{X}((Δ_{X}, Δ_{X}) ⊗ (Δ_{X}, Δ_{X}))(\binom{Δ_{X}}{∅} ⊗ \binom{∅}{Δ_{X}})
= ψ_{X}(Δ_{X} ⊗ Δ_{X}) ,

because (Δ_{X}, Δ_{X})\binom{Δ_{X}}{∅} = Δ_{X} and (Δ_{X}, Δ_{X})\binom{∅}{Δ_{X}} = Δ_{X}. Since Δ_{X} ⊗ Δ_{X} acts as the identity on

M′(X) ⊗_{k} M(X), we get ˜ ψ̂_{X} = ψ_{X}, as required.

Now let η : M′ × M → M′′ be a bilinear pairing. For any finite sets X and Y, the definition of ψ̂_{X,Y} applied to ψ = η̃ and the definition of η̃_{X⊔Y} yield

ˆη̃_{X,Y} = η̃_{X⊔Y} (\binom{Δ_{X}}{∅} ⊗ \binom{∅}{Δ_{Y}})
= (Δ_{X⊔Y} , Δ_{X⊔Y}) η_{X⊔Y,X⊔Y} (\binom{Δ_{X}}{∅} ⊗ \binom{∅}{Δ_{Y}})

= (Δ_{X⊔Y} , Δ_{X⊔Y}) ( \begin{matrix}\binom{Δ_{X}}{∅} & ∅ \\ ∅ & \binom{∅}{Δ_{Y}} \end{matrix} ) η_{X,Y} ,

the latter equality coming from the commutative diagram (3.4) for the sets X′ = Y′ = X ⊔ Y and the correspondences U = \binom{Δ_{X}}{∅} and V = \binom{∅}{Δ_{Y}}. Now it is easy to check that

(Δ_{X⊔Y} , Δ_{X⊔Y}) ( \begin{matrix}\binom{Δ_{X}}{∅} & ∅ \\ ∅ & \binom{∅}{Δ_{Y}} \end{matrix} ) = ( \begin{matrix}Δ_{X} & ∅ \\ ∅ & Δ_{Y} \end{matrix} ) ,

and this acts as the identity on M′(X) ⊗_{k} M(Y). Therefore ˆ η̃_{X,Y} = η_{X,Y}, as was to be shown.

Theorem 3.3 shows that we have a special case of the general construction due to Day, known as Day convolution [Da].

4. Lattices

We want to apply the tensor product construction to functors of the form F_{T}, where T is a finite lattice, as defined in Example 2.3. As in [BT3], we define the category kL of finite lattices as follows. Its objects are the finite lattices and Hom_{kL}(T, T′) is the free k-module with basis the set of all join-morphisms from T to T′.

The direct product T × T′ of two lattices is defined using componentwise operations. Our next lemma shows that there is also a direct product for morphisms in kL.

4.1. Lemma. Let S, T, S′, T′ be finite lattices.

(a) If f : S → T and f′ : S′ → T′ are join-morphisms, then f × f′ : S × S′ → T × T′ is a join-morphism.

(b) Extending this direct product by k-bilinearity defines a k-linear bifunctor kL × kL → kL.

Proof : Given a subset A ⊆ S × S′, let B ⊆ S (respectively B′ ⊆ S′) denote the first projection of A (respectively the second projection). Then

(f × f′)( ⋁_{(s,s′)∈A}(s, s′)) = (f × f′)( ⋁_{(s,s′)∈A} ((s, ˆ0) ∨ (ˆ0, s′)))

= (f × f′)(( ⋁_{s∈B}(s, ˆ0)) ∨ ( ⋁_{s′∈B′}(ˆ0, s′)))
= (f × f′)( ⋁_{s∈B} s , ⋁_{s′∈B′} s′)
= (f( ⋁_{s∈B} s) , f′( ⋁_{s′∈B′} s′))
= ( ⋁_{s∈B} f(s), ⋁_{s′∈B′} f′(s′))
= ( ⋁_{s∈B} f(s) , ˆ0) ∨ (ˆ0, ⋁_{s′∈B′} f′(s′))
= ( ⋁_{s∈B} (f(s), ˆ0)) ∨ ( ⋁_{s′∈B′} (ˆ0, f′(s′)))
= ⋁_{(s,s′)∈A} (f(s), f′(s′))

as required. The second assertion follows by bilinearity.

4.2. Theorem. The bifunctors

kL × kL ⟶ F_{k} , (T, T′) ↦ F_{T} ⊗ F_{T′}

and

kL × kL ⟶ F_{k} , (T, T′) ↦ F_{T ×T′}

are isomorphic.

Proof : Let T and T′ be finite lattices. For any finite set X, there is a unique isomorphism of k-modules

τ_{X} : (F_{T} ⊗ F_{T′})(X) = k(T^{X}) ⊗_{k} k(T^{′X}) ⟶ k(T × T′)^{X} ,

mapping φ ⊗ φ′ to φ × φ′. Here of course, the map φ × φ′ : X → T × T′ is obtained by direct product from the maps φ : X → T and φ′ : X → T′. If Y is a finite set and U ∈ C(Y, X), then for any y ∈ Y,

U(φ × φ′)(y) = ⋁_{(y,x)∈U} (φ(x), φ′(x))
= ( ⋁_{(y,x)∈U} φ(x), ⋁_{(y,x′)∈U} φ′(x′))
= (Uφ(y), Uφ′(y)) .

Thus U(φ × φ′) = Uφ × Uφ′, that is, U τ_{X}(φ ⊗ φ′) = τ_{Y} (Uφ ⊗ Uφ′). Therefore

τ : F_{T} ⊗ F_{T′} ⟶ F_{T ×T′}

is an isomorphism of correspondence functors.

If f : S → T and f′ : S′ → T′ are join-morphisms, then f × f′ : S × S′ → T × T′ is a join-morphism, by Lemma 4.1. Moreover, we claim that the diagram

\xymatrix{F_{S} ⊗ F_{S′} \ar[d]_{F_{f} ⊗F_{f′}} \ar[r]^{σ} & F_{S×S′} \ar[d]^{F_{f×f′}} \\ F_{T} ⊗ F_{T′} \ar[r]^{τ} & F_{T ×T′}}

is commutative, where σ : F_{S} ⊗ F_{S′} → F_{S×S′} denotes the corresponding isomorphism for the lattices S and S′. This is because, for any finite set X, any map φ : X → S, and any map φ′ : X′ → S′, we have

F_{f×f′} σ_{X}(φ ⊗ φ′) = F_{f×f′}(φ × φ′) = (fφ) × (f′φ′) = τ_{X}(F_{f}(φ) ⊗ F_{f′}(φ′)) .

It follows that the family of isomorphisms τ yields an isomorphism between the two bifunctors of the statement.

In the representation of a small category D, it can be shown that D has a monoidal structure if and only if the constant functor is representable and the tensor product of two representable functors is again representable. Thus the symmetric monoidal structure of C comes again into play. We now show in a direct way that the tensor product of representable functors is again representable.

4.3. Corollary. If E and E′ are finite sets, then

kC(−, E) ⊗ kC(−, E′) ≅ kC(−, E ⊔ E′) .

Proof : This follows from Theorem 4.2 applied to the lattice T of subsets of E and the lattice T′ of subsets of E′. Then F_{T} ≅ kC(−, E) and F_{T′} ≅ kC(−, E′), because a map from X to T corresponds to a subset of X × E. Moreover T × T′ is isomorphic to the lattice of subsets of E ⊔ E′.

5. More properties of tensor product

We first discuss projectivity. Recall that any correspondence functor M is isomorphic to a quotient of ⨁_{i∈I} kC(−, E_{i}) where each E_{i} is a finite set and I is some index set. This is because

if m_{i} ∈ M(E_{i}), Yoneda’s lemma gives a morphism ψ_{i} : kC(−, E_{i}) → M mapping Δ_{E_{i}} to m_{i}. Choosing a set {m_{i} | i ∈ I} of generators of M, the sum of the morphisms ψ_{i} yields a surjective morphism ⨁_{i∈I} kC(−, E_{i}) → M, as required. In particular, any projective correspondence functor is isomorphic to a direct summand of a direct sum of representable functors.

5.1. Proposition. Let M and N be correspondence functors over k. If M and N are projective, then so is M ⊗ N.

Proof : By the observation above, it suffices to assume that M = ⨁_{i∈I} kC(−, E_{i}) and N =

⨁_{j∈J} kC(−, F_{j}), where E_{i} and F_{j} are finite sets and where I and J are some index sets. By Corol-

lary 4.3, we obtain

M ⊗ N ≅ ⨁_{i∈I, j∈J} kC(−, E_{i}) ⊗ kC(−, F_{j}) ≅ ⨁_{i∈I, j∈J} kC(−, E_{i} ⊔ F_{j}) ,

so M ⊗ N is projective.

It should be observed that, since \underline{k}⊗M ≅ M for any correspondence functor M (Proposition 3.2) and since \underline{k} is projective (Example 2.2), tensoring with a projective functor does not yield a projective functor in general, contrary to the case of finite group representations.

Next we consider the functors L_{E,V} introduced in Example 2.1, where E is a finite set and V is an R_{E}-module. Recall that R_{E} := kC(E, E). There is an induction procedure

V ↑_{E}^{F} := kC(F, E) ⊗_{R_{E}} V ,

where F is any finite set. Clearly V ↑_{E}^{F} is a left R_{F} -module. Notice that we have L_{E,V} (F) = V ↑_{E}^{F} by the definition of L_{E,V}.

5.2. Theorem. Let E and F be finite sets and let G = E ⊔ F. Let V be a R_{E}-module and W be a R_{F} -module. Then there is an isomorphism of correspondence functors

L_{E,V} ⊗ L_{F,W} ≅ L_{G,V ↑_{E}^{G}⊗_{k}W ↑_{F}^{G}} ,

where the kC(G, G)-module structure on V ↑_{E}^{G} ⊗_{k}W ↑_{F}^{G} is induced by the comultiplication ν : kC(G, G) → kC(G, G) ⊗_{k} kC(G, G) defined by ν(U) = U ⊗ U for any relation U ∈ C(G, G).

Proof : Since (L_{E,V} ⊗ L_{F,W})(G) = L_{E,V} (G) ⊗_{k} L_{F,W} (G) = V ↑_{E}^{G} ⊗_{k} W ↑_{F}^{G}, the identity map V ↑_{E}^{G} ⊗_{k} W ↑_{F}^{G} → (L_{E,V} ⊗ L_{F,W})(G) corresponds, by the adjointness property of L_{G,V↑_{E}^{G}⊗_{k}W↑_{F}^{G}}, to a morphism

Φ : L_{G,V↑_{E}^{G}⊗_{k}W↑_{F}^{G}} ⟶ L_{E,V} ⊗ L_{F,W}

which we need to described explicitly. For a finite set X,

L_{G,V ↑_{E}^{G}⊗_{k}W ↑_{F}^{G}}(X) = kC(X, G) ⊗_{R_{G}} ((kC(G, E) ⊗_{R_{E}} V ) ⊗_{k} (kC(G, F) ⊗_{R_{F}} W ))

and

(L_{E,V} ⊗ L_{F,W})(X) = (kC(X, E) ⊗_{R_{E}} V ) ⊗_{k} (kC(X, F) ⊗_{R_{F}} W ) .

It is easy to check that the morphism Φ_{X} maps the element

C ⊗_{R_{G}} ((A ⊗_{R_{E}} v) ⊗_{k} (B ⊗_{R_{F}} w)) ∈ L_{G,V ↑_{E}^{G}⊗_{k}W ↑_{F}^{G}}(X)

to the element

(CA ⊗_{R_{E}} v) ⊗_{k} (CB ⊗_{R_{F}} w) ∈ (L_{E,V} ⊗ L_{F,W})(X) ,

where C ∈ C(X, G), A ∈ C(G, E), v ∈ V, B ∈ C(G, F), and w ∈ W.

TENSOR PRODUCT OF CORRESPONDENCE FUNCTORS 9

Conversely, there is a morphism

Ψ_{X} : (L_{E,V} ⊗ L_{F,W})(X) → L_{G,V ↑_{E}^{G}⊗_{k}W ↑_{F}^{G}}(X)

defined as follows. For any P ∈ C(X, E), v ∈ V, Q ∈ C(X, F), and w ∈ W, it maps the element

(P ⊗_{R_{E}} v) ⊗_{k} (Q ⊗_{R_{F}} w) ∈ (L_{E,V} ⊗ L_{F,W})(X)

to the element

(P, Q) ⊗_{R_{G}} ((\binom{Δ_{E}}{∅} ⊗_{R_{E}} v) ⊗_{k} (\binom{∅}{Δ_{F}} ⊗_{R_{F}} w)) ∈ L_{G,V ↑_{E}^{G}⊗_{k}W ↑_{F}^{G}}(X) ,

where (P, Q) ∈ C(X, E ⊔ F), and where \binom{Δ_{E}}{∅} ∈ C(E ⊔ F, E) and \binom{∅}{Δ_{F}} ∈ C(E ⊔ F, F).

The map Ψ_{X} is well defined, for if R ∈ R_{E} and S ∈ R_{F}, then

Ψ_{X}((P ⊗_{R_{E}} Rv) ⊗_{k} (Q ⊗_{R_{F}} Sw)) =

= (P, Q) ⊗_{R_{G}} ((\binom{Δ_{E}}{∅} ⊗_{R_{E}} Rv) ⊗_{k} (\binom{∅}{Δ_{F}} ⊗_{R_{F}} Sw))
= (P, Q) ⊗_{R_{G}} ((\binom{R}{∅} ⊗_{R_{E}} v) ⊗_{k} (\binom{∅}{S} ⊗_{R_{F}} w))
= (P, Q) ⊗_{R_{G}} ( \begin{matrix}R & ∅ \\ ∅ & S \end{matrix} ) ((\binom{Δ_{E}}{∅} ⊗_{R_{E}} v) ⊗_{k} (\binom{∅}{Δ_{F}} ⊗_{R_{F}} w))
= (P R, QS)((\binom{Δ_{E}}{∅} ⊗_{R_{E}} v) ⊗_{k} (\binom{∅}{Δ_{F}} ⊗_{R_{F}} w))
= Ψ_{X}((P R ⊗_{R_{E}} v) ⊗_{k} (QS ⊗_{R_{F}} w)) .

Moreover, if Y is a finite set and U ∈ C(Y, X), then

Ψ_{X}(U((P ⊗_{R_{E}} Rv) ⊗_{k} (Q ⊗_{R_{F}} Sw))) =

= Ψ_{X}((UP ⊗_{R_{E}} Rv) ⊗_{k} (UQ ⊗_{R_{F}} Sw))
= (UP, UQ) ⊗_{R_{G}} ((\binom{Δ_{E}}{∅} ⊗_{R_{E}} v) ⊗_{k} (\binom{∅}{Δ_{F}} ⊗_{R_{F}} w))
= UΨ_{X}((P ⊗_{R_{E}} Rv) ⊗_{k} (Q ⊗_{R_{F}} Sw)) .

It follows that the maps Ψ_{X} define a morphism of correspondence functors

Ψ : L_{E,V} ⊗ L_{F,W} → L_{G,V ↑_{E}^{G}⊗_{k}W ↑_{F}^{G}} .

Moreover, setting u = (P ⊗_{R_{E}} v) ⊗ (Q ⊗_{R_{F}} w), we have

Φ_{X}Ψ_{X}(u) = Φ_{X} ((P, Q) ⊗_{R_{G}} ((\binom{Δ_{E}}{∅} ⊗_{R_{E}} v) ⊗_{k} (\binom{∅}{Δ_{F}} ⊗_{R_{F}} w)))
= ((P, Q)\binom{Δ_{E}}{∅} ⊗_{R_{E}} v) ⊗_{k} ((P, Q)\binom{∅}{Δ_{F}} ⊗_{R_{F}} w)
= (P ⊗_{R_{E}} v) ⊗ (Q ⊗_{R_{F}} w) = u ,

so ΦΨ is equal to the identity morphism.

Similarly, setting s = C ⊗_{R_{G}} ((A ⊗_{R_{E}} v) ⊗_{k} (B ⊗_{R_{F}} w)),

Ψ_{X}Φ_{X}(s) = Ψ_{X}((CA ⊗_{R_{E}} v) ⊗_{k} (CB ⊗_{R_{F}} w))
= (CA, CB) ⊗_{R_{G}} ((\binom{Δ_{E}}{∅} ⊗_{R_{E}} v) ⊗_{k} (\binom{∅}{Δ_{F}} ⊗_{R_{F}} w))
= C(A, B) ⊗_{R_{G}} ((\binom{Δ_{E}}{∅} ⊗_{R_{E}} v) ⊗_{k} (\binom{∅}{Δ_{F}} ⊗_{R_{F}} w))
= C ⊗_{R_{G}} (A, B)((\binom{Δ_{E}}{∅} ⊗_{R_{E}} v) ⊗_{k} (\binom{∅}{Δ_{F}} ⊗_{R_{F}} w))

= C ⊗_{R_{G}} ((A ⊗_{R_{E}} v) ⊗_{k} (B ⊗_{R_{F}} w)) = s ,

so ΨΦ is also equal to the identity morphism.

Finally we consider finite generation. Recall from [BT2] that a correspondence functor M has bounded type if there is a finite set E such that M is generated by M(E), that is, M(X) = kC(X, E)M(E) for every finite set X. Moreover, M is finitely generated if there is a finite set E and a finite subset A of M(E) such that M is generated by A, that is, M(X) = kC(X, E)A for every finite set X (see Proposition 6.4 in [BT2]).

5.3. Theorem. Let M and N be correspondence functors over k.

(a) If M and N have bounded type, so has M ⊗ N.

(b) If M and N are finitely generated, so is M ⊗ N.

Proof : (a) Let E and F be finite sets such that M is generated by M(E) and N is generated by N(F). Then the counit morphisms L_{E,M(E)} → M and L_{F,N(F)} → N are surjective. Therefore M ⊗ N is isomorphic to a quotient of L_{E,M(E)} ⊗ L_{F,N(F)}. By Theorem 5.2,

L_{E,M(E)} ⊗ L_{F,N(F)} ≅ L_{G,M(E)↑_{E}^{G}⊗_{k}N(F)↑_{F}^{G}}

where G = E ⊔ F. Since L_{G,M(E)↑_{E}^{G}⊗_{k}N(F)↑_{F}^{G}} is generated by its evaluation at G, so is M ⊗ N.

(b) Assume now that M is generated by a finite subset A of M(E) and N is generated by a finite subset B of N(F). In particular, M(E) is generated by A as an R_{E}-module and N(F) is generated by B as an R_{F} -module. Therefore, as R_{G}-modules, M(E) ↑_{E}^{G} is generated by the finite set C(G, E) ⊗_{R_{E}} A and N(F) ↑_{F}^{G} is generated by the finite set C(G, F) ⊗_{R_{F}} B, where G = E ⊔ F as before. It follows that M(E) ↑_{E}^{G} ⊗_{k}N(F) ↑_{F}^{G} is generated as an R_{G}-module by the finite set

S := (C(G, E) ⊗_{R_{E}} A) ⊗_{k} (C(G, F) ⊗_{R_{F}} B) .

Since L_{G,M(E)↑_{E}^{G}⊗_{k}N(F)↑_{F}^{G}} is generated by its evaluation at G, namely the k-module M(E) ↑_{E}^{G} ⊗_{k}N(F) ↑_{F}^{G}, it is also generated by the finite set S. Now M ⊗ N is isomorphic to a quotient of L_{G,M(E)↑_{E}^{G}⊗_{k}N(F)↑_{F}^{G}}, so it is generated by the image of S. Thus M ⊗ N is finitely generated.

6. Internal hom

In the representation theory of a small category D, the tensor product M ⊗ N exists and therefore an internal hom H(M, N) also exists, given by

(6.1)  H(M, N)(E) = Hom(P_{E} ⊗ M, N)

where P_{E} denotes the representable functor P_{E} = kD(−, E). For our category C of correspondences, we have an additional symmetric monoidal structure and we use it to obtain another very simple description of the internal hom. The proof that both approaches produce isomorphic results will be given in Corollary 6.7. We first introduce a useful construction.

6.2. Definition. Let E be a finite set and let M be a correspondence functor over k.

(a) We let t_{E} : kC → kC be the endofunctor defined on objects by t_{E}(X) = X ⊔ E and on correspondences U ∈ C(Y, X) by

t_{E}(U) = U ⊔ id = ( \begin{matrix}U & ∅ \\ ∅ & Δ_{E} \end{matrix} ) .

(b) We denote by M_{E} the correspondence functor obtained from M by precomposition with the endofunctor t_{E} : kC → kC.

(c) Let F be a finite set and V ∈ C(F, E). We define M_{V} : M_{E} → M_{F} to be the morphism obtained by precomposition with the natural transformation id ⊔V : t_{E} → t_{F}.

Explicitly, we see that M_{E}(X) = M(X ⊔ E) and M_{V} : M(X ⊔ E) → M(X ⊔ F) is given by the action of the correspondence

id ⊔V = ( \begin{matrix}Δ_{X} & ∅ \\ ∅ & V \end{matrix} ) .

6.3. Definition. Let M and M′ be correspondence functors over k. We denote by H(M, M′) the correspondence functor defined on a finite set E by

H(M, M′)(E) = Hom_{F_{k}}(M, M′_{E}) ,

and for V ∈ C(F, E), by composition with M′_{V} : M′_{E} → M′_{F}.

6.4. Lemma. The assignment (M, M′) ↦ H(M, M′) is a k-linear bifunctor F_{k}^{op} × F_{k} → F_{k}, left exact in M and M′.

Proof : This is straightforward.

Now we prove the basic adjointness property which shows that H(M, M′) is an internal hom in the category F_{k}.

6.5. Theorem. There are isomorphisms of k-modules

Hom_{F_{k}}(M′ ⊗ M, M′′) ≅ Hom_{F_{k}} (M, H(M′, M′′))

natural in M, M′, M′′. In particular, for any correspondence functor M′ over k, the endofunctor

F_{k} ⟶ F_{k} , M ↦ M′ ⊗ M

is left adjoint to the endofunctor

F_{k} ⟶ F_{k} , M ↦ H(M′, M) .

Proof : Let ψ : M′ ⊗ M → M′′ be a morphism of correspondence functors. By Theorem 3.3, we get a bilinear pairing M′ × M → M′′, hence, for any finite sets X and Y, a k-linear map

ψ̂_{X,Y} : M′(X) ⊗_{k} M(Y) → M′′(X ⊔ Y) ,

or equivalently, a k-linear map

\overline{ψ}_{Y,X} : M(Y) → Hom_{k} (M′(X), M′′(X ⊔ Y))

defined by \overline{ψ}_{Y,X}(m)(m′) = ψ̂_{X,Y} (m′ ⊗ m), for m ∈ M(Y) and m′ ∈ M′(X).

Now M′′(X ⊔ Y) = M′′_{Y}(X). Moreover, for any finite set X′ and any U ∈ C(X′, X), the commutative diagram (3.4), for Y′ = Y and V = Δ_{Y}, becomes

[figure: M′(X) ⊗_{k} M(Y); U⊗Δ_{Y}; ψ̂_{X,Y}; M′′(X ⊔ Y); _{( \begin{matrix}U & ∅ \\ ∅ & Δ_{Y} \end{matrix} )}; =; M′′_{Y}(X); U; M′(X′) ⊗_{k} M(Y); ψ̂_{X′,Y}; M′′(X′ ⊔ Y); =; M′′_{Y}(X′)]

or in other words \overline{ψ}_{Y,X′}(m)(Um′) = U\overline{ψ}_{Y,X}(m)(m′) for any m ∈ M(Y) and m′ ∈ M′(X). Therefore, for a fixed set Y and a fixed m ∈ M(Y), the maps \overline{ψ}_{Y,X}(m) define a morphism of correspondence functors

\overline{ψ}_{Y} (m) : M′ ⟶ M′′_{Y} ,

hence an element of H(M′, M′′)(Y). Allowing m to vary, we obtain a k-linear map

\overline{ψ}_{Y} : M(Y) ⟶ H(M′, M′′)(Y) .

Now if Y′ is a finite set and V ∈ C(Y′, Y), the commutative diagram (3.4), for X′ = X and U = Δ_{X}, becomes

[figure: M′(X) ⊗_{k} M(Y); Δ_{X}⊗V; ψ̂_{X,Y}; M′′(X ⊔ Y); _{( \begin{matrix}Δ_{X} & ∅ \\ ∅ & V \end{matrix} )}; =; M′′_{Y}(X); M′′_{V}; M′(X) ⊗_{k} M(Y′); ψ̂_{X,Y ′}; M′′(X ⊔ Y′); =; M′′_{Y′}(X) ,]

and it follows that the maps \overline{ψ}_{Y} define a morphism of correspondence functors \overline{ψ} : M → H(M′, M′′). Conversely, a morphism of correspondence functors ξ : M → H(M′, M′′) is determined by maps

ξ_{Y} : M(Y) ⟶ H(M′, M′′)(Y) = Hom_{F_{k}}(M′, M′′_{Y}) ,

for all finite sets Y. Furthermore, for m ∈ M(Y), the morphism ξ_{Y} (m) is in turn determined by maps

ξ_{Y} (m)_{X} : M′(X) ⟶ M′′_{Y}(X) = M′′(X ⊔ Y)

for all finite sets X. We claim that the family of maps

[figure: ξ̊_{X,Y} : M′(X) ⊗_{k} M(Y); M′′(X ⊔ Y); m′ ⊗ m; ξ_{Y} (m)_{X}(m′)]

defines a bilinear pairing ξ̊: M′ × M → M′′. We must show that, for any finite sets X, Y, X′, Y′ and any correspondences U ∈ C(X′, X) and V ∈ C(Y′, Y), the diagram

\xymatrix{M′(X) ⊗_{k} M(Y) \ar[d]_{U⊗V} \ar[r]^{ξ̊_{X,Y}} & M′′(X ⊔ Y) \ar[d]^{_{( \begin{matrix}U & ∅ \\ ∅ & V \end{matrix} )}} \\ M′(X′) ⊗_{k} M(Y′) \ar[r]^{ξ̊_{X′,Y ′}} & M′′(X′ ⊔ Y′)}

is commutative. First observe that we have

( \begin{matrix}U & ∅ \\ ∅ & V \end{matrix} ) ξ̊_{X,Y} (m′ ⊗ m) = ( \begin{matrix}U & ∅ \\ ∅ & V \end{matrix} ) ξ_{Y} (m)_{X}(m′)
= ( \begin{matrix}Δ_{X′} & ∅ \\ ∅ & V \end{matrix} ) ( \begin{matrix}U & ∅ \\ ∅ & Δ_{Y} \end{matrix} ) ξ_{Y} (m)_{X}(m′)
= ( \begin{matrix}Δ_{X′} & ∅ \\ ∅ & V \end{matrix} ) ξ_{Y} (m)_{X′}(Um′) ,

because ξ_{Y} (m) is a morphism of correspondence functors M′ → M′′_{Y}. Now the action of the

correspondence ( \begin{matrix}Δ_{X′} & ∅ \\ ∅ & V \end{matrix} ) is the composition with M′′_{V} : M′′_{Y}(X′) → M′′_{Y′}(X′), which is in turn

the action of V within the correspondence functor H(M′, M′′). Therefore we obtain

( \begin{matrix}Δ_{X′} & ∅ \\ ∅ & V \end{matrix} ) ξ_{Y} (m)_{X}(Um′) = ξ_{Y} (V m)_{X′}(Um′) = ξ̊_{X,Y} (Um′ ⊗ V m) ,

using the fact that ξ : M → H(M′, M′′) is a morphism of correspondence functors. This proves the claim.

By Theorem 3.3, the pairing ξ̊defines a morphism of correspondence functors

ξ̌:= ˜ ξ̊: M′ ⊗ M ⟶ M′′ .

Now it is straightforward to check that the maps ψ ↦ \overline{ψ} and ξ ↦ ξ̌ are inverse isomorphisms between Hom_{F_{k}}(M′ ⊗ M, M′′) and Hom_{F_{k}}(M, H(M′, M′′)).

In the case of a representable functor, there is the following useful isomorphism.

6.6. Proposition. Let P_{E} = kC(−, E) be the representable functor represented by a finite set E and let N a correspondence functor over k. There is an isomorphism of correspondence functors H(P_{E}, N) ≅ N_{E}.

Proof : Let X be a finite set. By Yoneda’s lemma, we get

H(P_{E}, N)(X) = Hom_{F_{k}} (P_{E}, N_{X}) ≅ N_{X}(E) .

Moreover, N_{X}(E) = N(E ⊔ X) ≅ N(X ⊔ E) = N_{E}(X). It is straightforward to check that the resulting isomorphism

H(P_{E}, N)(X) ≅ N_{E}(X)

is compatible with correspondences, so that it yields an isomorphism of correspondence functors H(P_{E}, N) ≅ N_{E}.

We have mentioned in (6.1) a general construction of the internal hom. We can now show that both approaches coincide (up to isomorphism).

6.7. Corollary. Let M and N be correspondence functors and let P_{E} = kC(−, E) where E is a finite set. There is an isomorphism

H(M, N)(E) := Hom_{F_{k}}(M, N_{E}) ≅ Hom_{F_{k}}(P_{E} ⊗ M, N) .

Proof : By Proposition 6.6, we have an isomorphism Hom_{F_{k}}(M, N_{E}) ≅ Hom_{F_{k}} (M, H(P_{E}, N)). Now the adjunction of Theorem 6.5 yields Hom_{F_{k}} (M, H(P_{E}, N)) ≅ Hom_{F_{k}}(P_{E} ⊗ M, N).

In the case of the constant functor, we obtain two quite different results.

6.8. Proposition. Let M be a correspondence functor over k.

(a) There is an isomorphism of correspondence functors H(\underline{k}, M) ≅ M.

(b) There is an isomorphism of correspondence functors

H(M, \underline{k}) ≅ \underline{k} ⊗_{k} M(∅)^{♮}

where M(∅)^{♮} = Hom_{k} (M(∅), k). In particular, if k is a field and if M(∅) is finitedimensional, H(M, \underline{k}) is isomorphic to a direct sum of dim(M(∅)) copies of \underline{k}.

Proof : (a) Take E = ∅ in Proposition 6.6. Then kC(−, ∅) ≅ \underline{k} because C(X, ∅) is the set of subsets of ∅, which is a singleton. On the other hand we clearly have N_{∅} = N.

(b) Let E be a finite set. It is straightforward to see that \underline{k}_{E} ≅ \underline{k}. Therefore

H(M, \underline{k})(E) ≅ Hom_{F_{k}}(M, \underline{k}_{E}) ≅ Hom_{F_{k}}(M, \underline{k}) ≅ Hom_{k} (M(∅), k) = M(∅)^{♮} .

It is then easy to check that the action of a correspondence U ∈ C(F, E) yields the identity endomorphism of M(∅)^{♮}, so that we get the constant functor tensored with M(∅)^{♮}. If k is a field and if M(∅) is finite-dimensional, it follows that H(M, \underline{k}) is isomorphic to a direct sum of copies of \underline{k}, their number being dim(M(∅)^{♮}) = dim(M(∅)).

In the same vein as in Theorem 5.3, we now consider bounded type and finite generation.

6.9. Theorem. Let M and N be correspondence functors over k.

(a) Let E and F be finite sets. If M is generated by M(E), then M_{F} is generated by M_{F} (E). Therefore, if M has bounded type, so has M_{F}. If M is finitely generated, so is M_{F}.

(b) Assume that the ring k is noetherian. If M is finitely generated and if N has bounded type (respectively is finitely generated), then H(M, N) has bounded type (respectively is finitely generated).

Proof : (a) By assumption, M(X) = kC(X, E)M(E) for each finite set X. Replacing X by X ⊔ F gives

M(X ⊔ F) = kC(X ⊔ F, E)M(E) .

Therefore M(X ⊔ F) is k-linearly generated by the elements \binom{V}{W}(m), where V ∈ C(X, E),

W ∈ C(F, E), and m ∈ M(E), because any correspondence in C(X ⊔ F, E) can be written \binom{V}{W}.

But we have

\binom{V}{W}(m) = ( \begin{matrix}V & ∅ \\ ∅ & Δ_{F} \end{matrix} ) \binom{Δ_{E}}{W}(m)

and this is the image of \binom{Δ_{E}}{W}(m) ∈ M_{F} (E) by the correspondence V within the functor M_{F}. It

follows that

\binom{V}{W}(m) ∈ kC(X, E)M_{F} (E) .

Therefore M(X ⊔ F) is k-linearly generated by kC(X, E)M_{F} (E), that is,

M_{F} (X) = kC(X, E)M_{F} (E) ,

as was to be shown. The other two assertions follow immediately.

(b) Since M is finitely generated, there is a finite subset A ⊆ M(E) such that M = kC(−, E)A, so M is isomorphic a quotient of the finite direct sum ⨁_{a∈A} kC(−, E) of representable functors. Since

H(−, N) is exact by Lemma 6.4, we deduce an embedding

\xymatrix{H(M, N) \ar@{^{(}->}[r] & ⨁_{a∈A} H(kC(−, E), N) ≅ ⨁_{a∈A} N_{E} ,}

using also Proposition 6.6. If N has bounded type, then N_{E} has bounded type, by (a), so the finite direct sum ⨁_{a∈A} N_{E} also has bounded type. Therefore H(M, N) is isomorphic to a subfunctor of a

functor of bounded type. Since k is noetherian, this implies that H(M, N) has bounded type, by Corollary 11.5 in [BT2]. The same argument with “bounded type” replaced by “finitely generated” goes through, completing the proof.

7. Algebra functors

For any finite lattice T, the functor F_{T} has more structure, namely it is a commutative algebra in the tensor category F_{k} of all correspondence functors. This section is devoted to a closer analysis of this additional structure. By a k-algebra, we always mean an associative k-algebra with an identity element.

7.1. Definition. An algebra correspondence functor over k is a correspondence functor A with values in the category k-Alg of k-algebras satisfying the following two conditions. For any finite sets X and Y, and for any correspondence U ∈ C(Y, X), the diagrams

\xymatrix{A(X) ⊗_{k} A(X) \ar[r]^{μ_{X}} \ar[d]_{U⊗U} & A(X) \ar[d]^{U} & and & k \ar[r]^{ε_{X}} \ar[d]_{id} & A(X) \ar[d]^{U} \\ A(Y) ⊗_{k} A(Y) \ar[r]^{μ_{Y}} & A(Y) &  & k \ar[r]^{ε_{Y}} & A(Y)}

are commutative, where μ_{X} : A(X) ⊗_{k} A(X) → A(X) denotes the multiplication map of the kalgebra A(X) and ε_{X} : k → A(X) denotes the map λ ↦ λ · 1_{A(X)}.

An algebra correspondence functor A is called commutative if A(X) is a commutative k-algebra for any finite set X.

The commutativity of all the diagrams in the definition can be interpreted in two different ways :

(a) The action of any correspondence U ∈ C(Y, X) is a map of k-algebras A(X) → A(Y), mapping 1_{A(X)} to 1_{A(Y)}.

(b) The family of multiplication maps μ_{X} defines a morphism of correspondence functor μ : A ⊗ A → A and the family of maps ε_{X} defines a morphism of correspondence functors ε : \underline{k} ⟶ A, where \underline{k} denotes the constant functor of Example 2.2. In other words, the triple (A, μ, ε) is an algebra in the tensor category F_{k}.

7.2. Lemma. Let A be an algebra correspondence functor, let μ : A⊗A → A be the multiplication map, let μ̂ : A × A → A be the associated pairing (see Theorem 3.3), and let X and Y be finite sets.

(a) The map

μ̂_{X,Y} : A(X) ⊗_{k} A(Y) ⟶ A(X ⊔ Y) .

is obtained as the composite of the action of \binom{Δ_{X}}{∅} ⊗ \binom{∅}{Δ_{Y}} and the multiplication μ_{X⊔Y}.

(b) If A is commutative, then μ̂_{X,Y} is an algebra homomorphism.

(c) The composite (Δ_{X}, Δ_{X}) μ̂_{X,X} is equal to μ_{X}.

Proof : (a) This follows from the proof of Theorem 3.3.

(b) The action of \binom{Δ_{X}}{∅} ⊗ \binom{∅}{Δ_{Y}} is an algebra homomorphism and, whenever A is commutative, so is the multiplication μ_{X⊔Y}.

(c) Using (a), we obtain

(Δ_{X}, Δ_{X}) μ̂_{X,X} = (Δ_{X}, Δ_{X}) μ_{X⊔Y} (\binom{Δ_{X}}{∅} ⊗ \binom{∅}{Δ_{Y}})
= μ_{X}((Δ_{X}, Δ_{X}) ⊗ (Δ_{X}, Δ_{X}))(\binom{Δ_{X}}{∅} ⊗ \binom{∅}{Δ_{Y}})
= μ_{X}(Δ_{X} ⊗ Δ_{X}) = μ_{X} .

We consider now the functor F_{T} of Example 2.3, associated to a finite lattice T. 7.3. Proposition. Let T be a finite lattice and let ˆ0 be its least element.

(a) Commutative algebra. F_{T} is a commutative algebra correspondence functor, with respect to the multiplication maps

μ_{X} : F_{T} (X) ⊗_{k} F_{T} (X) → F_{T} (X) , φ ⊗ ψ ↦ φ ∨ ψ ,

where X is a finite set and φ, ψ : X → T are maps. Here φ ∨ ψ : X → T denotes the map defined by (φ ∨ ψ)(x) = φ(x) ∨ ψ(x) for every x ∈ X. The identity element of F_{T} (X) is the constant map onto ˆ0.

(b) Exponential property. F_{T} (∅) ≅ k and, for any finite sets X and Y, the bilinear pairing

μ̂_{X,Y} : F_{T} (X) ⊗_{k} F_{T} (Y) → F_{T} (X ⊔ Y)

associated with μ is an isomorphism of k-modules.

More precisely, for any maps φ : X → T and ψ : Y → T, the element μ̂_{X,Y} (φ ⊗ ψ) ∈ F_{T} (X ⊔Y) is the function X ⊔Y → T equal to φ on X and to ψ on Y, providing a bijection between the canonical bases of F_{T} (X) ⊗_{k} F_{T} (Y) and F_{T} (X ⊔ Y).

(c) Splitting property. If • denotes a set of cardinality one, then F_{T} (•) is isomorphic to a finite direct product k × k × … × k as k-algebras.

Proof : (a) We claim that the map

υ : T × T → T , υ(a, b) = a ∨ b ,

is a join-morphism. Given a subset C ⊆ T × T, let A ⊆ T be its first projection and B ⊆ T its second projection. Then

υ( ⋁_{(a,b)∈C}(a, b)) = υ( ⋁_{(a,b)∈C} ((a, ˆ0) ∨ (ˆ0, b)))

= υ(( ⋁_{a∈A}(a, ˆ0)) ∨ ( ⋁_{b∈B}(ˆ0, b)))
= υ(( ⋁_{a∈A} a , ⋁_{b∈B} b)
= ( ⋁_{a∈A} a) ∨ ( ⋁_{b∈B} b)
= ⋁_{(a,b)∈C}(a ∨ b)
= ⋁_{(a,b)∈C} υ(a, b) ,

proving the claim. By Example 2.3, the map υ induces a morphism F_{T ×T} → F_{T}. Since F_{T ×T} ≅ F_{T} ⊗ F_{T} by Theorem 4.2, we obtain a morphism μ : F_{T} ⊗ F_{T} → F_{T}. For any finite set X, the map

μ_{X} : F_{T} (X) ⊗_{k} F_{T} (X) → F_{T} (X)

is easily seen to be the map of the statement. Clearly μ_{X} is associative and commutative and the constant map onto ˆ0 is an identity element. We obtain in this way an algebra correspondence functor F_{T} because μ : F_{T} ⊗ F_{T} → F_{T} is a morphism of functors, and so is ε : \underline{k} → F_{T}.

(b) One checks easily that \binom{Δ_{X}}{∅}φ is the function from X ⊔ Y to T equal to φ on X, and to ˆ0

on Y. Similarly \binom{∅}{Δ_{Y}}ψ is the map equal to ˆ0 on X and to ψ on Y. Thus (\binom{Δ_{X}}{∅}φ) ∨ (\binom{∅}{Δ_{Y}}ψ) is the map equal to φ on X and to ψ on Y.

(c) Since • has cardinality one, F_{T} (•) is a free k-module with basis

{g_{t} | t ∈ T }

where g_{t} : • → T is defined by g_{t}(•) = t (with • being also the unique element of the set •). Moreover, on restriction to this basis, the multiplication map corresponds to the map υ of the beginning of the proof, namely g_{t}g_{t′} = g_{t∨t′}. In that case, there is a standard procedure for finding another k-basis consisting of orthogonal idempotents f_{t} ∈ T whose sum is the identity element (namely ˆ0). These idempotents are defined by

f_{t} = ∑_{\begin{matrix}s∈T \\ s≥t \end{matrix}} χ(t, s)g_{s} ,

where χ(t, s) denotes the Möbius function of the poset T (see the appendix in [BT1]). Then we obtain isomorphisms of k-algebras

F_{T} (•) ≅ ∏_{t∈T} k·f_{t} ≅ k × k × … × k ,

as required.

Our final main result asserts that, with a small assumption on k, the converse of Proposition 7.3 holds. It is rather remarkable to obtain such a strong connection between finite lattices and algebra correspondence functors.

7.4. Theorem. Assume that k does not contain a nontrivial idempotent (i.e. Spec(k) is connected). Let A be a correspondence functor over k and suppose that A has the following three properties :

(a) Commutative algebra. A is a commutative algebra correspondence functor (with multiplication written μ).

(b) Exponential property. For any finite sets X and Y, the associated bilinear pairing

μ̂_{X,Y} : A(X) ⊗_{k} A(Y) → A(X ⊔ Y)

is an isomorphism of k-modules (hence an isomorphism of k-algebras, by commutativity of A). Moreover, A(∅) ≅ k.

(c) Splitting property. If • denotes a set of cardinality one, then A(•) is isomorphic to a finite direct product k × k × … × k as k-algebras.

Then there exists a finite lattice T such that A ≅ F_{T} (isomorphism of algebra correspondence functors).

We need a preliminary lemma.

7.5. Lemma. With the assumptions above, define a comultiplication

δ_{•} : A(•) ⟶ A(•) ⊗_{k} A(•)

as the composition of the action of \binom{Δ_{•}}{Δ_{•}} : A(•) → A(• ⊔ •) and the isomorphism

μ_{•,•}^{−1} : A(• ⊔ •) ⟶ A(•) ⊗_{k} A(•) .

(a) δ_{•} is an algebra homomorphism.

(b) δ_{•} is coassociative and cocommutative.

(c) μ_{•}δ_{•} = id_{A(•)}.

(d) The map η_{•} : A(•) → A(∅) = k induced by the action of the empty correspondence ∅ ∈ C(∅, •) is a counit for the comultiplication δ_{•} and is an algebra homomorphism.

Proof : (a) The action of \binom{Δ_{•}}{Δ_{•}} : A(•) → A(• ⊔ •) is an algebra homomorphism. So is the map μ̂_{•,•} by Lemma 7.2, hence so is the composite μ_{•,•}^{−1}\binom{Δ_{•}}{Δ_{•}}.

(b) This is left as an exercise for the reader. For the cocommutativity, use both the twist of

A(•) ⊗_{k} A(•) and the action of the correspondence ( \begin{matrix}∅ & Δ_{•} \\ Δ_{•} & ∅ \end{matrix} ).

(c) We compute

μ_{•}δ_{•} = (Δ_{•}, Δ_{•})μ̂_{•,•}μ_{•,•}^{−1}\binom{Δ_{•}}{Δ_{•}} = (Δ_{•}, Δ_{•})\binom{Δ_{•}}{Δ_{•}} = id_{A(•)} .

(d) A special case of diagram (3.4) yields the commutative diagram

\xymatrix{ & A(•) ⊗_{k} A(•) \ar[d]_{η_{•}⊗Δ_{•}} \ar[r]^{μ̂_{•,•}} & A(• ⊔ •) \ar[d]^{(∅,Δ_{•})} \\ A(•) \ar[r]^{∼} & A(∅) ⊗_{k} A(•) \ar[r]^{μ̂_{∅,•}} & A(∅ ⊔ •) \ar[r]^{=} & A(•)}

and it easy to check that the bottom composite is the identity id_{A(•)}. We deduce the commutative diagram

[figure: A(•)); ^{\binom{Δ•}{Δ•}}; =; A(• ⊔ •); (∅,Δ_{•}); μ_{•,•}^{−1}; A(•) ⊗_{k} A(•); η_{•}⊗Δ_{•}; A(•); =; A(∅ ⊔ •); μ_{∅,•}^{−1}; A(∅) ⊗_{k} A(•); ∼; A(•)]

and since the composite of the first row is equal to δ_{•}, we obtain that (η_{•} ⊗ Δ_{•})δ_{•} is the identity id_{A(•)}, as required for a counit.

Proof of Theorem 7.4 : By the splitting property, the k-algebra A(•) contains a k-basis

{f_{t} | t ∈ T }

consisting of orthogonal idempotents such that ∑_{t∈T} f_{t} = 1_{A(•)}, where T is a finite index set. Our first aim is to show that T has a lattice structure.

Now {f_{a} ⊗ f_{b} | a, b ∈ T } is a k-basis of orthogonal idempotents of A(•) ⊗_{k} A(•), so we can write

δ_{•}(f_{t}) = ∑_{a,b∈T} λ_{a,b} f_{a} ⊗ f_{b} , λ_{a,b} ∈ k .

Since δ_{•} : A(•) ⟶ A(•) ⊗_{k} A(•) is an algebra homomorphism by Lemma 7.5, δ_{•}(f_{t}) is an idempotent of A(•) ⊗_{k} A(•). Therefore λ_{a,b} is an idempotent of k, hence λ_{a,b} ∈ {0, 1} by our assumption on k, and

δ_{•}(f_{t}) = ∑_{(a,b)∈B_{t}} f_{a} ⊗ f_{b}

where B_{t} is a subset of T × T. Since the idempotents f_{t} are orthogonal and sum to 1_{A(•)}, the idempotents δ_{•}(f_{t}) are orthogonal and sum to 1_{A(•)} ⊗ 1_{A(•)}. This shows that B_{s} ∩ B_{t} = ∅ for any pair (s, t) of distinct elements of T, and that ⨆_{t∈T} B_{t} = T × T. Hence we can define an operation ∧

on T by

a ∧ b = t ⟺ (a, b) ∈ B_{t} .

In other words, for any t ∈ T,

δ_{•}(f_{t}) = ∑_{\begin{matrix}a,b∈T \\ a∧b=t \end{matrix}} f_{a} ⊗ f_{b} .

Since δ_{•} is coassociative by Lemma 7.5, this operation ∧ is associative. Similarly, since δ_{•} is cocommutative, ∧ is commutative. Finally, by Lemma 7.5 again, for any t ∈ T, we have

f_{t} = μ_{•}δ_{•}(f_{t}) = ∑_{\begin{matrix}a,b∈T \\ a∧b=t \end{matrix}} f_{a}f_{b} = ∑_{\begin{matrix}a∈T \\ a∧a=t \end{matrix}} f_{a} ,

because f_{a}f_{b} = 0 if a ≠ b. It follows that t ∧ t = t for any t ∈ T. Hence T is a commutative idempotent semigroup. Equivalently, if we define a relation ≤ on T by

a ≤ b ⟺ a ∧ b = a ,

we get an order relation on T, and any pair {a, b} of elements of T has a greatest lower bound a ∧ b. Thus T is a meet semilattice.

Since the counit η_{•} : A(•) → k is an algebra homomorphism by Lemma 7.5 and since the only idempotents of k are 0 and 1, we have η_{•}(f_{t}) ∈ {0, 1} for any t ∈ T. Moreover

1 = η_{•}(1_{A(•)}) = η_{•}( ∑_{t∈T} f_{t}) = ∑_{t∈T} η_{•}(f_{t}) ,

so there exists u ∈ T such that η_{•}(f_{u}) = 1. This element u is unique because if u, v ∈ T are such that η_{•}(f_{u}) = η_{•}(f_{v}) = 1, then η_{•}(f_{u}f_{v}) = 1, so f_{u}f_{v} ≠ 0, hence u = v. Moreover, since η_{•} is a counit, we have, for any t ∈ T,

f_{t} = (η_{•} ⊗ id)δ_{•}(f_{t}) = ∑_{\begin{matrix}a,b∈T \\ a∧b=t \end{matrix}} η_{•}(f_{a})f_{b} = ∑_{\begin{matrix}b∈T \\ u∧b=t \end{matrix}} f_{b}

and it follows that u ∧ t = t, hence t ≤ u. Therefore T is a meet semilattice with a greatest element u, so it is a lattice. We write ∨ for its join operation and ˆ0 for its least element.

Now, for any t ∈ T, we define

g_{t} = ∑_{\begin{matrix}s∈T \\ s≥t \end{matrix}} f_{s} .

The elements {g_{t} | t ∈ T } form another basis of A(•), because the transition matrix is unitriangular. Now for any t, t′ ∈ T, we have

g_{t}g_{t′} = ∑_{\begin{matrix}s≥t \\ s′≥t′ \end{matrix}} f_{s}f_{s′} = ∑_{s≥t∨t′} f_{s} = g_{t∨t′}

and also

gˆ0 = ∑_{\begin{matrix}s∈T \\ s≥ˆ0 \end{matrix}} f_{s} = ∑_{s∈T} f_{s} = 1_{A(•)} .

This means that the map t ↦ g_{t}, from T to A(•), induces an algebra homomorphism

θ_{•} : F_{T} (•) = kT^{•} ⟶ A(•) , φ ↦ g_{φ(•)} , ∀ φ ∈ T^{•} ,

and this is an isomorphism because it maps a basis to a basis.

Whenever we have a disjoint union X = W ⊔ Z, there is a bilinear pairing

μ̂_{W,Z} : A(W) ⊗_{k} A(Z) → A(X) ,

which is an isomorphism by the exponential property. Decomposing X as a union of singletons, we obtain by induction an isomorphism of k-algebras

μ̂ : ⨂_{x∈X} A(•) ⟶ A(X) .

By part (a) of Lemma 7.2, the isomorphism μ̂ maps ⨂_{x} a_{x} to the product ∏_{x∈X} C_{x}a_{x}, where

C_{x} := {(x, •)} ⊆ X × •. We then obtain a sequence of isomorphisms of k-algebras

[figure: F_{T} (X); μ^{−1}; ∼; ⨂_{x∈X} F_{T} (•); ⊗_{x} θ_{•}; ∼; ⨂_{x∈X} A(•); μ̂; ∼; A(X) .]

It is easy to check that the first isomorphism μ^{−1} maps a function φ : X → T to the element ⨂_{x∈X} h_{φ(x)}, where h_{φ(x)} : • → T is defined by h_{φ(x)}(•) = φ(x). The second isomorphism maps

⨂_{x∈X} h_{φ(x)} to ⨂_{x∈X} g_{φ(x)}, which in turn maps to ∏_{x∈X} C_{x}g_{φ(x)} via the third isomorphism μ̂. Therefore

we get the composite isomorphism of k-algebras

λ_{X} : F_{T} (X) ⟶ A(X) , λ_{X}(φ) = ∏_{x∈X} C_{x}g_{φ(x)} , ∀φ ∈ T^{X} .

We are going to show that the isomorphisms λ_{X} are compatible with the action of correspondences, but we first need a lemma.

7.6. Lemma. Let Y be a finite set and let W and Z be subsets of Y × •. Let t ∈ T and let g_{t} ∈ A(•) as defined above. Then, in the k-algebra A(Y), we have an equality

(W g_{t})(Zg_{t}) = (W ∪ Z)g_{t} .

Proof : By (3.4), there is a commutative diagram

[figure: A(•); δ_{•}; A(•) ⊗_{k} A(•); μ̂_{•,•}; W ⊗Z; A(• ⊔ •); _{( \begin{matrix}W & ∅ \\ ∅ & Z \end{matrix} )}; A(Y) ⊗_{k} A(Y); μ̂_{Y,Y}; A(Y ⊔ Y); (Δ_{Y} ,Δ_{Y} ); A(Y)]

and we compute the image of g_{t} using both paths. For the top path, the definition of δ_{•} gives

μ̂_{•,•} δ_{•}(g_{t}) = \binom{Δ_{•}}{Δ_{•}}g_{t}

and since (Δ_{Y} , Δ_{Y})( \begin{matrix}W & ∅ \\ ∅ & Z \end{matrix} )\binom{Δ_{•}}{Δ_{•}} = W ∪ Z, we obtain the element

(W ∪ Z)g_{t} ∈ A(Y) .

For the bottom path, we first have

δ_{•}(g_{t}) = ∑_{s≥t} δ_{•}(f_{s}) = ∑_{s≥t} ∑_{\begin{matrix}a,b∈T \\ a∧b=s \end{matrix}} f_{a} ⊗ f_{b} = ∑_{\begin{matrix}a,b∈T \\ a∧b≥t \end{matrix}} f_{a} ⊗ f_{b} = ( ∑_{a≥t} f_{a}) ⊗ ( ∑_{b≥t} f_{b}) = g_{t} ⊗ g_{t}

and this is mapped to W g_{t} ⊗ Zg_{t} by the vertical map W ⊗ Z. Since the composition of the two bottom maps is (Δ_{Y} , Δ_{Y})μ̂_{Y,Y} = μ_{Y}, we obtain

μ_{Y} (W g_{t} ⊗ Zg_{t}) = (W g_{t})(Zg_{t}) ∈ A(Y) .

This completes the proof of the lemma.

Now we return to the proof of the theorem and we let U ⊆ Y × X and φ ∈ T^{X}. Since

(Uφ)(y) = ⋁_{\begin{matrix}x∈X \\ (y,x)∈U \end{matrix}} φ(x) and since g_{a∨b} = g_{a}g_{b} , ∀a, b ∈ T, we obtain

λ_{Y} (Uφ) = ∏_{y∈Y} C_{y} g_{(Uφ)(y)} = ∏_{y∈Y} C_{y} ∏_{\begin{matrix}x∈X \\ (y,x)∈U \end{matrix}} g_{φ(x)}

= ∏_{x∈X} ( ∏_{\begin{matrix}y∈Y \\ (y,x)∈U \end{matrix}} C_{y} g_{φ(x)}) = ∏_{x∈X} ( ⨆_{\begin{matrix}y∈Y \\ (y,x)∈U \end{matrix}} C_{y})g_{φ(x)}

using Lemma 7.6. Therefore

λ_{Y} (Uφ) = ∏_{x∈X} ( ⨆_{\begin{matrix}y∈Y \\ (y,x)∈U \end{matrix}} C_{y})g_{φ(x)} = ∏_{x∈X}(UC_{x})g_{φ(x)} = U ∏_{x∈X} C_{x}g_{φ(x)} = Uλ_{X}(φ) .

This proves that λ : F_{T} → A is an isomorphism of algebra correspondence functors.

7.7. Remark. In Theorem 7.4, the algebra structure on A(X) is uniquely determined from that of A(•). More precisely, since A(•) is isomorphic to a product of |T | copies of k, the exponential property, which is an algebra isomorphism by Lemma 7.2, implies that A(X) is isomorphic to a product of |T |^{|X|} copies of k. However, the structure of correspondence functor is not determined by the algebra structure of A(•). Explicitly, in order to recover the lattice T from A(•), one needs more than the algebra structure. In fact, the proof shows that the additional necessary ingredient is the comultiplication δ_{•} introduced in Lemma 7.5. In other words, if A and A′ are correspondence functors satisfying the three properties of Theorem 7.4 and if moreover there is an isomorphism of co-algebras A(•) ≅ A′(•), then A ≅ A′.

7.8. Remark. The assumption on k is necessary in Theorem 7.4. Let k = k_{1} × k_{2} be the direct product of two nontrivial rings and, for i = 1, 2, let F_{T_{i}}^{(k_{i})} be the functor over k_{i} associated to a lattice T_{i}, but viewed as a functor over k (with the other factor k_{3−i} acting by zero). It can be shown that, if T_{1} and T_{2} have the same cardinality but are not isomorphic, then F_{T_{1}}^{(k_{1})} × F_{T_{2}}^{(k_{2})} satisfies the three assumptions of Theorem 7.4 but is not isomorphic to F_{T} for any lattice T.

8. Examples

Finding the decomposition of tensor products is not straightforward, due in particular to fast increasing dimensions. We only give here a few small examples, based on [BT3] and [BT4]. We refer to those two papers for details. For simplicity, we assume that k is a field.

For any n ∈ ℕ, we let [n] = \{1, 2, …, n\} and \underline{n} = \{0\} ⊔ [n]. Then \underline{n} is a totally ordered lattice and [n] is its subset of irreducible elements. There is a simple correspondence functor 𝕊_{n} introduced in Section 11 of [BT3]. It appears as a direct summand of the functor F\underline{n} associated to the lattice \underline{n}.

8.1. Example. When n = 0, then F\underline{0} = 𝕊_{0} is the constant functor \underline{k} of Example 2.2. For any correspondence functor M, we have 𝕊_{0} ⊗ M ≅ M, by Proposition 3.2.

8.2. Example. When n = 1, we have F\underline{1} ≅ 𝕊_{0} ⊕ 𝕊_{1}, by Theorem 11.6 of [BT3]. Moreover, by Theorem 4.2, we know that

F\underline{1} ⊗ F\underline{1} ≅ F_{\underline{1}×\underline{1}} = F_{♢} ,

where ♢ := \underline{1} × \underline{1} denotes the lattice of subsets of a set of cardinality 2. Applying Example 8.1, we obtain

F_{♢} ≅ F\underline{1} ⊗ F\underline{1}
≅ (𝕊_{0} ⊕ 𝕊_{1}) ⊗ (𝕊_{0} ⊕ 𝕊_{1})
≅ 𝕊_{0} ⊕ 2𝕊_{1} ⊕ (𝕊_{1} ⊗ 𝕊_{1})

On the other hand, by Example 8.7 of [BT4], there is a direct sum decomposition

F_{♢} ≅ 𝕊_{0} ⊕ 3𝕊_{1} ⊕ 2𝕊_{2} ⊕ 𝕊_{∘∘} ,

where 𝕊_{∘∘} is the fundamental functor associated to the poset ∘∘ of cardinality 2 ordered by the equality relation.

We now have two expressions for F_{♢} and we apply the Krull-Remak-Schmidt theorem, which holds by Proposition 6.6 in [BT2]. It follows that

𝕊_{1} ⊗ 𝕊_{1} ≅ 𝕊_{1} ⊕ 2𝕊_{2} ⊕ 𝕊_{∘∘} .

8.3. Example. When n = 2, we have F\underline{2} ≅ 𝕊_{0} ⊕ 2𝕊_{1} ⊕ 𝕊_{2}, by Theorem 11.6 of [BT3]. Moreover, by Theorem 4.2, we know that

F\underline{1} ⊗ F\underline{2} ≅ F_{\underline{1}×\underline{2}} = F_{P} ,

where P := \underline{1} × \underline{2}. Applying the previous two examples, we obtain

F_{P} ≅ F\underline{1} ⊗ F\underline{2}
≅ (𝕊_{0} ⊕ 𝕊_{1}) ⊗ (𝕊_{0} ⊕ 2𝕊_{1} ⊕ 𝕊_{2})
≅ 𝕊_{0} ⊕ 3𝕊_{1} ⊕ 𝕊_{2} ⊕ 2(𝕊_{1} ⊗ 𝕊_{1}) ⊕ (𝕊_{1} ⊗ 𝕊_{2})
≅ 𝕊_{0} ⊕ 3𝕊_{1} ⊕ 𝕊_{2} ⊕ 2𝕊_{1} ⊕ 4𝕊_{2} ⊕ 2𝕊_{∘∘} ⊕ (𝕊_{1} ⊗ 𝕊_{2})
≅ 𝕊_{0} ⊕ 5𝕊_{1} ⊕ 5𝕊_{2} ⊕ 2𝕊_{∘∘} ⊕ (𝕊_{1} ⊗ 𝕊_{2})

On the other hand, by Example 8.11 of [BT4], there is a direct sum decomposition

F_{P} ≅ 𝕊_{0} ⊕ 5𝕊_{1} ⊕ 7𝕊_{2} ⊕ 3𝕊_{3} ⊕ 3𝕊_{∘∘} ⊕ 𝕊_{{∘ ∘ \atop \/} ; ∘} ⊕ 𝕊_{∘ ; {/\ \atop ∘ ∘}} ⊕ U ,

where 𝕊_{{∘ ∘ \atop \/} ; ∘} (respectively 𝕊_{∘ ; {/\ \atop ∘ ∘}}) denotes the fundamental functor associated to the poset indicated

as a subscript, and where U is an indecomposable projective functor of Loewy length 3 described in Example 8.11 of [BT4].

We now have two expressions for F_{P} and it follows from the Krull-Remak-Schmidt theorem that

𝕊_{1} ⊗ 𝕊_{2} ≅ 2𝕊_{2} ⊕ 3𝕊_{3} ⊕ 𝕊_{∘∘} ⊕ 𝕊_{{∘ ∘ \atop \/} ; ∘} ⊕ 𝕊_{∘ ; {/\ \atop ∘ ∘}} ⊕ U .

We note that both 𝕊_{1} and 𝕊_{2} are projective functors, hence so is 𝕊_{1} ⊗ 𝕊_{2} by Proposition 5.1.

Acknowledgments. The first author is grateful to Antoine Touzé for an inspiring talk about exponential functors, given at the Amiens group theory seminar, and to Robert Boltje, for fruitful discussions around Remark 7.7.

We also thank the referee for a few useful comments which have allowed us to make some improvements in the exposition.

References

[Bo] S. Bouc. Foncteurs d’ensembles munis d’une double action, J. Algebra 183 (1996), 664–736.

[BT1] S. Bouc, J. Thévenaz. The algebra of essential relations on a finite set, J. reine angew. Math. 712 (2016), 225–250.

[BT2] S. Bouc, J. Thévenaz. Correspondence functors and finiteness conditions, J. Algebra 495 (2018), 150–198.

[BT3] S. Bouc, J. Thévenaz. Correspondence functors and lattices, J. Algebra 518 (2019), 453–518.

[BT4] S. Bouc, J. Thévenaz. The algebra of Boolean matrices, correspondence functors, and simplicity, preprint, 2018, arXiv:1902.05422.

[Da] B. Day. On closed categories of functors, Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics Vol. 137. Springer-Verlag, 1970, pp 1–38.

Serge Bouc, CNRS-LAMFA, Université de Picardie - Jules Verne, 33, rue St Leu, F-80039 Amiens Cedex 1, France.

serge.bouc@u-picardie.fr

Jacques Thévenaz, Section de mathématiques, EPFL, Station 8, CH-1015 Lausanne, Switzerland.

Jacques.Thevenaz@epfl.ch