sign in sign up
<<
Home , Functor, Polynomial functor

: THEORY AND APPLICATIONS

CHRISTINE VESPA

Abstract. This text is a preliminary version of material used for a course at the University of Tokyo, April-June 2019.

Keywords: functor categories; polynomial ; functor homology; stable homology.

Contents 1. Categories of functors 1 1.1. Functor categories 2 1.2. Properties of the precomposition functor 3 1.3. Projective generators 5 1.4. products 6 2. Polynomial functors 7 2.1. Definition of polynomial functors with cross-effects 7 2.2. Basic properties and examples 8 2.3. Equivalent definitions 10 2.4. Description of polynomial functors 11 2.5. Exponential functors 13 3. Homology of functors: definitions and properties 14 3.1. Definition of Tor and Ext 14 3.2. Properties 14 4. Methods to compute functor homology 16 4.1. Functor homology and adjunction 16 References 18

1. Categories of functors The following examples of small categories will be particularly interesting in these lectures. Example 1.1. (1) Let G be a . We can define a with a single object and where the endomorphisms of this object is the underlying set of G. The composition of in this category is given by the binary operation on the group G. The identity is the identity element in G. This category associated to the group G will be denoted by G. Note that any morphism is an since each element in G has an inverse. (2) Let Fin be (the skeleton of) the category of finite sets with objects n = {1, . . . , n} and morphisms arbitrary functions of finite sets and FI the category of finite sets and morphisms injective maps. (3) Let Γ be (the skeleton of) the category of finite pointed sets with objects [n] = {0, 1, . . . , n} with basepoint 0 and morphisms functions of finite sets preserving basepoint (i.e. sending 0 to 0). (4) For R a , let R-mod be (a skeleton of) the category of finitely generated free left R-modules. The category Z-mod of finitely generated free abelian groups is also denoted by ab. (5) Let gr be (a skeleton of) the category of finitely generated free groups.

Date: May 20, 2019. 1 2 CHRISTINE VESPA

1.1. Functor categories. A category C is called ”small” if both objects and morphisms are actually sets and not proper classes. In these lectures C denotes a small category. For c and c0 two objects of C, the set of morphisms from c to c0 in C will be denoted by C(c, c0). For k a commutative ring, we denote by k-Mod the over k. For C a small category and k a commutative ring, we denote by F(C, k) the category of all functors from C to k-Mod having natural transformations as morphisms. An object of F(C, k) is called a C-module. Here are some examples of interesting objects in F(C, k) for C one of the categories considered in Example 1.1. Example 1.2. (1) For C a small category we denote by k the functor in F(C, k) which is constant and equal to k. We denote also by k the functor in F(Cop, k) defined similarly. (2) For C = k-mod, we denote by Id : k-mod → k-Mod the and by T n : k-mod → k-Mod the n-th functor (i.e. T n(G) = G⊗n). The symmetric group n n Sn acts naturally on T by permutation of the factors. We denote by S : k-mod → k-Mod n the functor obtained taking the coinvariants of T by the action of Sn. This functor is called the n-th symmetric power. We denote by Γn : k-mod → k-Mod the functor obtained taking n the invariants of T by the action of Sn. This functor is called the n-th divided power. The n-th exterior power functor Λn : k-mod → k-Mod is defined by: for V ∈ k-mod, Λn(V ) is n the quotient of T (V ) by the relations v1 ⊗ ... ⊗ vn = 0 if there exists i and j such that vi = vj. (3) For C = gr and k = Z we denote by a : gr → Ab the abelianization functor. One can postcompose a with any functor given in the previous example (for k = Z). (4) For G a group, we denote by I(G) the augmentation ideal of Z[G] (i.e. the of the map P P  : Z[G] → Z given by ( αgg) = αg). Let Qn : gr → Ab be the functor given by g∈G g∈G n+1 Qn(G) = I(G)/I (G) (this functor is called sometimes the n-th Passi functor). Note that Q1 ' a. Here are some examples of morphisms in F(k-mod, k). Example 1.3. (1) By definition, we have natural transformations T n → Sn, Γn → T n and T n → Λn. (2) The norm defines a N : Sn → Γn. If n! is invertible in k (in particular if k is a field of characteristic zero) then N is a natural isomorphism. A ring R is the same as a (i.e. a category which is enriched over the of abelian groups) having one object. A covariant (resp. contravariant) additive functor (i.e. an enriched functor over the monoidal category of abelian groups) from the preadditive category R to Z-Mod is a left (resp. right) module. Therefore, functor categories can be viewed as a generalization to several objects of modules over a ring. In particular, the usual notions used in modules theory can also be defined for functors. For example, a functor F ∈ F(C, k) is a of a functor G ∈ F(C, k) if for all C ∈ C, F (C) is a sub k-module of G(C). Example 1.4. The functor Γn ∈ F(k-mod, k) is a subfunctor of T n. A functor S ∈ F(C, k) is simple if it contains no nonzero proper subfunctors. Example 1.5. The functor T n is not simple whereas the functor Λn is simple.

A functor F ∈ F(C, k) is indecomposable if there is no nonzero subfunctors F1, F2 such that F is the direct sum F1 ⊕ F2. A sequence 0 → F → G → H → 0 is an in F(C, k) if, for all C ∈ C 0 → F (C) → G(C) → H(C) → 0 in exact in k-Mod. FUNCTOR HOMOLOGY: THEORY AND APPLICATIONS 3

Example 1.6. (1) In F(k-mod, k) we have a short exact sequence 0 → Γ2 → T 2 → Λ2 → 0. If char(k) = 2 this sequence does not split. If char(k) 6= 2 this sequence has a section s :Λ2 → 2 1 T given by sV (x ∧ y) = 2 (x ⊗ y − y ⊗ x) for V ∈ k-mod and x, y ∈ V . (2) For G ∈ gr, the following short exact sequence of abelian groups:

(1) 0 ,2 InG/In+1G ,2 IG/In+1G ,2 IG/InG ,2 0. gives a non split short exact sequence in F(gr, Z): n (2) 0 ,2 T ◦ a ,2 Qn ,2 Qn−1 ,2 0.

Proposition 1.7. The category F(C, k) is abelian. Proof. The limits and colimits in F(C, k) are computed pointwise and k-Mod is an .  1.2. Properties of the precomposition functor. Comparison of functor categories is one of the important tools used in the study of these categories. In this section we give some basic facts concerning the precomposition functor. Let C and C0 be small categories and F : C → C0 be a functor. We denote by F ∗ : F(C0, k) → F(C, k) the functor obtained by precomposition by F (i.e. for M ∈ F(C0, k), F ∗(M) = M ◦ F ). Exercise 1.8. Show that the functor F ∗ is exact. This section is concerned with the following question: When the functor F has property P what can we deduce for the functor F ∗? If F is an equivalence of categories, F ∗ is also an equivalence of categories. We will study below the previous question for conditions P weaker that to be an equivalence of category. We recall that a functor F : C → C0 is essentially surjective if for each C0 ∈ C0 there exists C ∈ C 0 0 such that F (C) = C ; F is faithful if, for all C,D in C the map fC,D : C(C,D) → C (F (C),F (D)) is injective and F is full if for all C,D in D the map fC,D is surjective. A functor which is fully faithful and essentially surjective is an equivalence of category. Proposition 1.9. If F is essentially surjective, then F ∗ is faithful.

Proof. As the functor F ∗ is exact, it is sufficient to prove that, if M is an object of F(C0, k) such that M ◦ F = 0, then M = 0. For an object C of C, there exists an object C0 of C0 such that F (C0) = C as F is essentially surjective. So M(C) ' M ◦ F (C0) = 0.  Proposition 1.10. If F is full and essentially surjective, then F ∗ is fully faithful.

Proof. Exercise.  Exercise 1.11. Show that if S ∈ F(C0, k) is simple then F ∗(S) ∈ F(C, k) is simple. Recall that F : C → C0 is left adjoint to G : C0 → C (or G is right adjoint of F ) if for any C ∈ C and C0 ∈ C0 there is an isomorphism: C0(F (C),C0) 'C(C,G(C0)) which is natural in C and C0. Equivalently, F is left adjoint to G if there are two natural transformation µ : IdC → G ◦ F ( of the adjunction) and ν : F ◦ G → IdC0 (counit of the adjunction) such that F µ νF (3) (F −−→ F GF −−→ F ) = IdF

µG Gν (4) (G −−→ GF G −−→ G) = IdG. Proposition 1.12. If F is left adjoint to G : C0 → C then G∗ : F(C, k) → F(C0, k) is left adjoint to F ∗. 4 CHRISTINE VESPA

Proof. We denote by µ : IdC → G◦F the unit of the adjunction between F and G and ν : F ◦G → IdC0 its counit. We define 0 ∗ ∗ µ : IdF(C,k) → F ◦ G 0 ∗ ∗ by the following way: for M ∈ F(C, k) the natural transformation µM : M → F ◦ G (M) = M ◦ G ◦ F 0 is defined, for C ∈ C, by (µM )C = M(µC ): M(C) → M(G ◦ F (C)). Similarly, we define 0 ∗ ∗ 0 ν : G ◦ F → IdF(C ,k) 0 0 0 ∗ ∗ 0 0 by the following way: for M ∈ F(C , k) the natural transformation νM 0 : G ◦ F (M ) = M ◦ F ◦ G → 0 0 0 0 0 0 0 0 0 M is defined, for C ∈ C , by (νM 0 )C0 = M (νC0 ): M ((F ◦ G)(C )) → M (C ). Relations (3) and (4) are satisfy by µ0 and ν0 since they are satisfy by µ and ν. (For example, applying M to relation (3) for µ and ν gives relation (3) for µ0 and ν0).  Example 1.13. Inclusion-projection adjunction. Let C be a small category. We denote objects d+1 d of C by ai, where i ∈ N. For 0 ≤ j ≤ d we consider the functor πj : C → C given on objects by the projection

πj(a0, . . . , ad) = (a0,..., aˆj, . . . , ad). d d+1 • If C has a terminal object (denoted by T ) we consider the functor ij : C → C given on objects by

ij(a0, . . . , ad−1) = (a0, . . . , aj−1, T, aj, . . . , ad−1).

Proposition 1.14. If C has a terminal object, the functor πj is left adjoint to ij. Proof. On the one hand d C (πj(a0, . . . , ad), (b0, . . . , bd−1)) 'C(a0, b0) × ... × C(aj−1, bj−1) × C(aj+1, bj) × ... × C(ad, bd−1) and the other hand d+1 C ((a0, . . . , ad), ij(b0, . . . , bd−1)) 'C(a0, b0)×...×C(aj−1, bj−1)×C(ai,T )×C(aj+1, bj)×...×C(ad, bd−1).

Since T is a terminal object C(ai,T ) is a set having one element. The adjunction follows.  Applying Proposition 1.12 we obtain.

∗ ∗ Corollary 1.15. If C has a terminal object, the functor (ij) is left adjoint to (πj) . d d+1 • If C has an initial object (denoted by I) we consider the functor ij : C → C given on objects by 0 ij(a0, . . . , ad−1) = (a0, . . . , aj−1, I, aj, . . . , ad−1). 0 Proposition 1.16. If C has an initial object, the functor ij is left adjoint to πj. Applying Proposition 1.12 we obtain.

∗ 0 ∗ Corollary 1.17. If C has an initial object,the functor (πj) is left adjoint to (ij) . The following adjunction is particularly important. It will be used in the proof of Pirashvili’s cancellation result. Example 1.18. The sum-diagonal and product-diagonal adjunctions. Let C be a category having finite denoted by q. We denote also by q : C ×C → C the functor given by q(C,C0) = C qC0. To define the functor on morphisms we use the of coproduct: more concretely for f : C → C0, q(f) = f q (f). Consider the δ : C → C × C which assigns to each object C the ordered pair (C,C) and to each morphism f : C → D the pair (f, f). We have the following result. Proposition 1.19 (Sum-diagonal adjunction). If C is a category having finite coproduct, then the functor q : C × C → C is left adjoint to the diagonal functor δ : C → C × C. FUNCTOR HOMOLOGY: THEORY AND APPLICATIONS 5

Proof. We have natural 0 0 0 0 0 HomC(q(C1,C2),C ) = HomC(C1qC2,C ) ' HomC×C((C1,C2), (C ,C )) = HomC×C((C1,C2), δ(C )) where the isomorphism comes from the fact that q is the coproduct of C.  We deduce from Proposition 1.12 that δ∗ is left adjoint to q∗ i.e. for B ∈ F(C × C, k) and F ∈ F(C, k), we have natural isomorphisms ∗ ∗ HomF(C,k)(δ (B),F ) ' HomF(C×C,k)(B, q F ). Similarly, if C is a category having finite product denoted by ×, we define the functor × : C × C → C by ×(C,C0) = C × C0). Proposition 1.20 (Product-diagonal adjunction). If C is a category having finite product, then the functor × : C × C → C is right adjoint to the diagonal functor δ : C → C × C.

We deduce from Proposition 1.12 that δ∗ is right adjoint to ×∗ i.e. for B ∈ F(C × C, k) and F ∈ F(C, k), we have natural isomorphisms ∗ ∗ HomF(C,k)(F, δ (B)) ' HomF(C×C,k)(× F,B). We can also iterate the construction to define δn : C → C×n. If C has finite coproduct qn : C×n → C is left adjoint to δn. We deduce from Proposition 1.12 that (δn)∗ is left adjoint to (qn)∗ i.e. for M ∈ F(C×n, k) and F ∈ F(C, k), we have natural isomorphisms n ∗ n ∗ ×n (5) HomF(C,k)((δ ) (M),F ) ' HomF(C ,k)(M, (q ) F ). and if C has finite product ×n : C×n → C is right adjoint to δn 1.3. Projective generators. Recall that a set of generators in an abelian category A is a set E of objects of A such that for all A ∈ A there exists an epimorphism from a direct sum of object in E to A. C Definition 1.21. For any C ∈ C, the functor PC ∈ F(C, k) is defined by: C PC (X) = k[C(C,X)] where k[−]: Set → k-Mod is the k-linearization functor (i.e. the left adjoint to the forgetful functor k-Mod → Set). We will prove that these functors form a set of projective generators in F(C, k). The proof relies on the that we recall. Lemma 1.22. • Set-theoretic version For all C ∈ C and F : C → Ens we have a natural bijection

HomF unc(C,Ens)(C(C, −),F ) ' F (C). • k-linear version For all C ∈ C and F : C → k-Mod we have a natural bijection C HomF(C,k)(PC ,F ) ' F (C). The following corollary shows that F(C, k) has enough projective objects (i.e. for every functor F : C → k-Mod there is an epimorphism P → F where P is projective). C Corollary 1.23. The set {PC }C∈C is a set of projective generators of the category F(C, k). C C Proof. Let α : A → B be an epimorphism in F(C, k) and f : PC → B. Since HomF(C,k)(PC ,B) ' C C B(C), HomF(C,k)(PC ,A) ' A(C) and A(C) → B(C) is surjective we deduce that there exists g : PC → A such that α ◦ g = f. Let F : C → k-Mod. We consider the natural transformation M C PC → F C∈C c∈F (C) 6 CHRISTINE VESPA

C where the component PC → F indexed by c ∈ F (C) is the morphism corresponding to c by the Yoneda C isomorphism. This map is surjective so the set {PC }C∈C is a set of generators in the category F(C, k).  In particular, any functor F : C → k-Mod admits a by direct sums of projective generators C PC . So we can do in the category F(C, k) as in the category of k-Mod (see section 3). 1.4. Tensor products. The aim of this section is to define three different tensor product functors on functor categories and to give their basic properties. • Pointwise tensor product. Definition 1.24. For F : C → k-Mod and G : C → k-Mod, F ⊗ G : C → k-Mod is defined by (F ⊗ G)(C) = F (C) ⊗ G(C) for all C in C. Proposition 1.25. If the category C admits coproduct denoted by t then C C0 C PC ⊗ PC0 ' PCtC0 . Proof. Direct consequence of the definition of coproduct.  • Exterior tensor product. Exterior tensor product of functors is the analogue in the setup of functor categories of exterior product of modules over different rings. Definition 1.26. Let C and C0 be small categories, F : C → k-Mod and F 0 : C0 → k-Mod be functors, 0 0 0 the exterior tensor product of F and F is the functor F  F : C × C → k-Mod such that for C ∈ C, C0 ∈ C0, we have: 0 0 0 0 (F  F )(C,C ) = F (C) ⊗ F (C ). Lemma 1.27. We have an isomorphism C C0 C×C0 PC  PC0 ' P(C,C0) which is natural in C ∈ C and in C0 ∈ C0. For F, G, H in F(C, k), the sum-diagonal adjunction (see Proposition 1.19) can be reformulated by the following natural isomorphisms: ∗ HomF(C,k)(F ⊗ G, H) ' HomF(C×C,k)(F  G, q H). • Tensor product over a category. Definition 1.28. Let F : C → k-Mod and G : Cop → k-Mod, we define G ⊗ F ∈ k-Mod by C

f ∗  L 0 L  G ⊗ F = Coeq G(c ) ⊗ F (c) ,2,2 G(c) ⊗ F (c) C f∈C(c,c0) f∗ c∈C where for f ∈ C(c, c0), x ∈ G(c0) and y ∈ F (c), f ∗(x ⊗ y) = x ⊗ F (f)(y) ∈ G(c0) ⊗ F (c0) and f∗(x ⊗ y) = G(f)(x) ⊗ y ∈ G(c) ⊗ F (c). Remark 1.29. The previous definition of tensor product over a category is a particular case of coends (see [ML98, IX 6]). Remark 1.30. The enriched tensor product of such a contravariant and covariant functor is exactly the classical tensor product of a left and a right module over R. Example 1.31. Let k be the constant functor and F : C → k-Mod. We have k ⊗ F = colim F. C C (Replacing G by k in Definition 1.28, we recover the description of colim F in terms of coproduct and C . See for example [ML98, Chapter V]). FUNCTOR HOMOLOGY: THEORY AND APPLICATIONS 7

Proposition 1.32. For F : C → k-Mod and G : Cop → k-Mod, we have natural isomorphisms Cop C PC ⊗ F ' F (C) and G ⊗ PC ' G(C). C C Another way to characterize the tensor product over a category is by the following adjunction. Proposition 1.33. For F : C → k-Mod, G : Cop → k-Mod and M ∈ k-Mod we have the following isomorphism

k-Mod(G ⊗ F,M) 'F(C, k)(F, Homk(G, M)) C where Homk(G, M): C → k-Mod is given by C 7→ k-Mod(G(C),M). 2. Polynomial functors The definition of cross-effects and polynomial functors comes from the work of Eilenberg and Mac Lane on homology of spaces thereafter linked to their names [EML54]. In this paper the authors take for C a category of finitely generated free modules over a ring. This definition can easily be extended to a small monoidal category where the unit 0 is the null object of C. In a there are very huge functors which are often out of control. The polynomial property should be viewed as a way to measure the complexity of a functor. We will see that polynomial functors are easier to understand that general functors. For example, if C is an additive category, the reduced polynomial functors of degree one correspond to additive functors and for C = R-mod additive functors have been described by Eilenberg and Watts in the 60’s. Polynomial functors should be thought as being an analogue for functors of polynomial functions for functions. 2.1. Definition of polynomial functors with cross-effects. Let (C, ⊕, 0) be a small monoidal category where the unit 0 is the null object of C. Example 2.1. (1) Any pointed category with finite is an example of such category. In particular, the following categories introduced in Example 1.1 are such categories. • The category Γ is pointed by [0] and its coproduct is given by the wedge product of pointed sets [m] q [n] = [m + n]. • The category k-mod is pointed by 0 and its coproduct is the ⊕. • The category gr is pointed by 0 and its coproduct is the free product of groups ∗. (2) The category of finite pointed noncommutative sets Γ(Ass) having the same objects as Γ and where a morphism from [m] to [n] is a morphism in Γ together with a total ordering on f −1(j) for all j ∈ {1, . . . , n}. The composition is induced by the ordered union of ordered sets. The category Γ(Ass) is pointed by [0] but has no coproduct. However the wedge product of pointed sets provides a symmetric monoidal structure on Γ(Ass). This category has been considered by Pirashvili [Pir02] and is a particular case of categories of pointed sets associated to an operad (here the associative operad). See [HPV15] for more details. Remark 2.2. All the previous examples give rise to symmetric monoidal categories. Let F : C → k-Mod be a functor. The functor F is said reduced if F (0) = 0. We have a canonical decomposition F ' F (0) ⊕ F¯ where F (0) is the constant functor equal to F (0) and F¯(C) = ker(F (C) → F (0)) ' coker(F (0) → F (C)). The functor F¯ is reduced and is called the reduced functor associated to F . 1X1 ⊕t Fo X1 ∈ C and X2 ∈ C, we denote by p1 the composition p1 : X1 ⊕X2 −−−−→ X1 ⊕0 ' X1 where t is the unique element in C(X2, 0) (0 is terminal by hypothesis). We define similarly p2 : X1 ⊕ X2 → X2. ×n Definition 2.3. The n-th cross-effect of F is a functor crnF : C → k-Mod (or a multi-functor) defined inductively by cr1F (X) = ker(F (0) : F (X) → F (0)) t cr2F (X1,X2) = ker((F (p1),F (p2)) : F (X1 ⊕ X2) → F (X1) ⊕ F (X2)) and, for n ≥ 3, by

crnF (X1,...,Xn) = cr2(crn−1F (−,X3,...,Xn))(X1,X2). 8 CHRISTINE VESPA

In other words, to define the n-th cross-effect of F we consider the (n − 1)-st cross-effect, we fix the n − 2 last variables and we consider the second cross-effect of this functor. n More generally, for X1,...,Xn ∈ C we denote by rk the composition n 1⊕...⊕t⊕...⊕1 ˆ rk : X1 ⊕ ... ⊕ Xk ⊕ ... ⊕ Xn −−−−−−−−−→ X1 ⊕ ... ⊕ 0 ⊕ ... ⊕ Xn ' X1 ⊕ ... ⊕ Xk ⊕ ... ⊕ Xn. We have the following alternative definition of cross-effect.

Proposition 2.4. For F : C → k-Mod, the n-th cross-effect of F is equal to the kernel of the natural homomorphism n F r : F (X1 ⊕ ... ⊕ Xn) → ⊕ F (X1 ⊕ ... ⊕ Xˆk ⊕ ... ⊕ Xn) k=1 F n n t where r is the map (F (r1 ),...,F (rn)) . Cross-effect should be viewed as being derivations for functors. This motivates the following definition.

Definition 2.5. A functor F : C → k-Mod is said to be polynomial of degree lower or equal to n if crn+1F = 0.

We denote by Poln(C, k) the full of F(C, k) consisting of reduced polynomial functors of degree lower or equal to n. We have a filtration of categories

...,→ Poln−1(C, k) ,→ Poln(C, k) ,→ ...,→ F(C; k). A reduced of degree 1 is called linear and a reduced polynomial functor of degree 2 is called quadratic. 2.2. Basic properties and examples. The following decomposition result is particularly important.

Proposition 2.6. Let F : C → k-Mod be a reduced functor. Then there is a natural decomposition n M M F (X1 ⊕ ... ⊕ Xn) ' crkF (Xi1 ,...,Xik ).

k=1 1≤i1<...

F (X1 ⊕ X2) ' F (X1) ⊕ F (X2) ⊕ cr2F (X1,X2)

F (X1⊕X2⊕X3) ' F (X1)⊕F (X2)⊕F (X3)⊕cr2F (X1,X2)⊕cr2F (X1,X3)⊕cr2F (X2,X3)⊕cr3F (X1,X2,X3). Example 2.7. (1) The functor Id : k-mod → k-Mod is reduced and Id(U ⊕V ) = Id(U)⊕Id(V ). By the previous proposition we deduce that cr2(Id)(U, V ) = 0, so Id is polynomial of degree 1. (2) The abelianization functor a is reduced (a(0) = 0) and a(G∗H) ' a(G)⊕a(H). By the previous proposition we deduce that cr2(a)(G, H) = 0, so a is polynomial of degree 1. (3) The functor T 2 : k-mod → k-Mod is reduced and we have: T 2(U ⊕ V ) = (U ⊕ V ) ⊗ (U ⊕ V ) = T 2(U) ⊕ T 2(V ) ⊕ (U ⊗ V ⊕ V ⊗ U). 2 By the previous proposition we deduce that cr2(T )(U, V ) = U ⊗ V ⊕ V ⊗ U. Furthermore, we have: 2 2 2 2 2 2 2 T (U⊕V ⊕W ) = (U⊕V ⊕W )⊗(U⊕V ⊕W ) = T (U)⊕T (V )⊕T (W )⊕cr2T (U, V )⊕cr2T (U, W )⊕cr2T (V,W ). 2 2 We deduce from the previous proposition that cr3T (U, V, W ) = 0. So T is a polynomial func- tor of degree 2.

Proposition 2.8. If F : C → k-Mod and G : C → k-Mod are polynomial functors, then F ⊕ G is polynomial and deg(F ⊕ G) = Max{deg(F ), deg(G)}. FUNCTOR HOMOLOGY: THEORY AND APPLICATIONS 9

Proof. Suppose that F is polynomial of degree m and G is polynomial of degree n with m ≤ n. By Proposition 2.6 we have

(F ⊕ G)(X1 ⊕ ... ⊕ Xn+1) = F (X1 ⊕ ... ⊕ Xn+1) ⊕ G(X1 ⊕ ... ⊕ Xn+1)

n+1 n+1 M M M M ' crkF (Xi1 ,...,Xik ) ⊕ crkG(Xi1 ,...,Xik )

k=1 1≤i1<...

m n M M M M = crkF (Xi1 ,...,Xik ) ⊕ crkG(Xi1 ,...,Xik )

k=1 1≤i1<...

Proposition 2.9. If F : C → k-Mod and G : C → k-Mod are polynomial functors, then F ⊗ G is polynomial and deg(F ⊗ G) ≤ deg(F ) + deg(G). Proof. Suppose that F is polynomial of degree m and G is polynomial of degree n. By Proposition 2.6 we have

(F ⊗ G)(X1 ⊕ ... ⊕ Xm+n+1) = F (X1 ⊕ ... ⊕ Xn+1) ⊗ G(X1 ⊕ ... ⊕ Xm+n+1)

m+n+1 m+n+1 M M M M ' crkF (Xi1 ,...,Xik ) ⊗ crkG(Xi1 ,...,Xik )

k=1 1≤i1<...

m n M M M M = crkF (Xi1 ,...,Xik ) ⊗ crkG(Xi1 ,...,Xik )

k=1 1≤i1<...

k=1 1≤i1<...

Since crm+n+1(F ⊗G)(Xi1 ,...,Xik ) has m+n+1 variables, we deduce by identification that crm+n+1(F ⊗ G) = 0 so deg(F ⊗ G) ≤ m + n.  Remark 2.10. The inequality in the previous proposition can be strict. For example, consider F : Z-mod → Z-Mod given by F (−) = − ⊗ Q/Z. This functor is polynomial of degree 1 but F ⊗ F = 0 since Q/Z ⊗ Q/Z = 0. Proposition 2.11. If k is an integral domain and F : C → k-Mod and G : C → k-Mod are polynomial functors taking torsion free values, then F ⊗ G is polynomial and deg(F ⊗ G) = deg(F ) + deg(G). The proof of this proposition relies on the following lemma. Lemma 2.12. Let R be a commutative domain, M and N be torsion free R-modules, then M ⊗ N = 0 R iff M = 0 or N = 0. 10 CHRISTINE VESPA

Proof. Assume that M 6= 0 and N 6= 0. Let K be the field of fractions of R. We have M ⊗ K = S−1M R −1 m where S = R\{0} and S M denotes the localisation. Since M is torsion free we have: 1 = 0 ⇔ m = 0. −1 m Then the morphism φ : M → S M given by φ(m) = 1 is injective so M ⊗ K contains M. We deduce R that M ⊗ K and N ⊗ K are non zero so (M ⊗ K) ⊗ (N ⊗ K) 6= 0. Since R R R K R

(M ⊗ K) ⊗ (N ⊗ K) ' (M ⊗ N) ⊗ K 6= 0 R K R R R we deduce that M ⊗ N 6= 0.  R

Proof of Proposition 2.11. Suppose that F is polynomial of degree m and G is polynomial of degree n. By Proposition 2.6 we have

m n M M M M (F ⊗G)(X1⊕...⊕Xm+n) ' crkF (Xi1 ,...,Xik )⊗ crkG(Xi1 ,...,Xik )

k=1 1≤i1<...

m+n M M ' crk(F ⊗ G)(Xi1 ,...,Xik ).

k=1 1≤i1<...

C(X1,...,Xm+n) = crmF (X1,...,Xm) ⊗ crnG(Xm+1,...,Xm+n) is a subfunctor of crm+n(F ⊗ G)(X1,...,Xm+n). Since crmF and crnG are non-zero and take torsion free values we deduce from Lemma 2.12 that C is non zero, hence crm+n(F ⊗ G) is non zero. Hence deg(F ⊗ G) ≥ m + n. The other inequality is given in Proposition 2.9. 

Example 2.13. For a commutative domain k, using Proposition 2.11 and Example 2.7(1) we can prove by induction that the functor T n : k-mod → k-Mod is polynomial of degree n.

×n Proposition 2.14. The functor crn : F(C; k) → F(C ; k) is exact for all n ≥ 1. Definition 2.15. A full subcategory C0 of an abelian category C is thick (or is a Serre subcatrgory of C) if it contains 0 and is closed under extensions i.e. for every exact sequence 0 → B → A → C → 0 in C, A ∈ C0 if and only if B and C are in C0.

Proposition 2.16. The subcategory Poln(C, k) of F(C; k) is thick.

Proof. Immediate consequence of Proposition 2.14. 

Example 2.17. Using the short exact sequence (2) in Example 1.6, the fact that a is polynomial of degree 1 and the previous proposition we can prove by induction that Qn : gr → Ab is a polynomial functor of degree n.

2.3. Equivalent definitions. Polynomial functors can be defined in several other ways which can be more or less useful depending on the context. There are definitions in terms of instead of kernel, idempotent or the difference functor. We refer the reader to [DV15, Section 2.2] for the equivalence between these definitions when C has a null object.

Remark 2.18. In [DV19] we extend the notion of polynomial functors from a symmetric monoidal category where the unit 0 is the initial object. FUNCTOR HOMOLOGY: THEORY AND APPLICATIONS 11

2.4. Description of polynomial functors. In this section we present several results concerning the description of polynomial functors. The polynomial functors of degree one are simple to describe but, in general, it is difficult to describe the categories Poln(C, k) for n > 1. However, we will see in Section ?? that the quotient categories Poln(C, k)/Poln−1(C, k) have simple descriptions. • Description of polynomial functors of degree 1 If C is an additive category, for F : C → k-Mod a reduced polynomial functor of degree one, by Proposition 2.6 we have a natural isomorphism F (X ⊕ Y ) ' F (X) ⊕ F (Y ). So reduced polynomial functors of degree one on an additive category are equivalent to additive functors. For C an additive category, we denote by Add(C, k) the full subcategory of F(C, k) of additive functors. If C = R-mod for R a ring, the Eilenberg-Watts theorem gives the following description of additive functors from R-mod to k-Mod. Theorem 2.19. [Eil60, Wat60] Let R and k be two rings. The evaluation on R induces an equivalence of categories ' op Add(R-mod, k) −→ (R ⊗ k)-Mod. For M ∈ (Rop ⊗ k)-Mod, the quasi-inverse of this functor is given by M 7→ M ⊗ −. R

Let C be a pointed category having finite coproducts denoted by q. For E ∈ C we denote by hEiC the full subcategory of C having as objects finite sums of copies of E.

Example 2.20. (1) For C = R-Mod and E = R, we have hEiC = R-mod. (2) For C = Gr and E = Z, we have hEiC = gr. (3) For C = Set∗ (the category of pointed sets) and E = [1], we have hEiC = Γ.

Eilenberg-Watts theorem has been generalized to linear functors on hEiC in [HV11]. We need some notations before to give the statement. ¯ A functor F : hEiC → k-Mod has a greatest linear quotient functor denoted by T1(F ). This functor is given explicitely by the following formula

T¯1(F )(V ) = Coker(F (p1) + F (p2) − F (s): F (V q V ) → F (V )) where p1, p2, s : V q V → V are respectively, the first projection, the second projection and the sum (given by the universal property of coproduct). Let ΛC(E) := T¯1(PE)(E) where PE is the projective generator associated to E (see Definition 1.21). Theorem 2.21. [HV11, Theorem 3.12] The evaluation on E induces an equivalence of categories ' Pol1(hEiC, Z) −→ ΛC(E)-Mod.

For M ∈ ΛC(E)-Mod, the quasi-inverse of this functor is given by M 7→ T¯1PE(−) ⊗ M. ΛC (E) Remark 2.22. This theorem can easily be extend to functors with values in k-modules. • Description of polynomial functors of degree n

In general it is difficult to give a complete description of the categories Poln(C, k). The case of quadratic functors has been studied for several categories C. For C = ab quadratic functors have been described by Baues in [Bau94], for C = gr they have been described by Baues and Pirashvili in [BP99]. For C = Γ quadratic functors can be described thanks to the Dold-Kan type theorem of Pirashvili as we will explain below. All these cases are covered by the complete description of the category Pol2(hEiC, k) given in [HV11]. For higher degrees, a description of polynomial functors of degree n, for all n, has been given in [BDFP01] for C = ab and in [HPV15] for C = gr. Polynomials functor on the category Γ can be described thanks to the following theorem. Theorem 2.23 (Dold-Kan type theorem of Pirashvili). [Pir00] Let Ω be the category of finite sets and surjections. The functor: cr : F(Γ; k) → F(Ω; k) 12 CHRISTINE VESPA given by cr(F )(n) = crnF ([1],..., [1]) for F ∈ F(Γ; k) is an equivalence of categories (where crnF is the n-th cross-effect of F , see Definition 2.3).

Let F(Ω; k)≤n the full subcategory of F(Ω; k) having as objects functors vanishing on sets X such that | X |> n. Corollary 2.24. The functor: cr : Poln(Γ; k) → F(Ω; k)≤n is an equivalence of categories. Remark 2.25. The classical Dold-Kan theorem gives an equivalence between simplicial sets in an abelian category A (i.e. F(∆op, A)) and the category of chain complexes of A. The category of chain complexes can be viewed as the category of functors from an over Γ preserving nul morphism. This justifies the name of the previous theorem. Remark 2.26. In [DPV] we extend this result to the PROP associated to a set-operad. The previous theorem corresponding to the PROP associated to the operad Com. Remark 2.27. Two rings are called Morita equivalent if they have equivalent categories of left modules. As functor categories are generalizations to several objects of the theory of modules over a ring this definition can be extended in the following form: two small categories C and C0 are Morita equivalent if the functors categories F(C, k) and F(C0, k) are equivalent. If C and C0 are equivalent they are obviously Morita equivalent but there are lots of examples of non equivalent categories which are Morita equivalent. For example Theorem 2.23 says that the categories Γ and Ω are Morita equivalent. In [S lo04] and [LS15] the authors give general conditions in order to obtain equivalences of functor categories. Their methods cover the classical Dold-Kan theorem but also the Dold-Kan type theorem of Pirashvili and its extension to PROP associated to the associative set-operad. • Description of polynomial functors of degree n modulo polynomial functors of degree n − 1

In order to understand polynomial functors of degree n we would like to describe the category of polynomial functors of degree ≤ n from the category of polynomial functors of degree ≤ n − 1 and another category which morally measures the difference between the functors of degree ≤ n and the functors of degree ≤ n − 1. For this we present in this section a general way to study an abelian category C from ”smaller” categories. More precisely, for C0 a subcategory of C having good properties, we define the C/C0. The study of C can be reduced to the study of C0 and C/C0. The original reference for this section is [Gab62, Chapitre III]. Remark 2.28. A quotient category is a particular case of the localisation of Gabriel-Zisman [GZ67] of a category relatively to a set of morphisms (here, we inverse the morphisms whose kernel and cokernel are in the subcategory C0). Proposition 2.29. Let F : C → A be a functor between abelian categories. If F is exact then the kernel of F (i.e. the full subcategory of C of objects which are sent to 0 by F ) is thick. The converse of this proposition is given by the notion of quotient category defined below. More precisely, if C0 is a thick subcategory of C, we define a new abelian category C/C0 called ”quotient category” such that there exists an F : C → C/C0 whose kernel is C0. Definition 2.30. Let C be an abelian category and C0 a thick subcategory of C. The quotient category C/C0 has as objects the objects of C and for X and Y two objects of C 0 0 0 C/C (X,Y ) = colim HomC(X ,Y/Y ) where the colimit runs through all subobjects X0 ⊂ X, Y 0 ⊂ Y such that X/X0 and Y 0 are objects in C0. FUNCTOR HOMOLOGY: THEORY AND APPLICATIONS 13

The composition of morphisms in C/C0 is given carefully in [Gab62]. 0 0 If f ∈ HomC(X,Y ) such that Ker(f) ∈ C (resp. Coker(f) ∈ C ) then f is a monomorphism (resp. epimorphism) in C/C0. 0 We have a quotient functor T : C → C/C given by T (X) = X for X ∈ C and for f ∈ HomC(X,Y ), 0 0 T (f) is the of f in colim HomC(X ,Y/Y ). Proposition 2.31. The functor T : C → C/C0 is exact and its kernel is C0. Moreover, for D an abelian category, F : C → D an exact functor which is trivial on C0, there exists a unique functor G : C/C0 → D such that G ◦ T = F .

By Proposition 2.16, the category Poln(C, k) is thick. We will describe below the quotient categories Poln(C, k)/Poln−1(C, k).

Proposition 2.32 (Pirashvili). The functor crn : Poln(ab, k) → k[Sn]-Mod, F 7→ crnF (Z,..., Z) induces an equivalence of categories:

Poln(ab, k)/Poln−1(ab, k) ' k[Sn]-Mod.

For n > 1 the categories Poln(ab, k) and Poln(gr, k) are not equivalent. However we have the following result. Proposition 2.33. [DV15, Corollaire 3.6] The abelianization functor a : gr → Ab induces an equiva- lence of categories:

Poln(gr, k)/Poln−1(gr, k) 'Poln(ab, k)/Poln−1(ab, k) Combinig the last two propositions we obtain the equivalence of categories

Poln(gr, k)/Poln−1(gr, k) ' k[Sn]-Mod.

Remark 2.34. In [DV15], we give more generally the description of Poln(hEiC, k)/Poln−1(hEiC, k) where C is a small pointed category with finite coproduct, E is a fixed object in C and hEiC is the full subcategory of C having as objects finite coproducts of E. More precisely, we give the recollement diagram between the categories Poln(hEiC, k), Poln−1(hEiC, k) and a category of modules. 2.5. Exponential functors. One of the most important property of exponential is that the exponential of a sum is the product of exponentials. In this section we introduce functors satisfying a similar property. We will see in section ?? that functor homology of exponential functors has interesting properties. Let C be a small category having finite coproducts denoted by t. Recall that (C, t, 0) (where 0 denotes the initial object corresponding to the empty coproduct) is a symmetric monoidal category.

Definition 2.35. (1) An exponential functor E of F(C, k) is a symmetric from C to k-Mod. In particular, E is equipped with natural isomorphisms E(C t C0) ' E(C) ⊗ E(C0) for C and C0 in C. (2) A graded functor E• = (En) where En : C → k-Mod is exponential if we have a natural isomorphism: n M En(U t V ) ' Ei(U) ⊗ En−i(V ). i=0 Example 2.36. The graded functors Λ•,S•, Γ• are exponential functors. The graded functor T • is not exponential but it is not far to be exponential in the sense that we have the following isomorphism n n M i n−i T (U ⊕ V ) ' (T (U) ⊗ T (V )) ⊗ Z[Sn]. S ×S i=0 i n−i Proposition 2.37. Let E• = (En) be a graded exponential functor such that E0 = k. Then En is polynomial of degree ≤ n. 14 CHRISTINE VESPA

Proof. By the exponential property and the hypothesis on E0 we have natural isomorphisms: E1(U t 1 1 1 V ) ' E (U) ⊕ E (V ). So cr2(E ) = 0. By induction, suppose that for all i ≤ n, deg(Ei) ≤ i. Using the exponential property and Proposi- tion 2.9 we obtain the result. 

Example 2.38. The functors Γn,Sn, Λn from k-mod to k-Mod are polynomial of degree n.

3. Homology of functors: definitions and properties The terms ”functor homology” denote homological algebra in functor categories. In this section we will explain in more details what does it mean, give some basic properties and several results concerning functor homology over gr.

3.1. Definition of Tor and Ext. • Tor groups. The functors − ⊗ F and G ⊗ − are right exact. They commute with colimits in each C C variables. We can derive these functors on the left.

Definition 3.1. For F ∈ F(C, k) and G ∈ F(Cop, k) we define:

C T ori (G, F ) = Hi(G ⊗ P•) C where P• is a projective resolution of F ∈ F(C, k). Remark 3.2. We could equally well resolve G.

Homology of a category

Definition 3.3. For F ∈ F(C, k) we define the graded k-module

C H∗(C,F ) = T or∗ (k,F ).

Remark 3.4. We have H0(C,F ) = k ⊗ F = colim(F ) by Example 1.31. C Example 3.5. If G is the category associated to a group G (see example 1.1), a functor F : G → k-Mod is a k[G]-module and the homology of the category G, H∗(G, F ), corresponds to the usual notion of homology of the group G with coefficients in a k[G]-module. We give two simple examples of computation of homologies of categories.

Example 3.6. If C has an initial object I then H0(C, k) = k and H∗(C, k) = 0 for ∗ > 0. C In fact, k = PI (−) = k[C(I, −)] so k is a projective object in F(C, k).

Example 3.7. If C has a terminal object T then H0(C; F ) = F (T ) and Hi(C; F ) = 0 for ∗ > 0. Cop op In fact, k = PT (−) = k[C(−,T )] so k is a projective object in F(C , k). • Ext groups.

Definition 3.8. For F ∈ F(C, k) and G ∈ F(C, k) we define: Exti (G, F ) = H (Hom (P ,F )) F(Ck) i F(C,k) • where P• is a projective resolution of G ∈ F(C, k). 3.2. Properties. FUNCTOR HOMOLOGY: THEORY AND APPLICATIONS 15

• Relation between Ext and Tor.

Proposition 3.9. Let G : Cop → k-Mod, F : C → k-Mod and I an injective k-module then we have a natural graded isomorphism C ∨ • T or• (G, F ) ' ExtF(C)(F, Hom(G, I)) ∨ where V = Homk(V,I) for a k-module V and Hom(G, I): C → k-Mod, C 7→ Homk-Mod(G(C),I). Proof. We use the characterization of Ext groups given in [ML63, Chapter III, Theorem 10.1] for n C ∨ Ex (F ) = T orn(G, F ) . We have 0 C ∨ ∨ Ex (F ) = T or0 (G, F ) = (G ⊗ F ) = Homk(G ⊗ F,I) ' HomC(F, Homk(G, I)) C C by Proposition 1.33. For a projective functor P we have n C ∨ Ex (P ) = T orn(G, P ) = 0 n Since I is an injective k-module, Homk(−,I) is exact and Ex send short exact sequence to long exact sequence. n n ∗ We deduce that Ex ' Ext (−,G ).  • Functoriality.

Lemma 3.10. Let C and D be two small category, ϕ : C → D, F : D → k-Mod and G : Dop → k-Mod be functors then ϕ induces a natural morphism C ∗ ∗ D ϕ• : T or• (ϕ (G), ϕ (F )) → T or• (G, F ).

Proof. To construct the maps ϕ• we use the notion of universal δ-functors (see [Wei94, Section 2.1] for the definition) and the following result (see [Wei94, Theorem 2.4.7 p47]) Let A and B be abelian categories. If A has enough projective objects, then for any right exact functor F : A → B, the derived functors LnF form a universal δ-functor. The categories F(C, k) and F(D, k) are abelian by Proposition 1.7 and have enough projective objects by Corollary 1.23. For G : Dop → k-Mod, the functor G ⊗ − : F(D, k) → k-Mod is right D exact. Since the functor ϕ∗ is exact (see Exercise 1.8), the functor ϕ∗(G)⊗− : F(C, k) → k-Mod is also C D C ∗ right exact. We deduce that the functors T orn (G, −) and T orn(ϕ (G), −) form universal δ-functors. C ∗ By the universal property of the δ-functor T orn(ϕ (G), −) the natural transformation ∗ ∗ ϕ0 : ϕ (G) ⊗ ϕ (−) → G ⊗ − C D gives morphisms C ∗ ∗ D ϕn : T orn(ϕ (G), ϕ (F )) → T orn (G, F ) extending ϕ0. Moreover these morphisms are unique if we require to have a morphism of δ-functor. 

Corollary 3.11. Let C and D be two small category, ϕ : C → D and F : D → k-Mod be functors then ϕ induces a natural morphism ∗ ϕ• : H•(C, ϕ (F )) → H•(D,F ).

Proof. Apply the previous lemma for G the constant functor equal to k.  One of the main ingredient of the method developed in [DV10] consists to compare functor homolo- gies of various categories. More precisely, we will see in Proposition ?? a criterion implying that the ∗ natural map ϕ∗ : H∗(C, ϕ (F )) → H∗(D,F ) obtained in the previous corollary is an isomorphism. The following lemma will be also useful. 16 CHRISTINE VESPA

Lemma 3.12. Let C and D be two small category, ϕ, ψ : C → D be functors, u : ϕ → ψ a natural transformation and F : D → k-Mod and G : D → k-Mod be functors, then the following diagram is commutative T orC (ψ∗F,F ◦u) C ∗ ∗ • C ∗ ∗ T or• (ψ G, ϕ F ) ,2 T or• (ϕ G, ϕ F )

C ∗ T or• (G◦u,ϕ F ) ψ•   T orC(ϕ∗G, ϕ∗F ) ,2 T orD(G, F ). • ϕ• •

4. Methods to compute functor homology In this section we present several useful tools to compute functor homology.

4.1. Functor homology and adjunction. We will prove that the adjunction between functor cat- egories coming from an adjunction of functors (see Proposition 1.12) can be derived to give natural isomorphisms between Ext. We will need the following lemma Lemma 4.1. If ψ : C → C0 is left adjoint to ϕ : C0 → C then, for each c in C

∗ C C0 ϕ PC ' Pψ(C). In particular ϕ∗ preserves projective objects.

∗ C C0 Proof. ϕ PC = K[HomC(C, ϕ(−))] ' K[HomC0 (ψ(C), −) = Pψ(C). 

Proposition 4.2. If ψ : C → C0 is left adjoint to ϕ : C0 → C then we have natural isomorphisms

∗ ∗ ∗ ∗ Ext 0 (ϕ F,G) ' Ext (F, ψ G) F(C ,k) F(C,k) of graded k-modules for F ∈ F(C, k) and G ∈ F(C0, k). ∗ ∗ Proof. Let P• → F be a projective resolution. Since ϕ is exact (see Exercice 1.8) and ϕ preserves ∗ ∗ ∗ projective objects by Lemma 4.1, ϕ (P•) → ϕ (F ) is also a projective resolution. Therefore, since ϕ is left adjoint to ψ∗ according to Proposition 1.12, we have:

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Ext 0 (ϕ F,G) = H (Hom 0 (ϕ (P ),G)) ' H (Hom (P , ψ G)) ' Ext (F, ψ G). F(C ,k) F(C ,k) • F(C,k) • F(C,k)  Application 1: Base change. Let K and K0 be two fields and K → K0 be a finite extension of degree d. Consider the forgetful functor F : P (K0) → P (K) and the base change T := K0 ⊗ − : K-mod → K K0-mod. Proposition 4.3. The base change functor T is left adjoint and right adjoint to the forgetful functor.

Proposition 4.4. For IK : P (K) → Ab and IK0 : P (K) → Ab the forgetful functors we have:

∗ ∗ ⊕d Ext 0 (I 0 ,I 0 ) ' Ext (I ,I ) . F(P (K ),k) K K F(P (K),k) K K ∗ ∗ 0 ⊕d Proof. We have F (IK ) = IK0 and T (IK0 ) ' K ⊗ IK ' IK . By Proposition 4.2 we have: K ∗ ∗ ∗ ∗ ∗ Ext 0 (I 0 ,I 0 ) = Ext 0 (I 0 ,F (I )) ' Ext (T I 0 ,I ) F(P (K ),k) K K F(P (K ),k) K K F(P (K),k) K K

' Ext∗ (I⊕d,I ) ' Ext∗ (I ,I )⊕d. F(P (K),k) K K F(P (K),k) K K  FUNCTOR HOMOLOGY: THEORY AND APPLICATIONS 17

Application 2: Pirashvili’s cancellation result. The following cancellation result is due to Pirashvili [Pir88]. Recall that, for C a category having a null object 0, a functor F : C → k-Mod is reduced if F (0) = 0. Proposition 4.5 (Pirashvili’s cancellation result). Let C be a small pointed category and n ∈ N. Let G0,...,Gn ∈ F(C, k) be reduced functors and F ∈ F(C, k) be a polynomial functor of degree ≤ n. Then: (1) If C has finite coproducts, then Ext∗ (G ⊗ ... ⊗ G ,F ) = 0 F(C,k) 0 n (2) If C is additive then Ext∗ (G ⊗ ... ⊗ G ,F ) = Ext∗ (F,G ⊗ ... ⊗ G ) = 0. F(C,k) 0 n F(C,k) 0 n Proof. To prove (1), since C has finite coproduct, we can apply Proposition 4.2 to the sum-diagonal adjunction (see Example 1.19) to obtain natural isomorphisms ∗ ∗ n+1 ∗ Ext (G ⊗ ... ⊗ G ,F ) ' Ext n+1 (G ... G , (q ) F ) F(C,k) 0 n F(C ,k) 0   n Since C has a null object and F is polynomial of degree ≤ n, by Proposition 2.6 n+1 n n+1 ∗ M M M M (q ) F (C0,...,Cn) = F (C0q...qCn) ' crkF (Ci1 ,...,Cik ) = crkF (Ci1 ,...,Cik )

k=1 0≤i1<...

(2) For all i ∈ {0, . . . , n} fix Ai = Id. We deduce from Proposition 4.5 (2) that for F a polynomial functor of degree ≤ n − 1, we have Ext∗ (T n,F ) = Ext∗ (F,T n) = 0. F(k-mod,k) F(k-mod,k) Example 4.7. Consider the category gr. This category has a null objects and finite coproducts (given by the free product) but has not finite product. For all i ∈ {0, . . . , n} fix Ai = a. We deduce from Proposition 4.5 (1) that for F a polynomial functor of degree ≤ n − 1, we have Ext∗ (T n ◦ a,F ) = 0. F(gr,k) 18 CHRISTINE VESPA

In particular for F = T i ◦ a with i < n we have Ext∗ (T n ◦ a,T i ◦ a) = 0. F(gr,k) In Theorem ?? we will give the complete computation of Ext∗ (T n ◦ a,T m ◦ a). F(gr,Z) References [Bau94] Hans Joachim Baues. Quadratic functors and metastable . J. Pure Appl. Algebra, 91(1-3):49–107, 1994. [BDFP01] Hans-Joachim Baues, Winfried Dreckmann, Vincent Franjou, and Teimuraz Pirashvili. Foncteurs polynomiaux et foncteurs de Mackey non lin´eaires. Bull. Soc. Math. France, 129(2):237–257, 2001. [BP99] Hans-Joachim Baues and Teimuraz Pirashvili. Quadratic endofunctors of the . Adv. Math., 141(1):167–206, 1999. [DPV] Aur´elienDjament, Teimuraz Pirashvili, and Christine Vespa. Cohomologie des foncteurs polynomiaux sur les groupes libres. arXiv:1409.0629. [DV10] Aur´elienDjament and Christine Vespa. Sur l’homologie des groupes orthogonaux et symplectiques `acoeffi- cients tordus. Ann. Sci. Ec.´ Norm. Sup´er.(4), 43(3):395–459, 2010. [DV15] Aur´elienDjament and Christine Vespa. Sur l’homologie des groupes d’automorphismes des groupes libres `a coefficients polynomiaux. Comment. Math. Helv., 90(1):33–58, 2015. [DV19] Aur´elienDjament and Christine Vespa. Foncteurs faiblement polynomiaux. Int. Math. Res. Not. IMRN, (2):321–391, 2019. [Eil60] Samuel Eilenberg. Abstract description of some basic functors. J. Indian Math. Soc. (N.S.), 24:231–234 (1961), 1960. [EML54] Samuel Eilenberg and . On the groups H(Π, n). II. Methods of computation. Ann. of Math. (2), 60:49–139, 1954. [Gab62] Pierre Gabriel. Des cat´egoriesab´eliennes. Bull. Soc. Math. France, 90:323–448, 1962. [GZ67] P. Gabriel and M. Zisman. Calculus of fractions and . Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35. Springer-Verlag New York, Inc., New York, 1967. [HPV15] Manfred Hartl, Teimuraz Pirashvili, and Christine Vespa. Polynomial functors from algebras over a set-operad and nonlinear Mackey functors. Int. Math. Res. Not. IMRN, (6):1461–1554, 2015. [HV11] Manfred Hartl and Christine Vespa. Quadratic functors on pointed categories. Adv. Math., 226(5):3927–4010, 2011. [LS15] Stephen Lack and Ross Street. Combinatorial categorical equivalences of Dold-Kan type. J. Pure Appl. Alge- bra, 219(10):4343–4367, 2015. [ML63] Saunders Mac Lane. Homology. Die Grundlehren der mathematischen Wissenschaften, Bd. 114. Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-G¨ottingen-Heidelberg, 1963. [ML98] Saunders Mac Lane. Categories for the working mathematician, volume 5 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1998. [Pir88] T. I. Pirashvili. Higher additivizations. Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR, 91:44– 54, 1988. [Pir00] Teimuraz Pirashvili. Dold-Kan type theorem for Γ-groups. Math. Ann., 318(2):277–298, 2000. [Pir02] Teimuraz Pirashvili. On the PROP corresponding to bialgebras. Cah. Topol. G´eom.Diff´er.Cat´eg., 43(3):221– 239, 2002. [S lo04] Jolanta S lomi´nska. Dold-Kan type theorems and Morita equivalences of functor categories. J. Algebra, 274(1):118–137, 2004. [Tou18] Antoine Touz´e.On the structure of graded commutative exponential functors. arXiv:1810.01623, 2018. [Wat60] Charles E. Watts. Intrinsic characterizations of some additive functors. Proc. Amer. Math. Soc., 11:5–8, 1960. [Wei94] Charles A. Weibel. An introduction to homological algebra, volume 38 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1994.

Universite´ de Strasbourg, Institut de Recherche Mathematique´ Avancee,´ Strasbourg, France. E-mail address: [email protected]