Homological Kernels of Monoidal Functors

HOMOLOGICAL KERNELS OF MONOIDAL FUNCTORS

KEVIN COULEMBIER

Abstract. We show that each rigid monoidal category a over a field defines a family of universal tensor categories, which together classify all faithful monoidal functors from a to tensor categories. Each of the universal tensor categories classifies monoidal functors of a given ‘homological kernel’ and can be realised as a sheaf category, not necessarily on a. This yields a theory of ‘local abelian envelopes’ which completes the notion of monoidal abelian envelopes.

Introduction Weak abelian envelopes. For a ﬁeld k, let a be a small k-linear rigid monoidal category in which the endomorphism algebra of the tensor unit is k (a rigid monoidal category over k). If such a category is also abelian, it is called a tensor category over k, see [De1, EGNO]. The 2-category of tensor categories and exact linear monoidal functors is denoted by Tens. A weak abelian envelope of a is a faithful k-linear monoidal functor a → T to a tensor category T which induces equivalences ′ ∼ ′ ⊗ Tens(T, T ) Ð→ [a, T ]faith, for all tensor categories T′ over k, where the right-hand side is the category of faithful linear monoidal functors. If a → T is also full, it is an abelian envelope. Examples of abelian envelopes have led to some spectacular applications in recent years, see for instance [BEO, CEH, De2, EHS, Os], and they proved to be a great source for new interesting tensor categories. This has prompted the developed of extensive theory in [BEO, Co2, Co3, CEOP] and references therein. The most natural question in the subject asks when a given category a admits a (weak) abelian envelope. We refer to [CEOP] for some partial and conjectural answers. Many rigid monoidal categories a over k, such as most tensor-triangulated ones, do not admit any faithful monoidal functors to tensor categories. Clearly these do not admit (weak) abelian envelopes. The current work establishes that as soon as there exist such faithful monoidal functors, then a always deﬁnes universal tensor categories, namely a non-empty set of local abelian envelopes, which together play the role of an envelope. In particular, this set has cardinality 1 if and only if a admits a weak abelian envelope. In other words, the question of whether a admits a weak abelian envelope is equivalent to the question of whether the 2-functor ⊗ arXiv:2107.02374v1 [math.CT] 6 Jul 2021 [a, −]faith from Tens to the 2-category of groupoids is representable. We will demonstrate that this 2-functor is in fact always “multi representable”. This thus greatly extends the potential of abelian envelopes as a source of tensor categories.

Multi representability of 2-functors. We adapt some terminology from [Di1] to the 2-categorical setting in a 1-dimensional way. Let M be a 2-category and F a 2-functor M → Cat to the 2-category of categories. For a family {Fγ, γ ∈ Γ} of such 2-functors, we can deﬁne the 2-functor ∐γ∈Γ Fγ which sends X ∈ M to the category ∐γ Fγ(X). Here, ∐γ Cγ

2010 Mathematics Subject Classification. 18D10, 18D15, 18F10, 18G55. Key words and phrases. universal tensor category, multi representability, flat functor, homotopy category, abelian envelope. 1 2 KEVIN COULEMBIER for a family of categories {Cγ} stands for the obvious category with class of objects given by ⊔γObCγ. A 2-functor F is multi-representable if there exists a family {Fγ, γ ∈ Γ} of representable 2-functors with F ≃ ∐γ Fγ. Clearly, such a set Γ = Γ(F ), when it exists, is uniquely defined. In particular, F is then representable if and only if SΓ(F )S = 1. The representing objects Xγ ∈ M of Fγ are the locally representing objects of F . As an intermediary result, we will demonstrate the following example of the above situation. Let M be the 2-category of AB5 abelian categories and exact faithful cocontinuous functors. For any small additive category a, we have a 2-functor Fa ∶ M → Cat which sends an AB5 category C to the category of additive functors a → C (we ignore that this lifts to a 2-functor to the 2-category of preadditive or even abelian categories). We prove that this 2-functor is multi representable; note that it is not representable unless a is the zero category. The locally representing objects are Grothendieck categories, and a subset of them comprise the sheaf categories on a for all additive Grothendieck topologies on a. Hence, the set Γ(Fa) is an extension of the set of topologies on a. We call this generalised notion homological Grothendieck topologies, and they can be interpreted in several ways as ordinary Grothendieck topologies, for instance on Kba or on the category obtained from a by freely adjoining kernels. An inclusion U ∶ M → N of a sub-2-category is multi reflexive if N (A, U−) ∶ M → Cat is multi representable for every A ∈ N . Following [Di2], we can then define SpecU A as the set Γ(N (A, U−)) labelling the locally representing objects of N (A, U−). Main result. We return to the original set up. Let RMon be the 2-category of rigid monoidal categories over k and faithful linear monoidal functors. Denote by U the 2-fully faithful forgetful 2-functor Tens → RMon. For θ ∈ RMon(a,UT), we define its homological kernel (we will see various alternative guises of this kernel, in particular it contains the b information of a homological topology) as the kernel of K a(−, 1) → T(−, 1), where the b 0 homological functor K a → T is the one induced from θ by taking H of the action of θ on chain complexes. It is clear that the 2-functor RMon(a,U−) decomposes into subfunctors labelled by these homological kernels, and our main result states that this decomposition corresponds to an instance of multi representability.

Theorem A. Consider a ∈ RMon. (1) The 2-functor RMon(a,U−) ∶ Tens → Cat is multi representable, that is (for some set Γ) RMon(a,U−) ≃ O Tens(Tγ, −). γ∈Γ b (2) The set Γ ∶= SpecU a can be identified with the set of subfunctors of K a(−, 1) which appear as homological kernels, and there exist no inclusions between the subfunctors in Γ. The functor θ ∈ RMon(a,UT) factors via Tγ if its homological kernel is γ. (3) The category a admits a weak abelian envelope if and only if precisely one subfunctor b of K a(−, 1) appears as a homological kernel (meaning SpecU (a) is a singleton). (4) Tens is a multi reflexive sub-2-category of RMon. Parts (3) and (4) are tautological consequences of (1) and (2). The latter results are proved in Theorem 5.2.4. Organisation of the paper. In Section 1 we recall some preliminary notions. The remainder of the paper is divided into Parts 1 and 2. Part 1 deals with the general theory of homological topologies and Part 2 applies that theory to our case of interest; tensor categories. Part 1: In Section 2, we study prexact functors, which are functors for which the left Kan extension to the presheaf category is exact. In particular we use them to show that HOMOLOGICAL KERNELS 3 the set of Grothendieck topologies on a given category appears as a labelling set for a multi- representable 2-functor. In Section 3 we introduce homological topologies to extend this result and in Section 4 we discuss the generalisation of the previous result to monoidal functors. Part 2: In Section 5 we prove Theorem A above. In Section 6 we explore examples. In particular we demonstrate that in certain cases SpecU (a) is a singleton, giving new sources of examples of abelian envelopes and that in other cases SpecU (a) can be an infinite set. We also realise the abelian categories of motives recently constructed in [Sc] as instances of our theory. Finally, in Appendix A we discuss some variations of the universal monoidal categories from [De2] which are used in Section 6.

1. Preliminaries 1.1. Abelian categories.

1.1.1. We follow the conventions from [Gr] and use conditions AB1-AB5 from [Gr, §1]. An abelian category is an additive category satisfying AB1 and AB2. An AB3 category is an abelian category which admits all (small) coproducts, i.e. a cocomplete abelian category. An AB5 category is an AB3 category in which direct limits of short exact sequences are exact. A Grothendieck category is an AB5 category which admits a generator in the sense of [Gr, §1.9].

1.1.2. A full additive subcategory B of an abelian category A is an abelian subcategory if B contains all kernels and cokernels in A of morphisms between objects in B. An AB3 subcategory of an AB3 category is an abelian subcategory closed under coproducts (and hence all small colimits).

Deﬁnition 1.1.3. An additive functor φ ∶ a → A from a small additive category a to an AB3 category A is AB3-tight if for every AB3 subcategory B ⊂ A which contains the image of φ, we have B = A. The following is an obvious consequence of the deﬁnition, but it will be useful to have it spelled out. Lemma 1.1.4. For a small additive category a and AB3 categories A and B, assume we have a commutative diagram of functors, a

φ ~ F B / A, where φ is AB3-tight and F is fully faithful, exact and cocontinuous. Then F is an equivalence. 1.2. Additive Grothendieck topologies. We refer to [BQ] for the theory of Grothendieck topologies on enriched categories. We are interested only in additive Grothendieck topologies on preadditive categories. For this case the required notions from [BQ] are written out in detail in [Co3, §1.4]. We fix the following notation. For a small preadditive category a, we denote the set of Grothendieck topologies on a by Top(a). For T ∈ Top(a), we denote the (Grothendieck) category of T -sheaves on a by Sh(a, T ) and the composition of the Yoneda embedding Y with the (exact) sheafification functor S is denoted as Y S Z ∶ a Ð→ PSha Ð→ Sh(a, T ). 4 KEVIN COULEMBIER

1.3. Tensor categories. Fix a field k. We say that an essentially small k-linear additive monoidal category (a, ⊗, 1) with k-linear tensor product is a rigid monoidal category over k if (1) The morphism k → End(1) is an isomorphism. ∗ ∗ (2) Every object X ∈ a has a left and right dual X and X. ∗ ∗ Recall that a left dual of X is actually a triple (X , evX , coX ) with morphisms evX ∶ X ⊗X → ∗ 1 and coX ∶ 1 → X ⊗ X satisfying the snake relations. If moreover, (3) a is abelian, we say that (a, ⊗, 1) is a tensor category over k. It then follows that 1 is simple. A standard example is the tensor category Veck of finite dimensional vector spaces. A tensor category over k is artinian if all objects have finite length (it then follows that all morphism spaces are finite dimensional too). We will refer to exact k-linear monoidal functors between tensor categories (over k or field extensions K~k) as tensor functors. By [CEOP, Theorem 2.4.1] or [Co3, Theorem 4.4.1], we can replace ‘exact’ with ‘faithful’ in this definition. We consider the 2-category Tens = Tensk of tensor categories over k, tensor functors and ↑ ↑ monoidal natural transformations. By Tens = Tensk we mean the 2-category of tensor categories over all field extensions K~k, tensor functors and monoidal natural transformations. ↑ ↑ We also consider the 2-categories RMon = RMonk (resp. RMon = RMonk) of rigid monoidal categories over k (resp. over field extensions K~k), k-linear faithful monoidal functors and monoidal natural transformations. By reformulating the definitions from [EHS, CEH, CEOP] we have the following terminology.

Definition 1.3.1. Let a be a rigid monoidal category over k. (1) The category a admits a weak abelian envelope if the 2-functor RMon(a, −) ∶ Tens → Cat is representable. The weak abelian envelope is then given by the faithful tensor functor a → T for the representing tensor category T which induces the equivalence RMon(a, −) ≃ Tens(T, −). (2) A weak abelian envelope as in (1) is an abelian envelope if a → T is full. 1.4. Notation and conventions. For an additive category A, we denote by AZ the category >i∈Z A and write X⟨i⟩ for the object X ∈ A interpreted in to be in the i-th copy of A in AZ. In Part 1, for two pradditive categories a, b, we denote by [a, b] the category of additive op functors a → b. For instance PSha = [a , Ab]. We will use subscripts to specify subcategories. For instance, if a is finitely complete, [a, b]lex is the category of left exact functors and if a is triangulated and b abelian, [a, b]hom is the category of homological functors. In Part 2 the same notation will be used for k-linear functors. For a ring R, we denote by R-free the category of free (left) R-modules of finite rank. By R-Mod we denote the category of all R-modules. We denote by Ab the category of abelian groups and by Abf the category of finitely generated abelian groups. For a category C of functors between two (symmetric) monoidal categories, C⊗ (Cs⊗) is the category of (symmetric) monoidal functors for which the underlying functors belong to C. We largely ignore set theoretic issues. In Part 1, it would be more precise to work with additive essentially small U-categories and abelian U-categories, and then extend the Gro- thendieck universe U when considering 2-categories. For Part 2 this becomes irrelevant as we only need essentially small 1-categories. HOMOLOGICAL KERNELS 5

Part 1. Homological Grothendieck topologies 2. Prexact functors and Grothendieck topologies Let a and b be essentially small preadditive categories and let C denote an AB3 (cocomplete abelian) category. 2.1. Deﬁnitions.

2.1.1. For any additive functor θ ∶ a → C, we can consider the left Kan extension LanYθ ∶ a → C along the Yoneda embedding Y ∶ a → PSha, which we will typically denote by Θ. For our purposes, we can actually define Θ as the unique, up to isomorphism, cocontinuous functor Θ ∶ PSha → C for which Θ ○ Y is isomorphic to θ, see for instance [Co3, Proposition 2.5.2], or as the right adjoint to C → PSha,M ↦ C(θ−,M). For an additive functor u ∶ a → b, we consider u! = − ⊗a b ∶= LanY(Y ○ u) ∶ PSha → PShb. By the above, u! is the left adjoint of − ○ u ∶ PShb → PSha. Definition 2.1.2. (1) An additive functor θ ∶ a → C is prexact if the cocontinuous functor Θ = LanYθ ∶ PSha → C is (left) exact. (2) An additive functor u ∶ a → b is flat if the cocontinuous functor u! ∶ PSha → PShb is (left) exact, or equivalently if Y ○ u is prexact. By 2.1.1, we can alternatively define prexact functors as the restrictions of exact cocontinuous functors PSha → C along Y. This yields the following examples. Example 2.1.3. (1) The functor Z ∶ a → Sh(a, T ) is prexact for every T ∈ Top(a). In fact, these will be the ‘universal prexact functors’. (2) If θ ∶ a → C is prexact, then so is F ○ θ for every exact cocontinuous functor F between AB3 categories. If F is also faithful then conversely F ○ θ prexact implies that θ must be prexact. (3) If θ ∶ a → C is prexact and u ∶ b → a is flat, then θ ○ u is prexact. 2.1.4. The topology of a prexact functor. Consider a prexact functor θ ∶ a → C. We denote by Tθ the following covering system on a. For X ∈ a and R ⊂ a(−,X), we have R ∈ Tθ(X) if and only if one of the following equivalent conditions is satisfied (1)Θ (R) → θ(X) is an isomorphism; (2) R contains a collection of morphisms Yβ → X such that ⊕βθ(Yβ) → θ(X) is an epimorphism. Then Tθ is a Grothendieck topology on a, as follows either directly from the definition ([Co3, 1.4.2]) or because, using the notation of [Co3, §2.2], we have Tθ = top(S(Θ)). Example 2.1.5. (1) For a given Grothendieck topology T on a and Z ∶ a → Sh(a, T ), we have TZ = T , by [BQ] see [Co3, Theorem 1.4.5]. (2) For a prexact functor θ ∶ a → C and a faithful exact cocontinuous functor F ∶ C → B to a second AB3 category B, it follows that Tθ = TF ○θ.

For sets {Xα S α ∈ A} and {Yβ S β ∈ B} of objects in a we call elements of ⎛ ⎞ PSha ? Y(Yβ), ? Y(Xα) ≃ M ? a(Yβ,Xα) ⎝ β α ⎠ β α ‘formal morphisms’ and write them as ⊔βYβ → ⊔αXα. Their composition is deﬁned canonically.

Lemma 2.1.6. For an additive functor θ ∶ a → C the following are equivalent: 6 KEVIN COULEMBIER

(a) θ is prexact. (b) For each formal morphism f ∶ ⊔βYβ → ⊔αXα in a, the sequence

? θ(Z) → ? θ(Yβ) → ? θ(Xα) g∶Z→⊔β Yβ ,f○g=0 β α is exact (acyclic) in C. Proof. Assume ﬁrst that θ is prexact. The sequence

? Y(Z) → ? Y(Yβ) → ? Y(Xα) g∶Z→⊔β Yβ ,f○g=0 β α is tautologically exact in PSha. The fact that Θ is exact and cocontinuous then implies that the sequence in (b) is exact. Hence (a) implies (b). If (b) is satisﬁed, it follows that for every N ∈ PSha we can construct an exact sequence

? Y(Zγ) → ? Y(Yβ) → ? Y(Xα) → N → 0 γ β α in PSha which is sent to an exact sequence in C by Θ. Indeed, we can start from a projective presentation of N and add an additional term by applying (b). That Θ is exact (so that (b) implies (a)) is then a standard consequence of this observation. Indeed, for a short exact sequence ′ ′′ 0 → N → N → N → 0 in PSha we can take such a partial projective resolution of N and N ′′, which also induces one for N ′. This allows to enlarge the above exact row to a commutative diagram where every row except the one displayed above is split exact and which has exact columns. Applying Θ then preserves (by assumption) all the exactness, except that, a priori, ′ ′′ 0 → ΘN → ΘN → ΘN → 0 is only right exact. However, it follows from diagram chasing that also that row is exact.

2.2. Multi representability for prexact functors and a partial function. The two results in this section will be completed in Section 3 when we introduce homological Gro- thendieck topologies.

2.2.1. Consider the 2-category AB3 which has as objects AB3 categories, as 1-morphisms faithful exact cocontinuous functors and as 2-morphisms all natural transformations. For an AB3 category C, we denote by [a, C]prex the category of prexact functors a → C. By Example 2.1.3(2), we can interpret this as a 2-functor

[a, −]prex ∶ AB3 → Cat. Unless a is the zero category, this 2-functor is not representable, as Example 2.1.5(2) implies the 2-functor decomposes as

[a, −]prex = O [a, −]prex,T T ∈Topa where [a, −]prex,T is the category of prexact functors φ with Tφ = T .

Remark 2.2.2. The above decomposition of [a, −]prex ignores the additive structure that the categories [a, C]prex have. In fact, for Fi ∈ [a, C]prex,Ti with i ∈ {1, 2}, we have

F1 ⊕ F2 ∈ [a, C]prex,T1∩T2 . HOMOLOGICAL KERNELS 7

Theorem 2.2.3. For each Grothendieck topology T on a, the 2-functor [a, −]prex,T ∶ AB3 → Cat is represented by Sh(a, T ). More concretely, composition with Z ∶ a → Sh(a, T ) yields an equivalence ∼ AB3(Sh(a, T ), C) → [a, C]prex,T . Proof. We let S be the class of all formal sequences ⊔γZγ → ⊔βYβ → X in a for which ⊕Y(Zγ) → ⊕Y(Yβ) → Y(X) is acyclic in PSha and for which ⊕Z(Yβ) → Z(X) is an epimorphism (or equivalently the sieve generated by ⊔βYβ → X is in the topology T ). ′ This is the pretopology S = pre (T ) from [Co3, §2.2.4]. By [Co3, Proposition 2.5.2], composition with Z yields an equivalence ∼ − ○Z ∶ [Sh(a, T ), C]cc → [a, C]S , (1) where the right-hand side is the category of additive functors θ ∶ a → C for which ⊕θ(Zγ) → ⊕θ(Yβ) → θ(X) → 0 is exact for each sequence in S. We will prove that this equivalence (1) restricts to the one in the theorem. By definition, AB3(Sh(a, T ), C) is a full subcategory of [Sh(a, T ), C]cc. We claim that [a, C]prex,T is a (full) subcategory of [a, C]S . Indeed, consider θ ∈ [a, C]prex. Then Θ is exact and cocontinuous, so it follows that ⊕θ(Zγ) → ⊕θ(Yβ) → θ(X) is acyclic. If furthermore θ ∈ [a, C]prex,T it follows that ⊕θ(Yβ) → θ(X) is an epimorphism. Now we prove that equivalence restricts to the relevant subcategories. By applying equivalence (1) to θ ∈ [a, C]prex,T we find an essentially unique cocontinuous functor T ∶ Sh(a, T ) → C with T ○ Z ≃ θ. We have to show that T is exact and faithful. Since we have Z = S ○ Y and θ ≃ Θ ○ Y, we find T ○ S ≃ Θ. Denote by I the inclusion of the full subcategory Sh(a, T ) in PSha, so S ⊣ I. Using S○I ≃ id then implies that T ≃ Θ○I. The right-hand side of the latter isomorphism is left exact, which shows that T is (left) exact. Now assume T (F ) = 0 for F ∈ Sh(a, T ), which is equivalent to Θ(I(F )) = 0. If F is non- zero there exists a non-zero morphism Y(X) → I(F ) for some X ∈ a, and hence an acyclic sequence ? Y(Yβ) → Y(X) → I(F ) β in PSha. Applying the exact continuous functor Θ shows that ⊕βθ(Yβ)) → θ(X) is an epimorphism, which means that ⊕βZ(Yβ) → Z(X) is also an epimorphism since TZ = T = Tθ. It follows in particular that the morphism Z(X) → F obtained by adjunction from Y(X) → I(F ) is zero, a contradiction. We have proved that T is exact and faithful as desired. Now assume that we have an exact faithful and cocontinuous functor F ∶ Sh(a, T ) → C. Then F ○ Z is prexact since Z is prexact, see Example 2.1.3, and faithfulness of F shows that TF ○Z = TZ = T , see Example 2.1.5. The following consequence is elementary, but it will be useful to have it written out.

Corollary 2.2.4. Consider prexact functors ξi ∶ a → Ci to AB3 categories Ci for i ∈ {1, 2}. If Tξ1 = Tξ2 , then we also have ker ξ1 = ker ξ2. 2.2.5. A partial function. Consider an additive functor u ∶ a → b. We define a (inclusion preserving) partial function ut ∶ Top(b) ⇀ Top(a). For a Grothendieck topology T on b, consider Z ∶ b → Sh(b, T ). In case Z ○ u is prexact, we define ut(T ) as TZ○u. In other words, when ut(T ) is defined, then R ⊂ a(−,A) is in ut(T ) if and only if it contains morphisms fβ ∶ Bβ → A for which the sieve on u(A) generated 8 KEVIN COULEMBIER by u(fβ) is in T . By Theorem 2.2.3, in case ut(T ) is defined, there exists a commutative diagram of functors u a / b (2)

Z Z Sh(a, ut(T )) / Sh(b, T ) where the lower arrow is a faithful exact cocontinuous functor. By Example 2.1.3(3), ut is a function (defined everywhere) if and only if u is flat. 2.3. Criteria for flat and prexact functors. In this section we assume that C is an AB5 category and that a and b are additive. Then we can improve on Lemma 2.1.6.

Theorem 2.3.1. For an additive functor θ ∶ a → C the following are equivalent: (a) θ is prexact. (b) For each f ∶ Y → X in a, the sequence ? θ(Z) → θ(Y ) → θ(X) g∶Z→Y,f○g=0 is exact (acyclic) in C. Proof. Based on Lemma 2.1.6, it is sufficient to prove that for an AB5 category C, exactness of all sequences in (b) implies exactness of all sequences in 2.1.6(b). We will prove this in increasing generality. Denote by A and B the sets to which α and β in 2.1.6(b) belong. Since a and θ are additive, when A and B are both finite, exactness of the sequence in 2.1.6(b) follows from (b). Now assume that B is finite but allow A to be infinite. By definition of formal morphisms, we can still interpret f as an actual morphism in a and 2.1.6(b) remains exact. Finally we also allow B to be infinite. Denote by K the kernel of the right morphism of the sequence in 2.1.6(b) and by I ⊂ K the image of the left morphism. For every finite subset E ⊂ B, by the previous case we know that the kernel KE ⊂ K of >β∈E θ(Yβ) → >α θ(Xα) is contained in I. However, we have K = ∪EKE, since C satisfies AB5, and hence I = K. It follows indeed that 2.1.6(b) is exact. Remark 2.3.2. The condition that C be AB5 abelian is necessary for Theorem 2.3.1. Indeed, it suffices to consider the inclusions of small abelian subcategories of AB3 (or even AB4) categories. These always satisfy 2.3.1(b), but need not be prexact. For a concrete op op example, take Abf ↪ Ab . For c an essentially small abelian category, with slight abuse of notation we will call an additive functor a → c prexact if the composite a → c → Indc is prexact. Corollary 2.3.3. Let c be an essentially small abelian category. An additive functor θ ∶ a → c is prexact if and only if for each f ∶ Y → X in a, there exists g ∶ Z → Y with f ○ g = 0 such that the sequence θ(g) θ(f) θ(Z) ÐÐ→ θ(Y ) ÐÐ→ θ(X) is exact (acyclic) in c. Proof. If there exists Z as in the corollary, then clearly the sequence in 2.3.1(b) is exact, so prexactness of θ follows from Theorem 2.3.1. On the other hand if θ is prexact then the sequence in 2.3.1(b) is exact. Moreover, since c is an abelian subcategory of Indc the kernel K of θ(f) is compact. Hence we can restrict to a finite coproduct in >g∶Z→Y,f○g=0 θ(Z) ↠ K while retaining an epimorphism. By additivity of a we arrive at a single Z → Y . Proposition 2.3.4. If a has weak kernels, then the following are equivalent conditions on an additive functor θ ∶ a → C: HOMOLOGICAL KERNELS 9

(a) θ is prexact. (b) For every morphism f ∶ Y → X, there is a weak kernel K → Y for which θ(f) θ(K) → θ(Y ) ÐÐ→ θ(X) is acyclic. (c) For every morphism f ∶ Y → X, the sequence θ(f) θ(K) → θ(Y ) ÐÐ→ θ(X) is acyclic for every weak kernel K → Y of f. Proof. Clearly (c) implies (b). That (b) implies (a) follows from Theorem 2.3.1. That (a) implies (c) follows again from Theorem 2.3.1 and the universality of a weak kernel. Corollary 2.3.5. (1) Assume that a is finitely complete (a has kernels) and θ ∶ a → C is an additive functor. Then θ is prexact if and only if it is left exact (i.e. θ preserves finite limits). (2) Let t be an essentially small triangulated category and θ ∶ t → C an additive functor. Then θ is prexact if and only if it is homological. Proof. By definition, prexact functors are always left exact. Part (1) therefore follows easily from Proposition 2.3.4. Now we prove part (2). By the rotation axiom of triangles, θ is homological if and only if θ(g) θ(f) θ(Z) ÐÐ→ θ(Y ) ÐÐ→ θ(X) is acyclic for every distinguished triangle g f Z Ð→ Y Ð→ X → Z[1]. Since g is a weak kernel of f, and since every morphism can be completed to a distinguished triangle, 2.3.4 (b)⇒(a) shows that homological functors are prexact. Similarly, 2.3.4 (a)⇒(c) shows that prexact functors are homological. Remark 2.3.6. Corollary 2.3.5(2) can alternatively be derived from Freyd’s universal homological functor, see [Fr2, Ve]. We conclude this section by applying the above results to obtain criteria for flatness. Corollary 2.3.7. The following conditions are equivalent on an additive functor u ∶ a → b. (a) u is flat. (b) For every pair of a morphism f ∶ Y → X in a and a morphism b ∶ B → u(Y ) in b with u(f) ○ b = 0, there exist morphisms g ∶ Z → Y with f ○ g = 0 and h ∶ B → u(Z) such that b = u(g) ○ h. (c) For every object B ∈ b, the comma category B~u, with objects given by the morphisms {B → u(A)S A ∈ a}, is cofiltered. Remark 2.3.8. By Theorem 2.3.1, a functor θ ∶ a → Ab is prexact if and only if the category of elements of θ is cofiltered. The latter category is equal to the comma category Z~θ. By Corollary 2.3.7, this demonstrates the (easily directly verifiable) fact that a flat functor to Abf is prexact; but also that we can construct prexact functors to Abf which are not flat. Corollary 2.3.9. Assume that a has weak kernels. The following conditions are equivalent on an additive functor u ∶ a → b: (a) u is flat. (b) For every morphism f ∶ Y → X, there is a weak kernel K → Y for which u(K) is a weak kernel of u(f). (c) u sends all weak kernels to weak kernels. 10 KEVIN COULEMBIER

3. The kernel category and homological topologies Let a be an essentially small additive category. 3.1. Some categories with weak kernels. 3.1.1. We will work extensively with the following categories and functors: b b a / K+a / K a

# Noya. Here Kba is the bounded homotopy category of a (the quotient of the category of bounded b complexes by the chain homotopy relation) and K+a is its full subcategory of complexes i b (X , d) for which X = 0 whenever i < 0. The functor a → K+a is the obvious embedding of a as the full subcategory of complexes contained in degree 0. We use the same notation for an object X ∈ a interpreted in a or as a chain complex contained in degree 0. We take the b convention that the shift functor [1] on K a acts as i i+1 (X [1]) = X , in particular X[i] for X ∈ a is a chain complex contained in degree −i. Definition 3.1.2. The category Noya has as objects morphisms in a. For morphisms f ∶ 0 1 0 1 X → X and g ∶ Y → Y in a, the set Noya(f, g) consists of the equivalence classes of 0 1 morphisms α ∶ X → X for which g ○ α factors via f (typically denoted as in equation (3) ′ ′ below), where α and α are equivalent if α − α factors via f. Denote by N ∶ a → Noya the fully faithful functor which sends A ∈ a to A → 0. Composition in Noya is defined in the obvious way and Noya is additive. We can also define the group Noya(f, g) via the following diagram:

0 0 φ1=(g○−) 0 1 a(X ,Y ) / a(X ,Y ) O O φ2=(−○f) φ3=(−○f)

1 0 1 1 −1 a(X ,Y ) a(X ,Y ) Noya(f, g) = φ1 (imφ3)~imφ2. For example, for morphisms f, g and an object A in a, we have Noya(f, NA) = coker a(f, A) and Noya(NA, g) = ker a(A, g). b 0 The functor K+a → Noya in the diagram in 3.1.1 is obtained by sending (X , d) to d ∶ 0 1 X → X and a class of chain maps with representative a ∶ X → Y to the class of morphisms 0 0 0 in Noya corresponding to a ∶ X → Y . The following proposition summarises how Noya corresponds to several categories in the literature, for instance in [Be, Fr1]. The equivalence between the categories in (2) and (3) is given in [Be, Corollary 3.9(i)]. Proposition 3.1.3. The category Noya is equivalent to each of the following three categories. (1) The quotient of the subcategory of Kba of complexes in degree 0, 1 with respect to the full subcategory of complexes in degree 1. (2) The quotient of the arrow category Arra with respect to its full subcategory of retracts i ∶ A → B in a. op (3) The category (fpa) , where fpa stands for the category of ﬁnitely presented functors in the module category [a, Ab]. HOMOLOGICAL KERNELS 11

Example 3.1.4. If a is the additive closure of the one object category of a finite dimensional algebra A over a field k, then Noya is the category of finite dimensional right A-modules. As observed in [Be, Fr1] the category Noya has kernels, as it is obtained from a by freely adjoining kernels. b b Lemma 3.1.5. (1) The categories K+a and K a have weak kernels. b b (2) The inclusion K+a → K a is flat. b (3) The category Noya has kernels and K+a → Noya is flat. b Proof. Since K a is triangulated it has weak kernels. Concretely, a weak kernel of f ∶ b b X → Y is given by Cone(f)[−1] → X . Moreover, if X , Y ∈ K+a, then Cone(f)[−1] ∈ K+a. b Consequently, also K+a has weak kernels. This proves part (1) and, by Corollary 2.3.9, also part (2). It follows easily that for a morphism in Noya represented by

f X0 / X1 (3)

α g Y 0 / Y 1 0 1 0 b the kernel is given by (f, α) ∶ X → X ⊕ Y . That K+a → Noya is ﬂat follows from Corollary 2.3.9. Remark 3.1.6. It is clear that N ∶ a → Noya does not preserve the kernels that might already exist in a. In particular, when a admits kernels it does not follow that a ≃ Noya. However, as proved in [Be, Lemma 3.3], N does preserve all cokernels that exist in a.

3.1.7. Let C be an abelian category. For an additive functor θ ∶ a → C, we use the same b b notation for the associated functor K a → K C and we will introduce the following functors θ∆ b b Z Z a / K+a / K a / C ∆ θ0

" " Noya / C. θ⃗

We let θ⃗ ∶ Noy(a) → C be the functor f ↦ ker θ(f), ∆ b for an arbitrary morphism f in a interpreted as an object in Noya. Similarly, θ0 ∶ K a → C is the functor 0 0 −1 (X , d) ↦ H (θX ) = ker θ(d )~imθ(d ). The functor θ∆ Kba CZ is deﬁned as Z ∶ → i (X , d) ↦ ? H (θX )⟨i⟩. i∈Z + b Finally, we will write θ0 for the functor K+a → C which can be obtained by composition in two equivalent ways in the above diagram, meaning the functor 0 (X , d) ↦ ker θ(d ). These functors have a number of universal properties. Part (1) below is essentially [Be, Corollary 3.2]. Theorem 3.1.8. Fix an abelian category C. 12 KEVIN COULEMBIER

(1) Composition with N ∶ a → Noya yields an equivalence ∼ [Noya, C]lex → [a, C] with inverse given by θ ↦ θ⃗. b + (2) Composition with a → K+a yields an equivalence (with inverse θ ↦ θ0 ) b ∼ [K+a, C]wker,0 → [a, C] b b where [K+a, C]wker,0 stands for the category of functors φ ∶ K+a → C which send weak kernels to acyclic sequences and satisfy one of the following equivalent additional conditions: (a) φ(X[i]) = 0 for all X ∈ a and i ∈ Z<0; b (b) φ(X [−1]) = 0 for all X ∈ K+a. b ∆ (3) Composition with a → K a yields an equivalence (with inverse θ ↦ θ0 ) b ∼ [K a, C]hom,0 → [a, C], b b where [K a, C]hom,0 stands for the category of homological functors φ ∶ K a → C which satisfy one of the following equivalent additional conditions: (a) φ(X[i]) = 0 for all X ∈ a and i ∈ Z=~0; b 0 (b) φ(X ) = 0 for all X ∈ K a with X = 0. If C is an AB5 category, the respective homological conditions in the left-hand sides of each equivalence can be replaced by the condition of being prexact. Proof. The interpretation in terms of prexact functors in case C is AB5 follows from Propo- sition 2.3.4 and Corollary 2.3.5. We start by proving part (1). It is readily veriﬁed that θ⃗ is left exact, for θ ∶ a → C. It is then also clear that θ ↦ θ⃗ yields a left inverse to φ ↦ φ ○ N. It thus suﬃces to show that ÐÐ→ ÐÐ→ for every φ ∈ [Noya, C]lex, we have φ ≃ φ ○ N functorially. A suitable isomorphism φ ⇒ φ ○ N follows from the sequence

0 f 1 0 1 0 → φ(X Ð→ X ) → φ(NX ) → φ(NX ), which is exact by left exactness of φ. Part (2) can be proved by a simpliﬁed version of the argument for part (3) below. Finally we prove part (3). That for homological functors condition (a) implies (b) follows by induction on the length of a complex. Composition in one direction of the proposed inverses is obviously isomorphic to the identity. We claim the composition in the other b direction is also the identity. Take therefore φ ∈ [K a, C]hom,0 and consider the following b commutative diagram, for any X ∈ K a: 0 0 O

≤0 >0 0 / φ(X ) / φ(τ X ) / φ((τ X )[1]) O

−1 0 1 φ(X ) / φ(X ) / φ(X ) O

<0 >1 φ((τ X )[−1]) φ((τ X )[2]). ≤i We used notation as τ X to denote the naive truncations of the complex X . The ﬁrst row >0 ≤0 >0 is exact, by the distinguished triangle τ X → X → τ X and the fact that φ(τ X ) = 0 HOMOLOGICAL KERNELS 13 by assumption (b). The two columns are similarly exact. By exactness and commutativity, 0 1 ≤0 it follows that the morphism from the kernel of φ(X ) → φ(X ) to φ(τ X ) factors via −1 0 φ(X ). This morphism restricts to zero on the image of φ(X ) → φ(X ) since the morphism −1 ≤0 b ∆ X → τ X is nullhomotopic (so zero in K a). This yields a morphism (φSa)0 (X ) → φ(X ). b Now consider a morphism X → Y in K a. We choose some lift in the category of complexes (since τ ≤0 is not a functor on Kba) which allows us to construct a commutative diagram combining the above diagram with the one for Y. This demonstrates that the morphism ∆ (φSa)0 (X ) → φ(X ) is natural in X . That this natural transformation is an isomorphism can be proved again by induction on the length of complexes and the 5-lemma.

3.2. Multi representability. In this section we define homological topologies on a simply as Grothendieck topologies on Noya. The justification for introducing a new term stems from the variety of alternative realisations of this set in Theorem 3.3.2 below. Definition 3.2.1. (1) The set of homological topologies on a is HTop(a) ∶= Top(Noya). (2) For a functor θ ∶ a → C to an AB5 category C, its homological topology HTθ ∈ HTop(a) is given by Tθ⃗ ∈ Top(Noya). (3) For a homological topology R ∈ HTop(a), we write HZ = Z ○ N ∶ a → Sh(Noya, R) =∶ HSh(a, R). Now we extend Theorem 2.2.3 from prexact functors to all additive functors. Denote by AB5 the 2-full 2-subcategory of AB3, from 2.2.1, consisting of AB5 categories.

Theorem 3.2.2. (1) The 2-functor [a, −] ∶ AB5 → Cat decomposes as

[a, −] = O [a, −]R, R∈HTop(a)

where [a, C]R is the category of functors θ ∶ a → C with HTθ = R. (2) For each R ∈ HTop(a), the 2-functor

[a, −]R ∶ AB5 → Cat

is represented by HSh(a, R). More concretely, for each C ∈ AB5 composition with HZ yields an equivalence ∼ AB5(HSh(a, R), C) → [a, C]R. (3) For every R ∈ HTop(a), the functor HZ ∶ a → HSh(a, R) is AB3-tight. ÐÐÐ→ Proof. For F ∈ AB5(C, B) and φ as above, we have F ○ φ⃗ = F ○ φ, and hence HTφ = HTF ○φ by Example 2.1.5(2). This proves part (1). By deﬁnition and Theorem 3.1.8(1), composition with N yields an equivalence ∼ [Noya, C]prex,R Ð→ [a, C]R. The proof of (2) is then concluded by applying Theorem 2.2.3 to Noya. For part (3), consider an AB3 subcategory A ⊂ Sh(Noya, R) which contains the image of HZ. Since A is is closed under kernels and since HZ⃗ ∶ Noya → Sh(Noya, R) is isomorphic to Z by Theorem 3.1.8(1), it follows that A contains the image of Z. Since every object in Sh(Noya, R) is a cokernel of a morphism between direct sums of objects in the image of Z, and A is closed under such operations, it follows indeed that A = Sh(Noya, R). 14 KEVIN COULEMBIER

To state the following theorem, we deﬁne a special type of sieves in Noya. For every 0 1 morphism f ∶ X → X in a viewed as an object in Noya, we let Rf ⊂ Noya(−, f) be the sieve generated by all morphisms

Y / 0 a f X0 / X1 in Noya to f from objects in the image of N ∶ a → Noya. Recall that by deﬁnition such morphisms satisfy f ○ a = 0.

Theorem 3.2.3. There exists an injection ι ∶ Top(a) ↪ HTop(a) with the following properties. (1) For each T ∈ Top(a), there is an equivalence

Sh(a, T ) ≃ HSh(a, ι(T )) which yields a commutative triangle with the functors Z and HZ out of a. (2) For a prexact functor θ ∶ a → C, we have ι(Tθ) = HTθ. (3) The image of ι consists of the Grothendieck topologies R on Noya for which Rf ∈ 0 1 R(f), for each object f ∶ X → X in Noya. (4) The partial function ut ∶ Top(b) ⇀ Top(a) from 2.2.5 induced by a functor u ∶ a → b is obtained from restricting source and target along ι of the function

HTop(b) → HTop(a), R ↦ HTHZ○u for HZ ∶ b → HSh(b, R).

Proof. We deﬁne a function ι ∶ Top(a) → HTop(a) which sends T to HTZ for Z ∶ a → Sh(a, T ). By Theorem 3.2.2(2), we ﬁnd a commutative diagram

HSh(a, ι(T )) 6 HZ

Z a / Sh(a, T ), where the vertical arrow is exact, faithful and cocontinuous. It follows that HZ is prexact with THZ = T . By Theorem 2.2.3, there is also an upwards exact faithful cocontinuous functor yielding a commutative diagram. The universalities in Theorems 2.2.3 and 3.2.2 show that the two functors are mutually inverse, proving (1). Part (2) follows from the same universality. By the above, to prove (3), it suﬃces to show that HZ ∶ a → HSh(a, R) is prexact if and only if R ∈ HTop(a) satisﬁes the condition in (3). By Theorem 3.1.8(1), HZ⃗ ≃ ZN , where we write ZN ∶ Noya → Sh(Noya, R) in order to avoid confusion with the Z in the previous paragraph. Application of Theorem 2.3.1 therefore shows that HZ = ZN ○ N is prexact if and only if for each object f ∶ Y → X in Noya, the morphism

? ZN (N(Z)) → ZN (f) g∶Z→Y,f○g=0 is an epimorphism, which is the same as saying Rf ∈ R(f) = TZN . Part (4) follows from deﬁnition and part (1).

Remark 3.2.4. The partial inverse to ι ∶ Top(a) ↪ HTop(a) is given by the partial function Nt ∶ Top(Noya) ⇀ Top(a) from 2.2.5. HOMOLOGICAL KERNELS 15

3.2.5. Extended example. We work out the universal Grothendieck categories predicted by Theorem 3.2.2 in a specific example. Let k be a field, consider the algebra of dual numbers 2 R = k[x]~x and set a = R-free. By Example 3.1.4, we know that Noya is equivalent to the category of finite dimensional R-modules. We have two indecomposable R-modules up to isomorphism: the projective module P ≃ R and the simple module L ≃ k. We can classify Grothendieck topologies on Noya directly. There turn out to be 4 possibilities for a topolgy R on R-mod: (0) The discrete topology R0: 0 ∈ R(X) for all X. (1) The minimal sieve in R1 on P is generated by L ↪ P and the only sieve on L in R1 is the representable functor itself. (2) The minimal sieve in R2 on L is generated by P ↠ L and the only sieve on P in R2 is the representable functor itself. (3) The trivial topology R3: the only sieves in R3 are the representable functors. By Theorem 3.2.3(3), the image of ι is {R0, R2}, so in particular, HSh(a, R2) ≃ R-Mod by Theorem 3.2.3(1). Obviously Sh(Noya, R0) = 0 and this category corresponds to zero functors out of a. A non-zero θ ∶ a → C induces homological topology R1 if and only x if the R Ð→ R is sent to zero. The universal property in Theorem 3.2.2 thus shows that Sh(Noya, R1) is the category of k-vector spaces. So Sh(Noya, R0) and Sh(Noya, R1) classify non-faithful functors and we summarise the results from the theory on faithful functors in the following proposition.

Proposition 3.2.6. Let C be a k-linear AB5 category with an object X ∈ C and non-zero 2 diﬀerential d ∶ X → X (i.e. d = 0). We consider the sequence d d X Ð→ X Ð→ X. (1) If the sequence is acyclic there exists an essentially unique faithful exact cocontinuous functor R-Mod → C with R ↦ X and x ↦ d. (2) If the sequence has non-zero homology there exists an essentially unique faithful exact cocontinuous functor PSh(R-mod) → C with Y(R) ↦ X and Y(x) ↦ d. Proof. The obvious universal property of a shows that we have an essentially unique (faithful) functor θ ∶ a → C with θ(x) = d. The morphism P ↠ L in Noya is sent to an epimorphism by θ⃗ precisely when the sequence in the proposition is exact. Part (2) is therefore an immediate application of Theorem 3.2.2. For part (1) we only need to add HSh(a, R2) ≃ R-Mod. Remark 3.2.7. (1) The universal category in 3.2.6(2) is the category of modules over the path algebra of the quiver a ( e h s b

with relation b○a = 0. In particular, if k = C the category is equivalent to IndO0(sl2), the ind-completion of the principal block in BGG category O of sl2(C). (2) For all i, the functor a → Sh(Noya, Ri) is full and for i > 1 it is faithful. 3.3. Alternative realisations. In this section we establish three alternative realisations of the universal category HSh(a, R). 3.3.1. Consider the following two sets of Grothendieck topologies: b b ● The set Top(K+a)0 of topologies R on K+a with 0 ∈ R(X[i]) for all i ∈ Z<0 and X ∈ a. b b ● The set Top(K a)0 of topologies R on K a with 0 ∈ R(X[i]) for all i ∈ Z=~0 and X ∈ a. 16 KEVIN COULEMBIER

b ˆ b b For R ∈ Top(K a)0, we consider the following covering system R on K a. For each X ∈ K a, ˆ a sieve S on X is in R if and only if S[i] ∈ R(X [i]), for all i ∈ Z. ˆ Since R is the intersection of a number of Grothendieck topologies (a countable number of b ‘shifts’ of the topology R), it is again a topology on K a. Theorem 3.3.2. (1) There exist bijections

b 1∶1 1∶1 b op K a 0 o op a / op K a 0 T ( + ) ρ1 HT ( ) ρ2 T ( )

such that, for each R ∈ HTop(a), we have a commutative diagram b b Noya o K+a / K a

b b Sh(Noya, R) o Sh(K+a, ρ1(R)) / Sh(K a, ρ2(R)) where the bottom line consists of equivalences. b b (2) For R ∈ Top(K a)0, the category Sh(K a, R) is equivalent to the minimal AB3 sub- b ˆ b ˆ category of Sh(K a, R) which contains the image of a → Sh(K a, R). (3) For an additive functor θ ∶ a → C to C ∈ AB5, the homological topology HTθ is determined by the topology of θ∆ Kba CZ and vice versa. Z ∶ → Remark 3.3.3. We will henceforth refer to an element of any of the sets Top(Noya), b b Top(K+a)0 or Top(K a)0 as a ‘homological topology on a’. Furthermore, we will inter- b b pret HSh(a, R) accordingly as a sheaf category on Noya, K+a or K a depending on what form is more convenient. We start the proof with the following general recognition result.

Lemma 3.3.4. Consider a homological Grothendieck topology R on a and φ ∈ [a, A]R for some AB5 category A for which φ ∶ a → A is AB3-tight. If − ○ φ ∶ AB5(A, C) → [a, C]R is dense for all C ∈ AB5, then φ is isomorphic to HZ ∶ a → HSh(a, R). Proof. We claim there exists faithful exact cocontinuous functors F,G for which the diagram a HZ HZ φ y F G % HSh(a, R) / A / HSh(a, R) is commutative. Indeed, G exists by assumption. Existence of F follows from Theo- rem 3.2.2(2). The latter result also implies that G ○ F is isomorphic to the identity functor. From this it follows that F is also full. That F is then an equivalence follows from Lemma 1.1.4. b b Proof of Theorem 3.3.2. The sets Top(K+a)0 and Top(K a)0 comprise precisely the topologies of the prexact functors φ which satisfy conditions (2)(a) and (3)(a) in Theorem 3.1.8. We can therefore repeat the proof of Theorem 3.2.2 to state multirepresentability of [a, −] b b in terms of sheaf categories on K+a or K a with respect to the above topologies. This shows that there must exist bijections between the three sets of topologies such that the bijection results in equivalences between the sheaf categories. By universality, explicitly constructing the bijections can be done as follows. For b, b′ two of the three categories b b Noya,K+a,K a and R a suitable topology on b, it suﬃces to consider the topology of the ′ functor b → Sh(b, R) obtained via Theorem 3.1.8 from a → b → Sh(b, R). The slightly HOMOLOGICAL KERNELS 17 stronger commutativity claim in part (1) follows from the principle in 2.2.5 applied to the u b v b ﬂat functors Noya ←Ð K+a Ð→ K a of Lemma 3.1.5 to get commutative diagrams b b b Noya o K+a K+a / K a

b b ′ b ′ Sh(Noya, R) o Sh(K+a, ut(R)) Sh(K+a, vt(R )) / Sh(K a, R ) ′ b for arbitrary R ∈ Top(Noya) and R ∈ Top(K a)0. Now we prove part (2). First we observe that, for θ ∶ a → C, if R is the topology ∆ ˆ corresponding to a prexact functor θ0 , then R is the topology corresponding to the prexact functor θ∆ Kba CZ. Let A denote the minimal AB3 subcategory of Sh Kba, ˆ which Z ∶ → ( R) b ˆ contains the image of a → Sh(K a, R). For every C ∈ AB5, we have functors θ↦θ∆ Z b Z −○Z b ˆ Z Z [a, C]R / [K a, C ]prex,Rˆ o ∼ AB5(Sh(K a, R), C ) / AB5(A, C ) , where the equivalence is from Theorem 2.2.3 and the right arrow is composition with the inclusion of A. Z Now take θ ∈ [a, C]R and consider the resulting functor F (θ) ∈ AB5(A, C ). The compos- θ ite of F (θ) and a → A is isomorphic to a Ð→ C ↪ CZ. Tightness of a → A then shows that also Z A → C actually takes values in C. The composite of the corresponding F0(θ) ∈ AB5(A, C) and and a → A is isomorphic to θ. In particular, a → A induces homological topology R. Now we can apply Lemma 3.3.4, since the functor AB5(A, C) → [a, C]R will send F0(θ) to θ, and therefore be dense. ∆ For part (3), by (1), we can replace HTθ with the topology R of the prexact functor θ0 . As discussed above, the topology of θ∆ is then ˆ (which is determined by ). The other Z R R direction is a consequence of part (2), but can also be demonstrated directly. 3.4. Homological kernel. We develop some notions to describe the kernel of θ⃗ eﬃciently. Fix A ∈ a. a Deﬁnition 3.4.1. The canonical kernel at A is the following sieve ΣA on NA ∈ Noya. For a f ∈ Noya,ΣA(f) consists of those morphisms α ∶ f → NA which compose to zero with all morphisms NY → f for all Y ∈ a, so

a ⎛ ⎞ ΣA(f) = ker Noya(f, NA) → M a(Y,A) . ⎝ g∶NY →f ⎠ 0 1 a For a morphism f ∶ X → X in a, we can also write the abelian group ΣA(f) without reference to Noya as

a ⎛ 1 h↦h○f 0 m→(m○g)g ⎞ ΣA(f) = H0 a(X ,A) ÐÐÐÐ→ a(X ,A) ÐÐÐÐÐÐ→ M a(Y,A) . ⎝ g∶Y →X0,f○g=0 ⎠ Remark 3.4.2. For a ﬁxed f ∈ Noya, we have the following extremal cases. (1) The collection of morphisms {NY → f S Y ∈ a} is jointly epimorphic in Noya if and a only if ΣA(f) = 0 for all A ∈ a. a (2) As a morphism in a, f is monic if and only if ΣA(f) = Noya(f, NA) for all A ∈ a. Example 3.4.3. For a ring R we set a = R-free. For r ∈ R interpreted as the morphism ⋅r R Ð→ R, we ﬁnd a ΣR(r) = {u ∈ R S tu = 0 for all t ∈ R with tr = 0}~rR. More concretely: 18 KEVIN COULEMBIER

a (1) If R is a domain and r =~ 0, then ΣR(r) = R~rR. 2 a (2) If R = k[x]~(x ) for a field k, then ΣR(r) = 0 for all r ∈ R. a Example 3.4.4. If a is an abelian category and I ∈ a injective, then ΣI = 0. Now we fix an additive functor θ ∶ a → C to an abelian category C. a Definition 3.4.5. The homological kernel at A of θ is the following sieve ΣAθ on a NA ∈ Noya. For f ∈ Noya, the subgroup ΣAθ(f) ⊂ Noya(f, NA) consists of the morphisms in the kernel of θ⃗, so a ΣAθ(f) = ker Noya(f, NA) → C(θf,⃗ θA) 1 0 = H0 a(X ,A) → a(X ,A) → C(ker θ(f), θA) . a The following proposition demonstrates why ΣA is the ‘canonical kernel’, that this canon- a ical kernel is achieved by faithful prexact functors and that {ΣAθ S A ∈ a} contains enough information to describe ker θ⃗ (whence the name homological kernel). a a Proposition 3.4.6. (1) There is an inclusion ΣAθ ⊂ ΣA of sieves on NA ∈ Noya if and only if θ ∶ a(−,A) → C(−, θA) is injective. a a (2) If C is an AB5 category and θ is prexact, then there is an inclusion ΣA ⊂ ΣAθ of sieves on NA ∈ Noya. (3) If C is an AB5 category and θ is faithful and prexact, then we have an equality a a ΣA = ΣAθ of sieves on NA ∈ Noya. a a (4) We have ΣA = ΣAY, for the Yoneda embedding Y ∶ a → PSha. 0 0 (5) A morphism f → g in Noya, represented by α ∶ X → Y as in equation (3), is in the 0 a kernel of θ⃗ if and only if the class of α in Noya(f, NY ) is in ΣY 0 θ(f). 0 0 Proof. Fix an object f ∶ X → Y in Noya and consider the commutative diagram

∏g∶Y →X0,f○g=0 a(Y,A) 6

1 0 * a(X ,A) / a(X ,A) ∏g∶Y →X0,f○g=0 C(θY, θA). 4

( C(ker θ(f), θA) The morphisms in the left-hand side are as above. The rightmost downwards arrow is given by θ. The rightmost upwards arrow is constructed using the universality of the kernel. Concretely, for each g, it follows that θ(g) factors via a unique morphism θY → ker θ(f), and the arrow is composition with said morphism in each factor. If C is an AB5 category and θ is prexact, then by Theorem 2.3.1, the latter arrow is injective. This proves part (2). Conversely, if θ is faithful on a(−,A), then the rightmost a a downwards arrow is injective, showing that ΣAθ ⊂ ΣA. Now we prove the other direction of part (1). If there exists some h ∶ B → A in a with ′ a ′ a ′ θ(h) = 0, then for f ∶ B → 0 in Noya, it follows that h ∈ ΣAθ(f ) but h ∈~ ΣA(f ). Part (3) is a consequence of parts (1) and (2). Part (4) follows immediately from the definitions. Alternatively it follows form part (3), as Y is faithful and by definition prexact. Conversely we can also use (4) to prove (2) immediately from the definitions. Since ker θ(f) → ker θ(g) is zero if and only if ker θ(f) → 0 θY is zero, part (5) follows. Remark 3.4.7. (1) The converse to Proposition 3.4.6(3) is not true. The property a a ΣAθ = ΣA for all A ∈ a for a functor θ ∶ a → C to an AB5 category C is not sufficient to conclude that θ is prexact, see Example 3.4.3(2) and 3.2.5. HOMOLOGICAL KERNELS 19

a (2) If F ∶ C → B is a faithful exact functor between abelian categories, then ΣAθ = a ΣA(F ○ θ). The homological kernel of a faithful functor is not always canonical, as the following examples show.

Example 3.4.8. (1) Consider a ring S with subring R. Set a = R-free and C = S-Mod ⋅r and let θ ∶ a → C be the functor S ⊗R −. For r ∈ R, seen as the morphism R Ð→ R, we ﬁnd a ΣRθ(r) = {u ∈ R S zu = 0 for all z ∈ S with zr = 0}~rR, a which is in general a proper subgroup of ΣR(r) from Example 3.4.3. 0 1 a a (2) If f ∶ X → X is monic in a and θ ∶ a → C faithful, then ΣAθ(f) = ΣA(f) if and 0 only if for every α ∈ a(X ,A), θ(α) restricts to zero on ker θ(f). Examples where is the latter condition is not satisﬁed will be given in Section 6.3.

4. Comments on monoidal and enriched versions Let (a, ⊗, 1) be a preadditive monoidal category and (C, ⊗, 1) an abelian monoidal category. We assume that both have additive tensor product − ⊗ −. 4.1. Multi representability.

4.1.1. Assume that C is an AB3 category for which the tensor product is cocontinuous in each variable. The 2-category of such monoidal categories, with 1-morphisms given by faithful exact cocontinuous monoidal functors and 2-morphisms natural transformations of ⊗ monoidal functors is denoted by AB3 . For a monoidal prexact functor θ ∶ a → C, it follows from [Co3, Theorem 3.5.4(c)] that the Grothendieck topology Tθ on a is ‘monoidal’ as deﬁned loc. cit. Consequently, Sh(a, Tθ) is a biclosed monoidal Grothendieck category (the functors N ⊗ − and − ⊗ N have right adjoints for every object N) and Z ∶ a → Sh(a, Tθ) is monoidal. It follows that the 2-functor ⊗ ⊗ [a, −]prex ∶ AB3 → Cat ⊗ decomposes into the 2-functors [a, −]prex,T , where T runs over all monoidal Grothendieck topologies.

Proposition 4.1.2. For each monoidal Grothendieck topology T on a, the 2-functor ⊗ ⊗ [a, −]prex,T ∶ AB3 → Cat is represented by the biclosed monoidal Grothendieck category Sh(a, T ). More concretely, composition with Z ∶ a → Sh(a, T ) yields an equivalence ⊗ ∼ ⊗ AB3 (Sh(a, T ), C) → [a, C]prex,T . Proof. By [Co3, Theorem 3.5.5], and with notation and results from the proof of Theo- rem 2.2.3, composition with Z yields an equivalence ⊗ ∼ ⊗ [Sh(a, T ), C]cc → [a, C]S . The proof that this restricts to an equivalence between subcategories yielding the proposition is identical to the proof of Theorem 2.2.3, as it only relates to the underlying functors and not their monoidal structure. 4.2. Monoidal structure on the kernel category. Assume a is additive. 20 KEVIN COULEMBIER

4.2.1. The category Kba is monoidal (rigid when a is rigid), with

i i−j j (X ⊗ Y) = ? X ⊗ Y . j∈Z The diﬀerential on the tensor product is given by

b a a a b (d⊗Y ,(−1) X ⊗d) a+1 b a b+1 X ⊗ Y ÐÐÐÐÐÐÐÐÐÐ→ X ⊗ Y ⊕ X ⊗ Y . b b b The functor a → K a is canonically monoidal. The subcategory K+a ⊂ K a is a monoidal subcategory. 0 1 0 1 Also Noya has a monoidal structure. The tensor product of f ∶ X → X and g ∶ Y → Y is given by 0 0 0 0 (f⊗Y ,X ⊗g) 1 0 0 1 X ⊗ Y ÐÐÐÐÐÐÐÐ→ X ⊗ Y ⊕ X ⊗ Y . b The functor K+a → Noya is then also monoidal. If a is braided (or symmetric), then so are all the above monoidal categories and functors (where we use the Koszul sign rule).

4.2.2. Consider a monoidal functor θ ∶ a → C. The left exact functor θ⃗ inherits a lax monoidal structure, coming from the obvious morphisms 0 0 0 1 1 0 ker θ(f) ⊗ ker θ(g) → ker(θX ⊗ θY → θX ⊗ θY ⊕ θX ⊗ θY ). If the tensor product in C is left exact, it follows that the lax monoidal structure on θ⃗ is actually monoidal.

Lemma 4.2.3. (1) Composition with the monoidal functor N ∶ a → Noya yields a fully faithful functor ⊗ ⊗ [Noya, C]lex → [a, C] . (2) If the tensor product in C is left exact, the functor in (1) is an equivalence.

⊗ ÐÐ→ Proof. If φ ∈ [Noya, C]lex, then the isomorphism between φ and φ ○ N from the proof of Theorem 3.1.8(1) is a natural transformation of (lax) monoidal functors. The assignment θ ↦ θ⃗ therefore provides an inverse to the functor in part (1), when restricted to the essential image of the latter functor. For part (2) it suﬃces to observe that the inverse is now deﬁned everywhere.

4.2.4. Now assume that the tensor product in C is right exact. Then for two bounded chain complexes X and Y there are canonical morphisms a b i ? H (X ) ⊗ H (Y) → H (X ⊗ Y). a+b=i These are isomorphisms if the tensor product is exact, by the K¨unneththeorem for chain complexes For a monoidal functor θ ∶ a → C, the homological functors + b ∆ b Z θ0 ∶ K+a → C and θZ ∶ K a → C (4) are therefore canonically lax monoidal, where we view CZ as a monoidal subcategory of KbC. If the tensor product on C is exact, then θ∆ is actually monoidal. Again, also the Z braiding of (lax) monoidal functors is carried over.

4.3. Homological kernels of monoidal functors. Assume a is additive and rigid. HOMOLOGICAL KERNELS 21

a a 4.3.1. The canonical homological kernel of a is the sieve Σ ∶= Σ1 on 1 ∈ Noya from 0 1 Deﬁnition 3.4.1. Concretely, for a morphism f ∶ X → X in a:

a ⎛ 1 0 ⎞ Σ (f) = H0 a(X , 1) → a(X , 1) → M a(Y, 1) . ⎝ g∶Y →X0,f○g=0 ⎠ a a Similarly, for a monoidal functor θ ∶ a → C, the sieve Σ θ = Σ1θ on 1 ∈ Noya given by a 1 0 Σ θ(f) = H0 a(X , 1) → a(X , 1) → C(ker θ(f), 1) , is the homological kernel of θ.

Lemma 4.3.2. For a monoidal functor θ ∶ a → C, a a (1) Σ θ ⊂ Σ if and only if θ is faithful. a a (2) Σ θ ⊃ Σ when θ is prexact (and C is AB3). Proof. Part (1) follows from Proposition 3.4.6(1) and the fact that a is rigid and θ monoidal. Part (2) is a special case of Proposition 3.4.6(2). For two posets (Λ, ≤), (P, ⪯), a function f ∶ Λ → P is an embedding if f(λ) ⪯ f(µ) ⇔ λ ≤ µ for all λ, µ ∈ Λ. Clearly this forces f to be injective. Proposition 4.3.3. There exist inclusion preserving functions

µ ν b {ideals in Noya} Ð→ {sieves on 1 ∈ Noya} ←Ð {ideals in K a} such that, with C the class of monoidal functors θ ∶ a → C to abelian monoidal categories C with exact tensor product: (1) µ restricts to an embedding on the subset of ideals of the form ker θ⃗, for θ ∈ C, (2) ν restricts to an embedding on the subset of ideals of the form ker θ∆, for θ C. Z ∈ (3) We have µ ker θ⃗ Σaθ ν ker θ∆ for all θ C. ( ) = = ( Z ) ∈ Proof. For an ideal I in Noya, the sieve µ(I) is deﬁned as the restriction µ(I) = I(−, 1). For an ideal J in Kba, we let a morphism in Noya represented by

f X0 / X1

1 / 0 b b be in ν(J) if and only if the corresponding morphism in K a (note Noya(f, 1) ≅ K a(X , 1) 0 1 canonically for X the complex 0 → X → X → 0) belongs to J. It follows easily that ν(J) is a sieve. Now we prove (1). Consider an arbitrary morphism α in Noya as in (3). It follows quickly from the deﬁnition of θ⃗ that α is in the kernel of θ⃗ if and only if the corresponding morphism 0 0 1 0 f → NY (the composition of α with (Y → Y ) → NY ) is in the kernel, see also 3.4.6(5). Since NY 0 has a left dual, it follows that α is in the kernel of θ⃗ if and only if the associated morphism Y 0 ∗ f 0 ∗ 0 ( ) ⊗ 0 ∗ 1 (Y ) ⊗ X / (Y ) ⊗ X

1 / 0 is in the kernel of θ⃗, or equivalently in µ(ker θ⃗). So µ(ker θ⃗) retains all information in ker θ⃗. 22 KEVIN COULEMBIER

Now we prove (2). Since Kba is rigid, the kernel of the monoidal functor θ∆ is determined Z by which morphisms , d 1 are sent to zero. Using the definition of θ∆, such a morphism (X ) → Z is sent to zero if and only if the morphism d0 1 in Noya is in ν ker θ∆ . → ( Z ) Part (3) follows from the definitions. 4.4. Enriched versions. In all previous sections we considered Z-linear (preadditive) categories. All results can be straightforwardly adapted to K-linear categories for commutative rings K. Note that for a K-linear category a, PSha is equivalent to the category of K-linear op functors a → K-Mod and K-linear Grothendieck topologies coincide with the additive topologies. ⊗ For example, if we define AB3 to be the 2-category of all monoidal K-linear AB3 categories with K-linear cocontinuous tensor product, let a be K-linear monoidal with K-linear tensor product and reserve the notation [−, −] for categories of K-linear functors, Proposi- tion 4.1.2 remains valid as stated. In the remainder of the paper we will use such enriched analogues of previous results without further comment.

Part 2. Local abelian envelopes For the remainder of the paper, we let k be a ﬁeld. All functors are assumed to be k-linear and correspondingly, from now on, the notation [−, −] will only be used to denote categories of k-linear functors.

5. Universal tensor functors of fixed homological kernel For the entire section, let a be a rigid monoidal category over k. By a tensor category we will understand a tensor category over some field extension K~k (including K = k). Unless further specified, T is assumed to be such a tensor category. 5.1. The tensor category associated to a prexact monoidal functor. Theorem 5.1.1. For a prexact monoidal functor θ ∶ a → T, there exists a tensor category U with IndU ≃ Sh(a, Tθ), over an intermediary field extension k < L < K, with a monoidal ′ functor θ ∶ a → U which yields an equivalence ′ ↑ ∼ ↑ − ○ θ ∶ Tens (U, T1) → RMon (a~ker θ, T1) for each tensor category T1. Here Tθ, so in particular also U, depends only on ker θ. Proof. By Proposition 4.1.2 we have the biclosed Grothendieck category Sh(a, Tθ) with the exact faithful monoidal functor T ∶ Sh(a, Tθ) → IndT such that T ○ Z ≃ θ (where we leave out the inclusion functor T → IndT). It follows from [CEOP, Proposition 3.3.2] that Sh(a, Tθ) ≃ IndU for some tensor category U. The universal property of θ′ then follows immediately from [Co3, Theorem 4.4.1]. This universal property shows that U only depends on ker θ. Alternatively, that Tθ depends only on ker θ is in [Co3, Theorem 4.4.1]. Corollary 5.1.2. For two prexact monoidal functors θ ∶ a → T1 and φ ∶ a → T2 to tensor categories, their kernels are either the same or incomparable. Proof. By Theorem 5.1.1 we can replace θ and φ by functors (θ′ and φ′) which are creators in the terminology of [Co3, 4.1.2]. The claim then follows from [Co3, Corollary 4.4.4]. Corollary 5.1.3. If there exists a faithful and prexact monoidal functor θ ∶ a → T, then every faithful monoidal functor from a to a tensor category is prexact. If furthermore T is a tensor category over k, then a admits a weak abelian envelope. Proof. Firstly we observe that θ′ in Theorem 5.1.1 is prexact. This follows either by construction, or from Remark 2.1.3(2). That all faithful monoidal functors to tensor categories are prexact now follows from the equivalence in Theorem 5.1.1 and again Remark 2.1.3(2). HOMOLOGICAL KERNELS 23

Remark 5.1.4. A direct construction of U (without reference to Tθ) is given in [Co3, §4]. To present one of the consequences, set b ∶= a~ker θ. The category U is a tensor category over k if and only if for each non-zero u ∶ U → 1 in b, the following sequence is exact: −○u −○(u⊗U−U⊗u) 0 → b(1, 1) ÐÐ→ b(U, 1) ÐÐÐÐÐÐÐÐ→ b(U ⊗ U, 1). Example 5.1.5. Consider a fully faithful monoidal functor θ ∶ a → T such that every object in T is a quotient of one in the image (or, more generally, θ satisﬁes the conditions in [Co3, Lemma 4.1.3]), then θ is prexact by Corollary 2.3.3. Proposition 5.1.6. The following conditions are equivalent for monoidal functors θ, φ from a to tensor categories. a a (a) Σ θ = Σ φ; (b) ker θ⃗ = ker φ⃗; (c) ker θ∆ ker θ∆; Z = Z (d) HTθ = HTφ. Proof. The equivalence between conditions (a), (b) and (c) is an immediate consequence of Proposition 4.3.3. Now we prove that (c) implies (d). Since Kba is rigid and θ∆ and φ∆ are Z Z prexact, ker θ∆ ker θ∆ implies that the topologies of θ∆ and φ∆ coincide, by Theorem 5.1.1. Z = Z Z Z The latter then implies that HTθ = HTφ, by Theorem 3.3.2(3). That (d) implies (b) follows from application of Corollary 2.2.4 to θ⃗ and φ⃗. Remark 5.1.7. Similarly one can prove that the kernels of θ∆ and θ∆ determine one another, 0 Z extending the list of equivalent conditions in Proposition 5.1.6. Theorem 5.1.1, together with the following example, allows us to recover several results from [BEO, Co2, EHS]. Recall that a is self-splitting if for every morphism f in a, there exists 0 =~ V ∈ a such that V ⊗ f is split. Example 5.1.8. If a is self-splitting, then every monoidal functor θ ∶ a → T is prexact. Indeed, for f ∶ Y → X in a, take 0 =~ V ∈ a for which there is a split exact sequence ′ ′ g V ⊗f 0 → Z Ð→ V ⊗ Y ÐÐ→ V ⊗ X in a. It follows easily that ∗ ′ ∗ ′ (evV ⊗Y )○(V ⊗g ) f V ⊗ Z ÐÐÐÐÐÐÐÐÐÐ→ Y Ð→ X is sent to an acyclic sequence under θ, so we can apply Corollary 2.3.3. 5.2. Universal tensor categories.

a ↑ 5.2.1. For a subsieve γ ⊂ Σ , see 4.3.1, we denote by RMonγ(a, T) the category of (faithful) monoidal functors with homological kernel γ, which is well deﬁned by Lemma 4.3.2(1). The a set of subsieves γ ⊂ Σ for which there exist such a faithful monoidal functor is denoted by HK(a). In other words, the set HK(a) contains the ‘admissible homological kernels’. We also consider the subset HK0(a) ⊂ HK(a) of those homological kernels which can be realised using faithful monoidal functors to tensor categories over k. ↑ ↑ Theorem 5.2.2. (1) The 2-functor RMon (a, −) ∶ Tens → Cat decomposes as ↑ ↑ RMon (a, −) = O RMonγ(a, −). γ∈HK(a) ↑ (2) For each γ ∈ HK(a), the 2-functor RMonγ(a, −) is representable by a tensor category Tγ. Moreover IndTγ ≃ HSh(a, R) for a monoidal Grothendieck topology R on Noya. a (3) There are no inclusions between the diﬀerent subfunctors of Σ in HK(a). 24 KEVIN COULEMBIER

Proof. Part (1) follows from Remark 3.4.7(2). Now take γ ∈ HK(a). By assumption, there exists a faithful monoidal functor θ ∶ a → T with homological kernel γ. Set R ∶= HTθ. By the combination of Lemma 4.2.3 and Proposition 4.1.2, the corresponding biclosed Grothendieck category Sh(Noya, R) admits a faithful exact monoidal functor to IndT and therefore is itself the ind-completion of a tensor category by [CEOP, Proposition 3.3.2]. We let Tγ be the latter tensor category. For an arbitrary tensor category T1, we claim there exist equivalences ↑ ⊗ ⊗ ⊗ Tens (Tγ, T1) ≃ [IndTγ, IndT]cc ≃ AB3 (Sh(Noya, R), IndT1) ≃ [Noya, IndT1]prex,R, which are all obtained from appropriate composition with the obvious functors. The ﬁrst equivalence is in the proof of [Co3, Theorem 4.4.1]. For the second equivalence, we only need to argue that the functors in the second term are automatically faithful and exact. By the ﬁrst equivalence we know that they are exact and faithful when restricted to Tγ, and that the extension to the ind-completion is still exact and faithful is then obvious. The third equivalence is Proposition 4.1.2. The conclusion then follows from the equivalences ⊗ ⊗ ↑ [Noya, IndT1]prex,R ≃ [a, IndT1]R ≃ RMonγ(a, T1).

The ﬁrst equivalence follows from Lemma 4.2.3(2). As T1 is equivalent to the category of rigid objects in IndT1, see [Co2, Lemma 1.3.7], the second equivalence follows from the equivalence of (a) and (d) in Proposition 5.1.6. Corollary 5.1.2 shows that there is no inclusion between ker θ∆ and ker φ∆ for monoidal Z Z ′ ′ functors θ ∶ a → T and φ ∶ a → T to tensor categories T and T . Part (3) then follows from Proposition 4.3.3. Corollary 5.2.3. Assume there exists a (faithful) monoidal functor θ ∶ a → T of canonical a a homological kernel, Σ θ = Σ . Then every faithful monoidal functor from a to a tensor category is of canonical homological kernel. Moreover, there exists such a functor a → T0 which induces an equivalence ↑ ∼ ↑ Tens (T0, T1) Ð→ RMon (a, T1) for every tensor category T1. Proof. By Theorem 5.2.2(3) and Lemma 4.3.2(1), every faithful monoidal functor from a to a tensor category must be of canonical homological kernel. The result is then an immediate application of Theorem 5.2.2(2). For completeness we formulate Theorem 5.2.2 restricted to tensor categories over k.

Theorem 5.2.4. (1) The 2-functor RMon(a, −) ∶ Tens → Cat decomposes as RMon(a, −) = O RMonγ(a, −). γ∈HK0(a)

(2) For each γ ∈ HK0(a), the 2-functor RMonγ(a, −) is representable by a tensor category Tγ over k. 5.3. Recognising a local abelian envelope. We give the following intrinsic, albeit te- dious, characterisation of the universal functors in Theorem 5.2.2.

Theorem 5.3.1. Consider a faithful monoidal functor θ ∶ a → T. Then θ is the universal functor of its homological strength if and only if the following three conditions are satisﬁed: 0 1 (G’) For every X ∈ T, there exists a morphism f ∶ A → A in a such that X is a quotient of ker θ(f). 0 1 0 1 (F’) For every two morphisms g ∶ B → B and h ∶ C → C in a and every morphism 0 1 0 0 α ∶ ker θ(g) → ker θ(h), there exists morphisms f ∶ A → A , u ∶ A → B and 0 0 v ∶ A → C in a such that HOMOLOGICAL KERNELS 25

(a) g ○ u and h ○ v factor via f; (b) θ(u) restricts to an epimorphism ker θ(f) → ker θ(g); (c) the following square is commutative 0 ker θ(f) / θA

α○θ(u) θ(v) 0 ker θ(h) / θC . 0 1 0 1 (FF’) For every two morphisms g ∶ B → B and h ∶ C → C in a and every morphism 0 0 ν ∶ B → C for which h ○ ν factors via g and θ(ν) restricts to zero on ker θ(g), there 0 1 0 0 exists morphisms f ∶ A → A and µ ∶ A → B such that (a) both g ○ µ and ν ○ µ factor via f; (b) θ(µ) restricts to an epimorphism ker θ(f) → ker θ(g). Proof. By construction, θ ∶ a → T is one of the universal functors if θ⃗ (composed with T ↪ IndT) corresponds to the canonical functor from Noya to a category of sheaves on Noya. For the latter property, we can use the intrinsic description in [Lo, Theorem 1.2]. Using the definition of θ⃗ and the fact that it takes values in the category of compact objects in IndT then yields the description. 5.4. Some considerations about the symmetric case. 5.4.1. All statements in Theorem 5.2.2 and Corollary 5.2.3 remain true if we restrict to braided (or symmetric) categories and functors. For this we can apply [Co3, Theorem 3.5.6]. Corollary 5.4.2. Assume that a is symmetric. (1) Consider field extensions K1~k and K2~k and faithful symmetric monoidal functors θi ∶ a → VecKi . Then θ1 and θ2 have the same homological kernel if and only if there exists a common field extension K1 ↪ K ↩ K2 such that the two monoidal functors (K ⊗Ki −) ○ θi ∶ a → VecK isomorphic. (2) Assume that k is algebraically closed. Every two non-isomorphic faithful symmetric monoidal functors a → Veck have different (and hence incomparable) homological kernel. (3) If there exists a faithful prexact symmetric monoidal functor a → Veck, then all faithful symmetric monoidal functors a → Veck are isomorphic.

Proof. For part (1), assume first that θ1 and θ2 have the same homological kernel γ. It follows from Theorem 5.2.2(2) that θ1 and θ2 are obtained by composition of a → Tγ with two tensor functors Tγ → VecKi . It follows from [De1, 1.10-1.13] that there exists a faithfully flat commutative K1 ⊗k K2-algebra R such that the two resulting monoidal functors Tγ → ModR are isomorphic. We take K to be the quotient of R with respect to a maximal ideal. Again, the two resulting monoidal functors Tγ → VecK are isomorphic and the same follows for the two functors a → VecK . The other direction in part (1) follows from Remark 3.4.7(2). For part (2) assume that two such functors have the same homological kernel γ. By Theorem 5.2.2(2) and 5.4.1, this leads to two non-isomorphic symmetric tensor functors Tγ → Vec. The latter is contradicted by work of Deligne, see [Co1, Theorem 6.4.1] and [De1, Corollaire 6.20]. By part (2), for part (3) it suffices to observe that all functors have canonical homological kernel, which follows from Lemma 4.3.2 and either Corollary 5.1.3 or Theorem 5.2.2(3). Recall from [De1, §8] that to a symmetric monoidal functor θ ∶ a → T we can associate ⊗ an affine group scheme Aut (θ) in T. This is a representable functor from the category of commutative algebras in IndT to the category of groups which sends an algebra R to 26 KEVIN COULEMBIER the automorphism group of the monoidal functor from a to the category of R-modules in IndT obtained by composing θ with extension of scalars. As an example of this we have the fundamental group π(T) of T. We refer to [De1] for details. Theorem 5.4.3. Assume that k is perfect and that a is symmetric and admits a faithful symmetric monoidal functor θ ∶ a → T of homological kernel γ to an artinian symmetric ⊗ tensor category T over k. Consider the affine group scheme G ∶= Aut (θ) in T equipped with the natural transformation ∶ π(T) → G sending η ∶ idT ⇒ idT to ηθ. Then there is an equivalence of symmetric tensor categories

Tγ ≃ RepT(G, ).

Proof. By construction of Tγ, there exists a tensor functor ω ∶ Tγ → T. Such that θ ≃ ω ○ φ, for the universal functor φ ∶ a → Tγ. By [De1, §8], we have an equivalence between Tγ ′ ′ ′ ⊗ ′ ′ and RepT(G , ). Here G is the affine group scheme Aut (ω) and ∶ π(T) → G the corresponding homomorphism. It thus suffices to show that for every commutative ind- algebra R in T, composition with φ yields an isomorphism ⊗ ⊗ Aut (R ⊗ − ○ ω) → Aut (R ⊗ − ○ θ). The latter follows since the functors ⊗ ⊗ ⊗ AB3 (IndTγ, (IndT)R) → [Noya, (IndT)R]prex → [a, (IndT)R] are fully faithful by Proposition 4.1.2 and Lemma 4.2.3(1). 6. Examples For the entire section, let a be a symmetric rigid monoidal category over k. Tensor categories are assumed to be over k or some field extension. We will freely use notation and results pfrom Appendix A. 6.1. Prexact monoidal functors. Throughout this subsection we assume that k is algebraically closed.

Theorem 6.1.1. Assume that char(k) = 0 and algebraically closed. Every faithful symmetric monoidal functor a → Veck is prexact. Before preparing the proof of the theorem, we note the following consequence, which we obtain by combining the theorem with Corollary 5.1.3.

Corollary 6.1.2. If char(k) = 0 and there exists a faithful symmetric monoidal functor a → Veck, then a admits a weak abelian envelope. Lemma 6.1.3. Consider an aﬃne group scheme G~k. Every fully faithful symmetric monoidal functor a → RepkG is prexact. Proof. By [CEOP, Theorem 4.2.1(1) and (2)], there exists an aﬃne group scheme H~k and a fully faithful monoidal functor a → RepH such that every object in RepH is a quotient of one in the image. That a → RepH is prexact thus follows from Example 5.1.5. That the original functor a → RepG is also prexact follows from Corollary 5.1.3. Lemma 6.1.4. Assume that char(k) = 0 and n ∈ N. For any ∆ ∈ kN and any symmetric monoidal functor φ ∶ EN (∆) → Veck, the faithful functor EN (∆)~ker φ → Veck is prexact. Proof. By Lemma A.3.1, choosing the functor φ corresponds to choosing an appropriate endomorphism A of a vector space V . Denote by GA the centraliser of A in GL(V ). There is a clear lift through the forgetful functor RepGA → Vec of φ to EN (∆) → RepGA. It follows from [KP, Theorem 6.1] that the latter functor is full. Hence, by Lemma 6.1.3, the functor EN (∆)~ker φ → RepGA is prexact, which implies that EN (∆)~ker φ → Vec is prexact. HOMOLOGICAL KERNELS 27

Proof of Theorem 6.1.1. Consider a functor θ ∶ a → Vec as in the theorem. For a given f ∶ Y → X in a we need to prove exactness of the sequence 2.3.1(b). By taking a direct sum of f with the zero morphism X → Y , we can assume without loss of generality that X = Y . We can consider the monoidal functor ψ ∶ EN (∆) → a, corresponding to this endomorphism f ∈ a(X,X) under the equivalence in Lemma A.3.1. Set n = δ0 = dim X. By Lemma 6.1.4, the composite θ EN (∆)~ ker ψ → a Ð→ Vec is prexact. We denote the equivalence class of a morphism m in EN (∆) in the quotient by [m]. By Theorem 2.3.1 we ﬁnd ? θ(ψ(B)) → θ(X) → θ(X) h∶B→●, [ε]○[h]=0 is exact. In particular, we can rewrite the exact sequence as

? θ(ψ(B)) → θ(X) → θ(X). h∶B→●, f○ψ(h)=0 This shows that also the sequence 2.3.1(b) must be exact, which concludes the proof.

Proposition 6.1.5. For every n ∈ N and every symmetric monoidal functor θ ∶ OBk(n) → Veck, the faithful functor OBk(n)~ker θ → Veck is prexact. Proof. It follows from the ﬁrst fundamental theorem of invariant theory, see for instance [KSX, Theorem 1.2], that the corresponding functor OB(n) → RepGL(n) is full. That OB(n)~ker θ → RepGL(n), and therefore also OB(n)~ker θ → Vec, is prexact thus follows from Lemma 6.1.3. Remark 6.1.6. Alternatively, it follows also easily from [Co3, CEH, CEOP] that RepGL(n) is the abelian envelope of the quotient in Proposition 6.1.5.

6.2. The universal case in prime characteristic. We assume that char(k) = p > 0 and show that OBk(δ) admits monoidal functors of many diﬀerent homological strengths. Recall from [Co1] that for a symmetric tensor category T over k and X ∈ T, we deﬁne Fr+X as the image of the composite p 0 p p p p Γ X = H (Sp, ⊗ X) ↪ ⊗ X ↠ H0(Sp, ⊗ X) = Sym X. We will also use the p-adic dimension Dim+ from [EHO]. For X ∈ T, the p-adict integer j Dim+X ∈ Zp contains the information of all categorical dimensions dim Sym X and dim X is the image of Dim+X under Zp ↠ Fp.

Theorem 6.2.1. Set a ∶= OBk(δ) for some δ ∈ Fp ⊂ k. Assume p > 2. (1) There is a surjective function

HK(a) ↠ Zp × {0, 1}, which sends γ to the following pair. With Fγ ∶ a → Tγ the universal functor of homological strength γ, the element of Zp is the image of Dim+Fγ(Vδ) under Zp ↠ Zp, i i−1 ∑i≥0 aip ↦ ∑i>0 aip . For the second factor, we take 0 when Fr+(Fγ(Vδ)) = 0 and 1 otherwise. (2) If k is ﬁnite or algebraically closed, the above restricts to a surjective function

HK0(a) ↠ Zp × {0, 1}. For p = 2 the statements remain true if we just consider HK(a) → Z2.

Remark 6.2.2. For δ ∈ kFp we have HK(OB(δ)) = ∅, by [EHO, Lemma 2.2]. The remainder of this section is devoted to the proof of the theorem. 28 KEVIN COULEMBIER

6.2.3. We refer to [Ha] for details on all of the constructions in this paragraph. Let U be U a non-principal ultrafilter on N. We have the ultrapower K = k which is a field extension of k. If k is a finite field, then K ≃ k canonically and if k is algebraically closed then K ≃ k non-canonically by Steinitz’ theorem. Since the set of p-adic integers Zp is compact for the p-adic topology, we can define the ultralimit limU an of any sequence {an ∈ Zp S n ∈ N}. This is a ∈ Zp for which, for every l l ∈ N, the set of n ∈ N for which p S(a − an) is in U. We set t(U) = limU n ∈ Zp and let N d(U) ∈ Fp ⊂ K denote the class of (0, 1, 2, ⋯) ∈ k in K. This is the same as the image of t(U) under Zp ↠ Fp. We define the ultrapower V(U) ∶= ∏U Veck, which is a (non-artinian) symmetric tensor n ×N category over K. We let W ∈ V(U) be the object corresponding to (k )n ∈ Veck . In particular dim W = d(U). As observed in [EHO, §3.1], we have Dim+W = t(U) and by construction Fr+W ≃ W . ′ We also define the ultrapower of the category of supervector spaces V (U) ∶= ∏U sVeck. ′ ′ 0Sn ×N ′ We let W ∈ V (U) be the object corresponding to (k )n ∈ sVeck . In particular dim W = −d(U) and Dim+W = −t(U) and by construction Fr+W = 0.

Proof of Theorem 6.2.1. Since Fr+ and Dim+ are preserved by symmetric tensor functors, by Theorem 5.2.2 it suffices to show that OBk(δ) admits faithful symmetric monoidal functors to tensor categories which send Vδ to objects with the appropriate properties. For part (1), by the discussion in 6.2.3, it thus suffices to observe the following two facts. Firstly, via the axiom of choice, for any t ∈ Zp we can choose an ultrafilter U with t(U) = t. Secondly, for d = d(U), the symmetric monoidal functors OBk(d) → V(U), Vd ↦ W and ′ ′ OBk(−d) → V (U), V−d ↦ W are faithful. The latter follows from the observation that n m 0Sm kSn → Endk(⊗ V ) is injective if V = k or V = k with m ≥ n. 6.2.4. It seems difficult to show directly that the homological kernel of a faithful symmetric monoidal functor F from OBk(δ) determines the p-adic dimension of F (Vδ). On the other hand, we can easily show that it determines whether Fr+(F (Vδ)) vanishes as follows. Consider the following morphism in a ∶= OBk(δ): p ∗ p p ∗ p 2p−2 f ∶ (⊗ Vδ ) ⊗ (⊗ Vδ) → ((⊗ Vδ ) ⊗ (⊗ Vδ)) p ∗ p where the 2p−2 endomorphisms of (⊗ Vδ )⊗(⊗ Vδ) which make up f are given by (id−si)⊗id and id ⊗ (id − si), with {si S 1 ≤ i < p} the generators of Sp acting via the symmetric braiding. a Proposition 6.2.5. For f as above, we have Σ (f) ≃ k. For a faithful symmetric monoidal functor θ ∶ a → T, Vδ ↦ X to a symmetric tensor category T, we have ⎧ a ⎪ 0 if Fr+(X)= ~ 0, Σ θ(f) = ⎨ ⎪ k if Fr+(X) = 0. ⎩⎪ p ∗ p Proof. Consider a morphism g ∶ Y → (⊗ Vδ ) ⊗ (⊗ Vδ) for which f ○ g = 0. It follows that ′ ′ g = Q (σ ⊗ σ ) ○ g ′ σ,σ ∈Sp ′ p ∗ p p ∗ p for some g ∶ Y → (⊗ Vδ )⊗(⊗ Vδ). It then follows that c○g = 0 for all c ∶ (⊗ Vδ )⊗(⊗ Vδ) → 1. This implies that

a p ∗ p 2p−2 p ∗ p Σ (f) = cokera(((⊗ Vδ ) ⊗ (⊗ Vδ)) , 1) → a((⊗ Vδ ) ⊗ (⊗ Vδ), 1), which we can reformulate as a ×2(p−1) Σ (f) ≃ coker (kSp) → kSp , HOMOLOGICAL KERNELS 29 where the 2(p − 1) maps kSp → kSp are given by left and right multiplication with 1 − si. It a then follows that Σ (f) ≃ k. p ∗ p a We have ker θ(f) = Γ X ⊗ Γ X. It now follows that Σ θ(f)= ~ 0 if and only if the composition p ∗ p p ∗ p ev⊗pX Γ X ⊗ Γ X ↪ ⊗ X ⊗ ⊗ X ÐÐÐ→ 1, is zero. The latter is equivalent with Fr+X = 0. Remark 6.2.6. In [Co2, Theorem 2.3.1] it is proved that for a ﬁeld F of characteristic zero and d ∈ F , the category OBF (d) is self-splitting. By Example 5.1.8, the same is not true for OBk(δ) with δ ∈ Fp ⊂ k. 6.3. More strictly subcanonical homological kernels.

Lemma 6.3.1. Consider a faithful monoidal functor θ ∶ b → T from a rigid monoidal category b over k to a tensor category T. If there is a monomorphism ι in b which is not sent to a monomorphism in T, then the homological kernel of θ is not canonical. ∗ Proof. Let m ∶ X → Y denote the image of ι ∶ A → B under θ. Since f ∶= A ⊗ ι is also a monomorphism, by Example 3.4.8(2) it suffices to show that there exists a morphism ∗ ∗ α ∶ A ⊗ A → 1 in a for which θ(α) does not restrict to zero on ker(X ⊗ m). We take α = evA, which θ sends to evX . By adjunction, the relevant restriction is zero if and only if ker(m) → X (the inclusion of the kernel) is zero. This proves the lemma. We give concrete examples of the above situation. Example 6.3.2. In the following two cases, the monomorphism µ from Lemma A.2.1 is not sent to a monomorphism. (1) Assume that char(k) = 0 and take δ ∈~ Z. The symmetric monoidal functor θ ∶ 1 1 MO(2δ, 2δ) → OB (δ), to the semisimple tensor category OB (δ), which sends µ to id● ⊕ 0 ∶ ● ⊕ ● → ● ⊕ ● is faithful. (2) Let k be arbitrary and let U be a non-principal ultrafilter on N. We use the notation U ×N of 6.2.3. For δ ∈ K = k given by the class of (n)n ∈ k , the symmetric monoidal functor MOk(δ, δ − 1) → V(U) which sends µ to a morphism represented by some n n−1 surjections {k ↠ k Sn ∈ N} is faithful. Example 6.3.3. The monomorphism ι in Seq from Lemma A.4.1 is never sent to a monomorphism under any faithful monoidal functor to a tensor category (as it is sent to a morphism of the form (a, b) ∶ A ⊕ B → C for an epimorphism b ∶ B → C and a non-zero morphism a ∶ A → C). Concrete examples of such faithful monoidal functors are given by the obvious functor 1 Seq → OB (δ), for char(k) = 0 and δ ∈~ Z, or can be constructed for arbitrary k via ultrafilters. 6.4. Motives. Let K be a field of characteristic zero and consider a Weil cohomology theory H●, on the category of smooth projective varieties over a field F , with values in K. Let MotH (F ) denote the category of Chow motives modulo homological equivalence. This is a ● rigid symmetric monoidal category over k = Q such that H extends to a faithful Q-linear Z symmetric monoidal functor H ∶ MotH (F ) → VecK . In [Sc, §3.2], Schäppiconstructed a tensor category MH (F ) yielding a commutative diagram

MotH (F ) / MH (F )

H % z Z VecK of symmetric monoidal functors where the right downwards arrow is a tensor functor. 30 KEVIN COULEMBIER

Lemma 6.4.1. The tensor category MH (F ) is the universal tensor category associated to Z MotH (F ) and the homological kernel of H ∶ MotH (F ) → VecK in Theorem 5.2.2. Proof. Set a ∶= MotH (F ). Tracing through the construction of MH (F ) in [Sc, §2.2 and §3.1], we can conclude the following. In the notation of the current paper, MH (F ) is the b smallest full subcategory of compact objects in Sh(K a, T ), for T the Grothendieck topology associated to H∆ Kba VecZ Z, which contains the image of a and is closed under ﬁnite Z ∶ → ( K ) direct sums, (co)kernels, tensor products and duals. By Lemma 6.4.4 below, we can leave out the last two requirements. That MH (F ) is b equivalent to the category of compact objects in Sh(K a, R), with R the Grothendieck ∆ b Z topology of H0 ∶ K a → VecK then follows quickly from Theorem 3.3.2(2). Remark 6.4.2. The new construction of MH (F ) as an instance of Theorem 5.2.2 reveals the following: Z (1) MH (F ) has a universal property which extends the existence of MH (F ) → VecK from [Sc]. (2) We can describe (the ind-completion of) MH (F ) directly as a category of sheaves b on NoyMotH (F ) or K MotH (F ), rather than as a certain maximal subcategory of a b category of sheaves on K MotH (F ). Remark 6.4.3. The observations from this section extend to any ‘basic tannakian factori- sation’ as in [Sc, §3.1]. Lemma 6.4.4. Consider a monoidal functor θ ∶ b → T from a rigid monoidal category b to a tensor category T. Let A ⊂ T be the minimal abelian subcategory that contains the image of θ. Then A is a tensor category, with tensor product inherited from T.

Proof. For some B ∈ b, we define A0 ⊂ A the full subcategory of objects X ∈ A for which X ⊗ θ(B) ∈ A. This is an abelian subcategory of T which contains the image of θ and so A0 = A. This way we can prove that A is closed under tensor product with objects in the image of θ. Subsequently we can prove similarly that A is a monoidal subcategory of T. Finally, that A is closed under taking duals follows from considering the full subcategory of all (left or right) duals of the objects in A. It is an abelian subcategory of T containing the image of θ, so it contains A, and it follows easily that this must in fact be an equality. Appendix A. Some universal monoidal categories The oriented Brauer category OB from [BCNR] was introduced as the universal rigid symmetric monoidal category on one object in [De2]. We pretend that OB stands for ‘object’, and introduce the universal rigid symmetric monoidal categories on one morphism and on one endomorphism as MO and EN . Consider a field extension K~k (potentially K = k). For any category C and a class of objects C ⊂ ObC, the groupoid corresponding to C is the subcategory of C with objects C and morphisms given by the isomorphisms in C. 0 A.1. One object. Fix δ ∈ k. We denote by OBk(δ), the specialisation of the oriented Brauer 0 category OB of [BCNR, §1] at δ. Concretely, objects in OBk(δ) are words in the alphabet {●, ○}, and the morphism space between two such words is spanned by all pairings of the letters such that we only pair the same colour on different lines and different colours on the same line, when we place the target on a horizontal line above the source. An example of a morphism from ● ○ ● ○ ●○ to ● ○ ○ ● ● ○ ○● is ● ○ ○ ● ● ○ ○ ●

● ○ ● ○ ● ○ HOMOLOGICAL KERNELS 31

Composition of morphisms corresponds to concatenation of the diagrams with evaluation of loops at δ. The tensor product corresponds to juxtaposition of diagrams. We refer to [BCNR, De2] for a full description. We denote by OBk(δ), or OB(δ) if k is clear from the 0 context, the additive envelope of OBk(δ). We will also typically denote ● by Vδ. The following well-known lemma follows as in [De2, Proposition 10.3].

Lemma A.1.1. Let a be a rigid symmetric monoidal category over K. Evaluation at Vδ = ● s⊗ yields an equivalence between [OBk(δ), a] and the groupoid corresponding to the objects in a of dimension δ in a.

Lemma A.1.2. [De2, Th´eor`eme10.5] Assume char(k) = 0 and δ ∈~ Z. The Karoubi envelope 1 (idempotent completion) OB (δ) of OB(δ) is a semisimple tensor category over k. 0 A.2. One morphism. Fix δ, t ∈ k. We denote by MOk(δ, t) the following strict rigid 0 monoidal category. Objects in MOk(δ, t) are words in the alphabet {●, ○, ∎, ◻}, and the space of morphisms between two such words has basis given by all pairings of the letters such that: ● Any occurrence of a symbol can be paired with an occurrence of the same symbol on the other line. ● We can pair ● with ○, and ∎ with ◻, if they are on the same line. ● We can pair ● on the lower line with ∎ on the top line; and ◻ on the lower line with ○ on the top line. An example of a diagram representing a morphism from ● ○ ● ○ ●◻ to ● ○ ○ ● ● ◻ ○∎ is ● ○ ○ ● ● ◻ ○ ∎

● ○ ● ○ ● ◻ Composition of morphisms corresponds to concatenation of the diagrams with evaluation of loops containing ● (and hence also ○) at δ and loops containing ∎ at t. The tensor product corresponds to juxtaposition of diagrams. We denote by MOk(δ, t), or MO(δ, t) if 0 k is clear from the context, the additive envelope of MOk(δ, t). We denote the morphism corresponding to the unique diagram from ● to ∎ by µ. Lemma A.2.1. The morphism µ ∶ ● → ∎ in MO(δ, t) is a monomorphism. 0 Proof. It suffices to show that µ is a monomorphism in MO (δ, t). For any word w in {●, ○, ∎, ◻}, composition with µ yields an isomorphism between Hom(w, ●) and the subspace of Hom(w, ∎) spanned by diagrams in which ∎ on the top row is paired with some ●. Lemma A.2.2. Let a be a rigid symmetric monoidal category over K. Evaluation at µ s⊗ yields an equivalence between [MOk(δ, t), a] and the groupoid corresponding to the objects in Arra which are morphisms in a from an object of dimension δ to one of dimension t. A.3. One endomorphism. Fix ∞ ∆ = (δ0, δ1, δ2, ⋯) ∈ M k. i=0 0 Denote by EN k(∆) the specialisation of the associated graded of the affine oriented Brauer category (see [BCNR, §3.4]), where we specialise a loop with i dots to δi. Concretely, objects 0 in EN k(∆) are finite words in the alphabet {●, ○} and the morphism space between two such words has as basis all ‘decorated’ pairings, where the rules for the pairings are as in A.1, and each pairing is decorated with a natural number i ∈ N. We can depict this graphically by drawing an appropriate number of indicators on each strand in the diagram. Composition 32 KEVIN COULEMBIER of morphisms corresponds to concatenation of the diagrams, where the number of indicators on a strand are simply added, with evaluation of loops with i dots at δi. In this category we then have for instance r End(⊗ ●) ≃ k[x1, x2, ⋯, xr]#Sr, for the canonical action of the symmetric group Sr on k[x1, x2, ⋯, xr]. We denote the morphism ● → ● corresponding to a line with one indicator (‘x1’ in the above equation) by ε. The tensor product corresponds to juxtaposition of diagrams. We refer to [BCNR] for a full description. We denote by EN k(∆), or EN (∆) if k is clear from the context, the 0 additive envelope of EN k(∆). The following is observed in [BCNR, Corollary 3.6]. Lemma A.3.1. Let a be a rigid symmetric monoidal category over K. Evaluation at ε s⊗ yields an equivalence between [EN k(∆), a] and the groupoid corresponding to the objects i in Arra which are endomorphisms f in a that satisfy Tr(f ) = δi for i ∈ N. A.4. One object, no braiding. We review the (k-linear version of the) universal (un- 0 braided) rigid monoidal category from [CSV]. Let Seqk be the strict rigid monoidal category over k where objects are words in the alphabet {Xi S i ∈ Z}. The morphism space between two words is spanned by all pairings which can be written diagrammatically without crossings and where each letter can be paired with itself on the other line; on the lower line Xi can only be paired with some Xi+1 appearing to its left or Xi−1 on its right. On the upper line, Xi can only be paired to Xi+1 appearing on its right or Xi−1 on its left. Composition and tensor product is again defined via concatenation and juxtaposition. We denote by Seqk the 0 additive envelope of Seqk. Consider a morphism

ι ∶= (coX0 ,X0 ⊗ evX1 ⊗ X1) ∶ 1 ⊕ X0 ⊗ X2 ⊗ X1 ⊗ X1 → X0 ⊗ X1. The following lemma follows from the diagrammatic calculus.

Lemma A.4.1. The morphism ι is a monomorphism in Seqk. The following lemma is proved as [CSV, Proposition 6].

Lemma A.4.2. Let a be a rigid monoidal category over K. Evaluation at X0 yields an ⊗ equivalence between [Seqk, a] and the groupoid corresponding to all objects in a. Acknowledgements. The author thanks Richard Garner and Amnon Neeman for interesting discussions. The research was partially supported by ARC grant DP210100251.

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