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CONFLUENTESMATHEMATICIBruno KAHNExactness and faithfulness of monoidal functorsTome 14, no 2 (2022), p. 45–51.https://doi.org/10.5802/cml.86© Les auteurs, 2022.Certains droits réservés.Cet article est mis à disposition selon les termes de la Licence Internationaled’Attribution Creative Commons BY-NC-ND 4.0https://creativecommons.org/licenses/by-nc-nd/4.0/Confluentes Mathematici est membre duCentre Mersenne pour l’édition scientifique ouvertehttp://www.centre-mersenne.org/e-ISSN : 1793-7434Confluentes Math.14, 2 (2022) 45-51EXACTNESS AND FAITHFULNESS OF MONOIDAL FUNCTORSBRUNO KAHNAbstract. Inspired by recent work of Peter O’Sullivan, we give a condition under which afaithful monoidal functor between abelian ⊗-categories is exact.IntroductionA well-known result of Deligne ([5, Prop. 1.19], [6, Cor. 2.10], [7, remark in 0.9])says that an exact ⊗-functor from a rigid abelian symmetric monoidal categoryin which End(1) is a field to a (nonzero) abelian symmetric monoidal category isfaithful. Coulembier recently proved that the converse is true, by topos-theoreticmethods ([3, Cor. 4.4.4], see also [4, Th. 2.4.1]). In [8, Lemma 10.7], O’Sulliv-an gives a different proof of this converse in the special case of super Tannakiancategories. These converses are remarkable, as they give a drastically simple wayof filling the gap in Saavedra’s proof that Tannakian categories are classified bygerbes (see [5, end of Question (3.15)]).The aim of this note is to generalise Deligne’s and O’Sullivan’s results by weak-ening their hypotheses, in particular, removing a rigidity hypothesis in the secondcase. The first case (Proposition 1.2) is very simple and is a footnote to Deligne’sargument; the second, Theorem 2.18 d), is more elaborate. Parts of this work applyto the quasi-abelian situation of [1], see Example 2.9 and Remark 2.14.The condition for a monoidal functor F to preserve monomorphisms or epimor-phisms turns out to be all-pervasive here. The novel fact is the existence of a notionof super-dimension in a very general context (Definition 2.12 and Lemma 2.13): itplays a key rôle in the proof of Theorem 2.18.1. On Corollary 2.10 of [6]Lemma 1.1. — Let A, B be two monoidal categories, with A rigid, and letF : A → B be a strong monoidal functor. Then F is faithful ⇐⇒ F separatesmorphisms with domain 1 ⇐⇒ F separates morphisms with codomain 1.Proof. — Let f, g : M ⇒ N be two parallel morphisms in A, and f̃ , g̃ : 1 ⇒M∨ ⊗ N be the morphisms corresponding to f and g by adjunction, where M∨ isthe dual of M . Suppose that F (f) = F (g). Then F (f̃) = F̃ (f) = F̃ (g) = F (g̃),hence f̃ = g̃ and f = g. Dual reasoning for “codomain”. □Proposition 1.2. — In Lemma 1.1, let A, B and F be additive (the tensorproducts are biadditive). Suppose thata) any nonzero morphism of A with domain 1 is a monomorphism;b) F preserves monomorphisms.2020 Mathematics Subject Classification: 18M05.Keywords: Monoidal categories.4546 B. KahnThen F is faithful.Same conclusion when replacing “monomorphism” by “epimorphism”.Proof. — The condition of Lemma 1.1 is verified. □Corollary 1.3. — If A, B are abelian, B ≠ 0, EndA(1) is a field and F pre-serves monomorphisms (equivalently, epimorphisms), it is faithful.Proof. — 1 is simple [5, Prop. 1.17], hence (i) holds in Proposition 1.2. □(See Lemma 2.3 d) and Proposition 2.11 c), e) for partial converses.)2. On Lemma 10.7 of [8]All categories are Q-linear, pseudo-abelian symmetric monoidal categories, andall functors are Q-linear ⊗-functors (= strong symmetric monoidal functors). Werefer to [7] for Schur functors.2.1. Property S.Definition 2.1. — We say that a category A has property S if the followingcondition holds:A morphism f : M → N is mono if and only if (S(f) = 0 ⇒S(M) = 0) for any Schur functor S.Remark 2.2. — Aop has property S if and only if the following condition holds:A morphism f : M → N is epi if and only if (S(f) = 0 ⇒ S(N) = 0)for any Schur functor S.We call this property Sop.Lemma 2.3. — Let F : A → B be faithful.a) If B verifies the “if” part of property S, so does A.b) If F preserves monomorphisms and B verifies the “only if” part of propertyS, so does A.c) If F preserves monomorphisms and B has property S, so does A.d) If A verifies the “only if” part of property S and B verifies the “if” part ofproperty S, then A has property S and F preserves monomorphisms.e) If A verifies the “if” part of property S and B verifies the “only if” part ofproperty S, then F reflects monomorphisms.Proof. — This is a simple matter of logic, but we give details to avoid anyconfusion.a) Let f : M → N . Suppose that F (f) is not mono. By hypothesis, there is aSchur functor S such that S(F (f)) = 0 but S(F (M)) ̸= 0. Then S(M) ̸= 0but S(f) = 0 by the faithfulness of F . If now f is not mono, then F (f) isnot mono again by the faithfulness of F , so the above applies.b) Let f : M → N which is mono, so that F (f) is mono. Let S be a Schur functorsuch that S(f) = 0. Then S(F (f)) = 0, hence S(F (M)) = F (S(M)) = 0 byhypothesis, and S(M) = 0 by the faithfulness of F .c) follows immediately from a) and b).EXACTNESS AND FAITHFULNESS OF MONOIDAL FUNCTORS 47d) The first claim follows from a). For the second one, let f : M → N which ismono. If F (f) is not mono, then by the proof of a) there is a Schur functorS such that S(f) = 0 but S(M) ̸= 0. This contradicts the “only if” part ofproperty S.e) Suppose that F (f) is mono. Let S be such that S(f) = 0. Then S(F (f)) =F (S(f)) = 0, hence F (S(M)) = S(F (M)) = 0 by the “only if” part ofproperty S. Thus S(M) = 0 by the faithfulness of F . This shows that f ismono by the “if” part of property S. □Proposition 2.4. — For any A,a) Kernels and cokernels are stable under tensor product by dualisable objects.b) Monomorphisms with dualisable domain and epimorphisms with dualisablecodomain are stable under tensor products, tensor powers and the applicationof Schur functors.c) If A is rigid, it verifies the “only if” part of properties S and Sop and its⊗-structure is exact.Proof. —a) More generally, cokernels are stable under tensor product with an object Psuch that the internal Hom functor R 7→ Hom(R ⊗ P, Q) is representable forany Q, because this tensor product is then right exact as a left adjoint. Simi-larly for kernels, assuming that internal coHom’s (Moh objects in Saavedra’sterminology) are representable.b) follows from a) by using the symmetry of the tensor product and the fact thatmonomorphisms and epimorphisms are stable under composition (the samegeneralisation as in a) applies);c) follows trivially from a) and b) (closed and coclosed would be sufficient). □Proposition 2.5. — Suppose A abelian semi-simple, Schur-finite (any objectis killed by some Schur functor) and such that M ⊗ N ̸= 0 if M ̸= 0 and N ̸= 0(see Remark 2.10 below). Then A has properties S and Sop. □Proof. — We only deal with property S. Let f : M → N be a morphism in A.Only if : by semi-simplicity, f mono has a retraction g, thus S(f) has the retrac-tion S(g) for all S.If : still by semi-simplicity, write f as a direct sum of an isomorphism f ′ : M ′ ∼−→N ′ and a 0 morphism f ′′ : M ′′ → N ′′. By [7, Cor. 1.7], the set of partitions λ suchthat Sλ(M ′) = 0 is a sieve for the opposite order to the inclusion of Young diagrams[λ]; since M ′ is Schur-finite, we may therefore find λ such that Sλ(M ′) = 0 butSµ(M ′) ̸= 0 for any µ such that [µ] ⊊ [λ]. By [7, (1.5.1) et (1.8.1)], Sλ(M) containsSµ(M ′) ⊗ M ′′ as a direct summand for such µ with |µ| = |λ| − 1. If f is not mono,i.e. if M ′′ ̸= 0, this summand is nonzero by hypothesis, so Sλ(M) ̸= 0. Let n = |λ|.Then f⊗n is the direct sum of the isomorphism f ′⊗n and a 0 morphism; applyingthe idempotent defining Sλ to this decomposition, we get Sλ(f) = 0. □Corollary 2.6. — The ⊗-category Vec±K of finite-dimensional super-vectorspaces over a field K of characteristic 0 has properties S and Sop. □The following lemma is obvious:48 B. KahnLemma 2.7. — Let M ∈ A. Write Σ(M) for the (possibly empty) set of Schurfunctors S such that S(M) = 0. Then Σ(M) ⊆ Σ(F (M)) for any ⊗-functorF : A → B, with equality if F is faithful. □2.2. Pre-tannakian categories.Definition 2.8. — Let A be a Q-linear symmetric monoidal category.a) A is proto-tannakian if it admits a faithful ⊗-functor ω : A → Vec±K , whereK is a field of characteristic 0. We call ω a fibre functor.b) A is left (resp. right) pre-tannakian if we can choose ω preserving monomor-phisms (resp. epimorphisms).c) A is pre-tannakian if it is left and right pre-tannakian.Examples 2.9. — a) By [1, Lemma 2.1.5 (2)], a quasi-tannakian category inthe sense of loc. cit., Def. 2.1.1 is pre-tannakian.b) If A is abelian, rigid and Schur-finite (super Tannakian in the sense of [8,Def. 10.2]), with k = EndA(1) a field, it is pre-tannakian. Indeed, we mayassume k algebraically closed; by [7, Prop. 2.1], A admits an exact ⊗-functorω to the category of super-modules over a super commutative k-algebra R,whose values are (by rigidity) projective. Thus we may replace R with a fieldK by specialisation. We then get the faithfulness of ω from Corollary 1.3.c) By [8, Th. 10.10], b) extends to the case where A is only integral in the senseof loc. cit., beg. §3, i.e. f ⊗ g = 0 ⇒ f = 0 or g = 0 (pseudo-Tannakianin the terminology of loc. cit.). (Conversely, any proto-tannakian category isclearly integral and Schur-finite.)Remark 2.10 (cf. proofs of Lemma 1.1 and Cor. 1.3). — If A is abelian and rigidand EndA(1) is a field, it is integral. Indeed, let f : A → A′ and g : B → B′ be twononzero morphisms. Their adjoints f̃ : 1 → A∨ ⊗ A′, g̃ : 1 → B∨ ⊗ B′ are nonzero,hence monomorphisms since 1 is simple [5, Prop. 1.17]. By the exactness of thetensor product (Proposition 2.4 c)), f̃ ⊗ g̃ is a monomorphism, hence nonzero. Butit is the adjoint of f ⊗ g. In particular, the last hypothesis of Propositon 2.5 isautomatic if A is rigid (or contained in a rigid abelian category).On the other hand, O’Sullivan proves in [8, Lemma 3.1] a very strong result: Ais fractionally closed in the sense of loc. cit., Sect. 3, bot. p. 6.Proposition 2.11. — a) Any left (resp. right) pre-tannakian category hasproperty S (resp. Sop).b) If A is left pre-tannakian and B verifies the “if” (resp. the “only if”) partof property S, any faithful functor F : A → B preserves (resp. reflects)monomorphisms. Same statement when replacing left by right, monomor-phism by epimorphism and S by Sop.c) If A and B are pre-tannakian, any faithful functor F : A → B (in particular,any fibre functor ω : A → Vec±K) preserves and reflects monomorphisms andepimorphisms.d) If A is pre-tannakian, its tensor product preserves monomorphisms and epi-morphisms.e) Any rigid proto-tannakian category is pre-tannakian.Proof. — a) follows from Lemma 2.3 c) and Corollary 2.6. In b), “preserves”follows from a) and Lemma 2.3 d), while “reflects” follows from a) and Lemma 2.3EXACTNESS AND FAITHFULNESS OF MONOIDAL FUNCTORS 49e). c) follows from b) and Corollary 2.6. d) follows from c) and the exactness ofthe tensor product in Vec±K ; e) follows from Proposition 2.4 c), Corollary 2.6 andLemma 2.3 d). □2.3. Super-dimensions.Definition 2.12. — Let A be proto-tannakian, and let M ∈ A. We definethe super-dimension dim±(M) of M as the super-dimension of ω(M) for ω as inDefinition 2.8.Lemma 2.13. — dim±(M) does not depend on the choice of ω. It is preservedby any faithful ⊗-functor (between proto-tannakian categories). Moreover:a) dim±(M) is additive in M ; M = 0 ⇐⇒ dim±(M) = 0.b) M invertible ⇒ dim±(M) = 1|0 or 0|1; the converse is true if M is dualisable.c) If A is left (resp. right) pre-tannakian, then dim±(M) ⩽ dim±(N) (resp.dim±(M) ⩾ dim±(N)) for any monomorphism (resp. epimorphism) f : M →N .d) If there is equality in c), then f is also epi (resp. mono).e) If dim±(M) = m|n and M is dualisable, then tr(1M ) = m − n.Proof. — By [7, Cor. 1.9], dim±(M) is the pair m|n such that Sλ(ω(M)) = 0for a partition λ if and only if [λ] contains the rectangular diagram with n + 1 rowsand m + 1 columns; the two claims then follow from Lemma 2.7. Next, a), c) ande) hold in a category of the form Vec±K , hence in general; d) follows in the sameway by using Proposition 2.11 c). Finally, in b) the implication ⇒ is trivial; forthe converse, the unit 1 → M∨ ⊗ M and the counit M ⊗ M∨ → 1 become inverseisomorphisms after applying ω, hence are inverse isomorphisms by faithfulness. □Remark 2.14. — By Example 2.9 a), Proposition 2.11 c) and Lemma 2.13 c) givean elementary proof of [1, Lemma 2.1.5 (2)]. Similarly, Proposition 2.4 a) gives anelementary proof of part a) of this lemma.In the sequel, we assume that A is pre-tannakian.Lemma 2.15. — Let E : 0 → M ′ i−→ M p−→ M ′′ → 0 be a 0 sequence in A, withi a mono and p an epi. Then we havedim±(M) ⩾ dim±(M ′) + dim±(M ′′) (2.1)with equality if and only if ω(E) is exact for some ( ⇐⇒ any) fibre functor ω :A → Vec±K .Proof. — This follows from Proposition 2.11 c). □Definition 2.16. — A sequence E as in Lemma 2.15 is dim-exact if (2.1) is anequality.Proposition 2.17. — Assume that A is abelian and that its tensor productis right exact (e.g. that A has internal Homs). Then dim±(M) ⩽ dim±(M ′) +dim±(M ′′) for any exact sequence E : M ′ → M → M ′′ → 0.Proof. — (cf. [7, proof of Prop. 1.19].) Let first Ei : M ′i → Mi → M ′′i → 0 bea family of exact sequences, for i ∈ {1, . . . , n}. Using the right exactness of thetensor product, a standard diagram chase yields an exact sequenceM ′1 ⊗ M2 ⊕ M1 ⊗ M ′2 → M1 ⊗ M2 → M ′′1 ⊗ M ′′2 → 050 B. Kahnhence, by induction, an exact sequenceM ′1 ⊗ M2 ⊗ · · · ⊗ Mn ⊕ M1 ⊗ M ′2 ⊗ · · · ⊗ Mn ⊕ · · · ⊕ M1 ⊗ M2 ⊗ · · · ⊗ M ′n→ M1 ⊗ · · · ⊗ Mn → M ′′1 ⊗ · · · ⊗ M ′′n → 0.Taking E1 = · · · = En = E , we thus get an exact sequenceM ′ ⊗ M⊗n−1 ⊕ M ⊗ M ′ ⊗ M⊗n−2 ⊕ · · · ⊕ M⊗n−1 ⊗ M ′→ M⊗n → M ′′⊗n → 0. (2.2)For the action of Sn on (2.2) by permutations, the first term is isomorphicto IndSnSn−1(M′ ⊗ M⊗n−1). Iterating, we see that M⊗n admits a Sn-equivariantfiltration of length n + 1 whose p-th quotient is isomorphic to IndSnSp×Sn−p(M′⊗p ⊗M ′′⊗n−p), except for p = n where it is only a quotient of M ′⊗n, cf. [7, bot. p.232]. As in loc. cit., Prop. 1.8, it follows that if λ is a partition of n, Sλ(M) has afiltration of length n + 1 whose p-th quotient is isomorphic to[M ′, M ′′]p =⊕µ,ν(Sµ(M ′) ⊗ Sν(M ′′))[λ:µ,ν]where |µ| = p, |ν| = n − p, except for p = n when it is only a quotient of Sλ(M ′).Here, [λ : µ, ν] is the multiplicity described in [7, 1.5].Now we have, by [7, Prop. 1.8]:ω( n⊕p=0[M ′, M ′′]p)≃ Sλ(ω(M ′ ⊕ M ′′)) ≃ ω(Sλ(M ′ ⊕ M ′′))henceSλ(M) ̸= 0 ⇒n⊕p=0[M ′, M ′′]p ̸= 0 ⇒ ω( n⊕p=0[M ′, M ′′]p)̸= 0⇐⇒ ω(Sλ(M ′ ⊕ M ′′)) ̸= 0 ⇒ Sλ(M ′ ⊕ M ′′) ̸= 0whence dim±(M) ⩽ dim±(M ′ ⊕ M ′′) = dim±(M ′) + dim±(M ′′) (Lemma 2.13 a)),as desired. □2.4. Main theorem. Recall that A is pre-tannakian. Let E be as in Lemma 2.15.Theorem 2.18. — a) For any faithful functor F : A → B (with B pre-tannakian) E dim-exact (Definition 2.16) ⇐⇒ F (E) dim-exact.b) Suppose that i has a cokernel. If E is dim-exact, so is the sequence 0 → M ′ i−→M → Coker i → 0, and the induced morphism Coker i → M ′′ is mono-epi.Same statement, mutatis mutandis, assuming that p has a kernel.c) If A is abelian, dim-exact ⇒ exact, and the converse is true if its tensorstructure is right exact.d) Let A, B be abelian pre-tannakian categories. If the tensor structure of A isright exact, any faithful ⊗-functor F : A → B is exact.Proof. —a) follows from Proposition 2.11 a) and b), and Lemma 2.13.EXACTNESS AND FAITHFULNESS OF MONOIDAL FUNCTORS 51b) The morphism Coker i → M ′′ is clearly epi, hence dim± Coker i ⩾ dim± M ′′by Lemma 2.13 c); if E is dim-exact, we must have equality, which shows thefirst claim. The second one follows from Lemma 2.13 d). Same argumentassuming that p has a kernel.c) ⇒ follows from b) and ⇐ follows from Proposition 2.17.d) now follows from a) and c). □Remarks 2.19. — a) The proof of Theorem 2.18 d) is inspired from the keyidea in the proof of [8, Lemma 10.7]. We recover this lemma when A and Bare rigid with End(1) a field, see Proposition 2.4 c) and Example 2.9 b).b) I don’t know if the right exactness hypothesis is necessary in Theorem 2.18c) and d) (see Proposition 2.11 d)); it holds for the abelian envelope Ab(A)of an additive ⊗-category A from [2, Prop. 1.8 and 1.10] (but Ab(A) maynot be pre-tannakian even if A is.)c) I don’t know if Theorem 2.18 c) and d) extend to the quasi-tannakian cate-gories of [1] (see Example 2.9).References[1] Y. André Slope filtrations, Confluentes Math. 1 (2009) 1–85.[2] L. Barbieri-Viale, A. Huber, M. Prest Tensor structure for Nori motives, Pacific J. Math. 306,2020, 1–30.[3] K. Coulembier Additive Grothendieck pretopologies and presentations of tensor categories,preprint, 2020, https://arxiv.org/abs/2011.02137.[4] K. Coulembier, P. Etingof, V. Ostrik, B. Pauwels Monoidal Abelian Envelopes with a quotientproperty, J. Reine Angew. Math. 794 (2023), 179 – 214.[5] P. Deligne, J. S. Milne Tannakian categories, in Hodge cycles, motives, and Shimura varieties,Lect. Notes in Math. 900, Springer, 1982, 101–228.[6] P. Deligne Catégories tannakiennes, in The Grothendieck Festschrift, Vol. II, Progr. Math.87, Birkhäuser, 1990, 111–195.[7] P. Deligne Catégories tensorielles, Moscow Math. J. 2 (2002), 227–248.[8] P. O’Sullivan Super Tannakian hulls, preprint (2020), https://arxiv.org/abs/2012.15703.Manuscript received 2nd December 2021,revised 14th September 2022,accepted 15th September 2022.Bruno KAHNIMJ-PRG, Case 247, 4 place Jussieu, 75252 Paris Cedex 05, Francebruno.kahn@imj-prg.fr

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