Non-canonical isomorphisms

Non-canonical isomorphis ms Stephen Lac k ∗ Sc ho ol of Computing and Mathematics Univ ersit y of W estern Sydney Lo c k ed Bag 1797 P enrith South DC NSW 1797 Australia email: s.lack@uw s.edu.au Abstract W e giv e tw o e xamples of catego rical axioms asserting that a canonically defined natural transforma- tion is inve rtible where the inv ertibilit y of any n atural transformation implies that t h e canonical on e is inv ertible. The fi rst example is distribu t ive categories, th e second (semi-)additive ones. W e show that eac h follow s from a general result ab out monoidal functors. In any category D with finite pro ducts and copr o ducts there is a natura l family of maps X × Y + X × Z δ X,Y ,Z / / X × ( Y + Z ) induced, via the universal pro p erty of the co pro duct X × Y + X × Z , by the mo r phisms X × i and X × j , where i a nd j deno te the co pro duct injections o f Y + Z . Such a D is said to be di stributive [2 , 3] if the canonical ma ps ar e invertible; in other words, if the functor X × − : D → D pr eserves binary co pr o ducts, for all ob jects X . As obse r ved by Co ck ett [3], it follows that X × 0 ∼ = 0, so that X × − in fact preser ves finite copr o ducts. Claudio Pis ani has asked whether the existence of any natural fa mily of isomorphisms X × Y + X × Z ψ X,Y ,Z / / X × ( Y + Z ) might imply that D is dis tributive. Such ψ are the non-ca nonical iso mo rphisms of the title. He sugges ted that this was probably not the case, a nd this was also m y immediate reaction. But in fac t it is true! This is the first result of the pap er. The sec o nd r e sult is an analog ue for semi-additive categ ories. Recall that a ca tegory is p ointed when it has an initial ob ject which is also terminal (1 = 0), and that for any a ny tw o ob jects Y and Z in a p ointed category there is a unique mor phism from Y to Z which factorizes thro ugh the zero ob ject; this morphism is called 0 Y ,Z or just 0 . If the categ ory has finite pro ducts and copr o ducts, then there is a natur al family of morphisms Y + Z α Y ,Z / / Y × Z induced b y the identities on Y and Z a nd the zero morphisms 0 : Y → Z a nd 0 : Z → Y . The catego r y is semi-a dditive when these α Y ,Z are inv ertible [6, VI I.2]. A se mi- additive category a dmits a canonical enrichmen t ov er comm utative monoids; conv ersely , an y categ ory enriched ov er commutative monoids which has either finite products or finite copro ducts is se mi- additive. O ur result for semi-additiv e categories a sserts once again that the existence of any natur a l isomorphism Y + Z ∼ = Y × Z implies that the categ o ry is semi- additive. ∗ The support of the Australian Research Counc il and DETY A i s gratefully ac kno wledged. 1 W e a lso show that the common par t of the tw o arg ument s follows from a genera l result ab out monoidal functors; since the individua l results a r e so easy to pr ov e, how ever, we g ive them first, in Sections 1 and 2 resp ectively , b efore turning to the gener al result in Section 3. 1 Non-canonical distributivit y isomorphisms This section in volv es, as in the in tro duction, a categor y D with finite pro ducts and copro ducts and a natural family o f isomo rphisms X × Y + X × Z ∼ = X × ( Y + Z ). First we show, in the following Lemma, that such a D will b e distributive if X × 0 ∼ = 0. Later on, we shall see tha t this Lemma follows fro m a more general result ab out co pro duct-preser ving functors due to Cacca mo and Winskel; and that this in tur n is a sp ecia l case of a still more g eneral result ab out mono idal functors: this is o ur Theore m 6 b elow. Lemma 1 Su pp o se t hat as ab ove t hat we have natur al isomorphisms X × Y + X × Z ψ / / X × ( Y + Z ) and that X × 0 ∼ = 0 . Then the c ate go ry D is distributive. Proof: If X × 0 ∼ = 0, then ϕ X,Y , 0 gives an isomor phism X × Y + X × 0 ∼ = X × ( Y + 0), which we can rega r d as simply b eing an iso mo rphism X × Y ∼ = X × Y . By naturality , the dia gram X × Y ψ X,Y , 0 / / i   X × Y X × i   X × Y + X × Z ψ X,Y ,Z / / X × ( Y + Z ) commutes, a nd similar ly w e hav e a commutativ e diagra m X × Z ψ X, 0 ,Z / / i   X × Y X × i   X × Y + X × Z ψ X,Y ,Z / / X × ( Y + Z ) and now combining these we get a commutativ e diagr a m X × Y + X × Z ψ X,Y , 0 + ψ X, 0 ,Z / / X × Y + X × Z . δ X,Y ,Z   X × Y + X × Z ψ X,Y ,Z / / X × ( Y + Z ) In this last dia gram, the ψ ’,s ar e all inv ertible, hence so is δ .  Recall tha t a n ob ject T is called subterminal if for any ob ject X there is at most one morphism from X to T . (If, as here, a terminal ob ject ex ists, this is equiv alent to saying that the unique map T → 1 is a monomorphism.) Thu s to pro ve our r esult ab out non-canonical distr ibutivit y is o morphisms, w e must show that the as- sumption that X × 0 ∼ = 0 made in the Lemma is unnecessary . The remainder of this sec tio n will b e devoted to doing s o. 2 Prop ositio n 2 The pr o duct 0 × 0 is initial, and so 0 is subterminal. Proof: F or the firs t par t, observe tha t ψ 0 , 0 , 1 gives a n isomo rphism 0 × 0 + 0 × 1 ∼ = 0 × (0 + 1), and that 0 + 1 ∼ = 1 a nd 0 × 1 ∼ = 0. F o r the second, we hav e an isomorphism 0 ∼ = 0 × 0, and since 0 is initial, this can only b e the diagona l ∆ : 0 → 0 × 0. Thus an y mor phism X → 0 × 0 factor iz e s through the diag onal, and so any tw o mor phisms X → 0 ar e equal.  Next we consider the sp ecial ca se where D is p ointed (0 = 1); ultimately we shall reduce the general case to this. In a p ointed catego ry , e very ob ject has a (unique) morphism into 0; but in a distributive categor y , any morphism in to 0 is invertible [2, Prop os ition 3.4]. It follows tha t a ny categor y which is pointed and distributive is equiv a le nt to the terminal categ o ry 1. O ur nex t result shows that the same conclusion holds under the assumption of p ointedness and a non-cano nical distributivity isomorphism. Prop ositio n 3 If D is p ointe d then D is e quivalent to the t erm inal c ate gory 1. Proof: T aking Y = Z = 1 gives a natural fa mily θ X = ψ X, 1 , 1 : X + X ∼ = X . By natur ality , the diagr am X + X + X + X θ X + θ X / / θ X + X   X + X θ X   X + X θ X / / X commutes, a nd now since θ X is inv ertible θ X + θ X = θ X + X . The diagra m X + X ∇   θ X / / i X + X ( ( P P P P P P P P P P P P X i # # G G G G G G G G G X + X + X + X θ X + θ X θ X + X / / ∇ + ∇   X + X ∇   X i ( ( P P P P P P P P P P P P P P X + X θ X / / X commutes, wher e i X + X denotes the injection of the firs t t wo co pies o f X into X + X + X + X . But now θ X = ∇ iθ X = θ X i ∇ is inv ertible, so ∇ : X + X → X is a mono mo rphism; s ince it a lso has a section i (and j ) it is in vertible. This prov es that any t w o maps X → Y m ust b e equa l. On the other hand, there is alwa ys at least one such map, since D is p o inted; th us there is exactly one, a nd so X ∼ = 0. Since X w as a rbitrary , the res ult follows.  Theorem 4 If D is a c ate gory with finite pr o duct s and c opr o ducts, and with a natu r al family ψ X,Y ,Z : X × Y + X × Z ∼ = X × ( Y + Z ) of isomorphisms, then D is distributive. Proof: By Lemma 1, it will suffice to show that X × 0 ∼ = 0. Since we have the pro jection X × 0 → 0 , and the comp osite 0 → X × 0 → 0 is certainly the identit y , we need only show tha t the o ther co mpo site e : X × 0 → 0 → X × 0 is the identit y . This is a n endomorphism in the slice categ ory D / 0. So if D / 0 is trivial, then this comp osite e will b e the identit y , and X × 0 will b e isomor phic to 0. Since 0 is s ubterminal, the pro jection D / 0 → D is fully faithful, and prese rves finite pro ducts a s well as copro ducts. Thu s the isomorphis ms ψ X,Y ,Z restrict to D / 0, th us equipping D / 0 with non-standa rd distributivity isomorphisms. By P rop osition 3, D / 0 is tr ivial, and s o D is distributive.  3 2 Non-canonical semi-additivit y isomorphisms W e now give an analo gous result for semi-additivity . An interesting featur e is that this do e s not require us to assume that the catego ry is p ointed, although that will of cour se b e a conseq uence. Theorem 5 If A is a c ate gory with fi n ite pr o ducts and c opr o ducts and with a natur al family ψ Y ,Z : Y + Z ∼ = Y × Z of isomorphisms, then A is semi-additive. Proof: T a k ing Y = 1 and Z = 0 gives a n iso mo rphism ψ 1 , 0 : 1 ∼ = 1 × 0; co mpo sing with the pro jection 1 × 0 → 0 gives a morphism 1 → 0. By uniq ue ne s s of morphis ms into 1 a nd out of 0, this is inv erse to the unique map 0 → 1, and so A is p ointed. T aking one of Y and Z to b e 0 gives natural isomorphisms ψ Y , 0 : Y ∼ = Y and ψ 0 ,Z : Z ∼ = Z . By naturalit y of the ψ Y ,Z , the diagr a ms Y ψ Y , 0 / / i   Y ( Y 0 )   Z ψ 0 ,Z / / j   Z ( 0 Z )   Y + Z ψ Y ,Z / / Y × Z Y + Z ψ Y ,Z / / Y × Z commute, and so a ls o Y + Z ψ Y , 0 + ψ 0 ,Z / / Y + Z α Y ,Z   Y + Z ψ Y ,Z / / Y × Z commutes. Just as in the pro of of the le mma , ψ Y , 0 + ψ 0 ,Z and ψ Y ,Z are inv ertible, hence so is α Y ,Z .  3 Non-canonical isomorphisms for monoidal func tors In this section we prove a general r esult on monoidal functors, whic h could b e us ed in the proof of b o th of the other theor ems. Recall that if A and B be monoidal categorie s , a mono idal functor F : A → B consists of a functor (also called F ) equipp ed with maps ϕ Y ,Z : F Y ⊗ F Z → F ( Y ⊗ Z ) and ϕ 0 : I → F I which need not b e inv er tible, but which ar e natural a nd c oherent [4]. The mono idal functor is said to b e str ong if ϕ Y ,Z and ϕ 0 are inv ertible, and normal if ϕ 0 is inv ertible. Given such an F and another monoidal functor G : A → B with str ucture ma ps ψ X,Y and ψ 0 , a natural tra nsformation α : F → G is monoidal if the diag rams F Y ⊗ F Z α X ⊗ α Y / / ϕ X,Y   GY ⊗ GZ ψ X,Y   I ϕ 0 / / ψ 0   ? ? ? ? ? ? ? ? F I αI   F ( Y ⊗ Z ) α X ⊗ Y / / G ( Y ⊗ Z ) GI commute. Recall further [5] that if C is braided monoidal, then the functor ⊗ : C × C → C is strong monoidal, with structure maps W ⊗ X ⊗ Y ⊗ Z W ⊗ γ ⊗ D / / W ⊗ Y ⊗ X ⊗ Z I λ / / I ⊗ I where γ denotes the bra iding and λ the canonical isomo rphism. 4 Theorem 6 L et A and B b e br aide d monoidal c ate gories, and F = ( F, ϕ, ϕ 0 ) : A → B a normal monoidal functor (so that ϕ 0 is invertible). S u pp ose further t hat we have a monoidal isomorphism A × A F × F / / ⊗   B × B ⊗         ψ A F / / B Then ϕ is invertible, and so F is st r ong monoidal. Proof: The fact that ψ is mono idal means in particular that the diagr am F W ⊗ F X ⊗ F Y ⊗ F Z ψ W,X ⊗ ψ Y ,Z / / 1 ⊗ γ ⊗ 1   F ( W ⊗ X ) ⊗ F ( Y ⊗ Z ) ϕ W ⊗ X,Y ⊗ Z   F W ⊗ F Y ⊗ F X ⊗ F Z ϕ W,Y ⊗ ϕ X,Z   F ( W ⊗ X ⊗ Y ⊗ Z ) F (1 ⊗ γ ⊗ 1)   F ( W ⊗ Y ) ⊗ F ( X ⊗ Z ) ψ W ⊗ Y ,X ⊗ Z / / F ( W ⊗ Y ⊗ X ⊗ Z ) commutes. T a king X = Y = I and twice using the isomor phism ϕ 0 gives commutativit y of F W ⊗ F Z 1 ⊗ ϕ 0 ⊗ ϕ 0 ⊗ 1   S S S S S S S S S S S S S S S S S S S S S S S S S S S S F W ⊗ F I ⊗ F I ⊗ F Z ψ W,I ⊗ ψ I ,Z / / ϕ W,I ⊗ ϕ I ,Z   F W ⊗ F Z ϕ W,Z   F W ⊗ F Z ψ W,Z / / F ( W ⊗ Z ) in which all a rrows except ϕ W ,Z are inv ertible; thus ϕ W ,Z to o is inv ertible.  Remark 7 In the pro o f o f Theorem 6, we have use d rather less than was a ssumed in the statement. F or example, we do no t use the nullary part of the a ssumption that the natur al transfo r mation is monoida l. The following corolla ry a ppe ared (in dual form) as [1, Theor em 3.3]: Corollary 8 (Caccamo-Winsk el) L et A and B b e c ate gories with finite c opr o ducts, and F : A → B a functor which pr eserves the initial obje ct. If ther e is a natur al family of isomorphisms F X + F Y ψ X,Y / / F ( X + Y ) then F pr eserves fin ite c opr o ducts. Proof: In this case F ha s a unique monoida l structure, and ψ is alwa ys monoidal.  In particular if D has finite pr o ducts and copro ducts , we ma y apply the Coro lla ry to the functor X × − : D → D a nd recov er Lemma 1. Section 2 in volves the case where th e categor ies A and B are the same, but the monoidal structure on A is ca rtesian and that on B is co cartesian. The functor F is the identit y . One proves 0 → 1 is inv ertible, a s in the pro of of Theorem 5; and then the ide ntit y 1 : A → A has a unique norma l monoidal structure, with binary par t precisely the canonical morphism α : Y + Z → Y × Z . F urthermo re, any natural isomor phism ψ Y ,Z : Y + Z ∼ = Y × Z is monoidal. 5 References [1] Mario Cac camo and Glynn Winskel. Limit preser v ation from na tur ality . In Pr o c e e dings of t he 10th Confer enc e on Cate gory The ory in Computer Scienc e (CTCS 2004) , volume 1 22 o f Ele ctr on. Notes The or. Comput. Sci. , pag es 3–22, Amsterda m, 2005 . Els evier. [2] Aurelio Carb oni, Stephen Lack, and R. F. C. W alters. In tro duction to e xtensive and distr ibutiv e cate- gories. J. Pur e Appl. Algebr a , 84(2):1 45–15 8, 1993. [3] J. R. B. Co ckett. Intro duction to distributive ca tegories. Math. Structu r es Comput. Sci. , 3(3 ):2 77–30 7, 1993. [4] Samuel E ilenberg and G. Max Ke lly . Clo sed categor ies. In Pr o c. Conf. Cate goric al A lgebr a (L a Jol la, Calif. , 1965) , page s 421 – 562. Spr inger, New Y ork, 1 966. [5] Andr´ e Joy al and Ros s Street. Braided tenso r catego ries. A dv. Math. , 102(1):2 0–78, 1993. [6] Saunders Ma c Lane. Cate gories for the working mathematician . Springe r-V erla g, New Y or k, 197 1. 6