aa r X i v : . [ m a t h . C T ] D ec Non-canonical isomorphisms
Stephen Lack ∗ School of Computing and MathematicsUniversity of Western SydneyLocked Bag 1797 Penrith South DC NSW 1797Australiaemail: [email protected]
Abstract
We give two examples of categorical axioms asserting that a canonically defined natural transforma-tion is invertible where the invertibility of any natural transformation implies that the canonical one isinvertible. The first example is distributive categories, the second (semi-)additive ones. We show thateach follows from a general result about monoidal functors.
In any category D with finite products and coproducts there is a natural family of maps X × Y + X × Z δ X,Y,Z / / X × ( Y + Z )induced, via the universal property of the coproduct X × Y + X × Z , by the morphisms X × i and X × j ,where i and j denote the coproduct injections of Y + Z . Such a D is said to be distributive [2, 3] if thecanonical maps are invertible; in other words, if the functor X × − : D → D preserves binary coproducts,for all objects X . As observed by Cockett [3], it follows that X × ∼ = 0, so that X × − in fact preservesfinite coproducts.Claudio Pisani has asked whether the existence of any natural family of isomorphisms X × Y + X × Z ψ X,Y,Z / / X × ( Y + Z )might imply that D is distributive. Such ψ are the non-canonical isomorphisms of the title. He suggestedthat this was probably not the case, and this was also my immediate reaction. But in fact it is true! This isthe first result of the paper.The second result is an analogue for semi-additive categories. Recall that a category is pointed when ithas an initial object which is also terminal (1 = 0), and that for any any two objects Y and Z in a pointedcategory there is a unique morphism from Y to Z which factorizes through the zero object; this morphismis called 0 Y,Z or just 0. If the category has finite products and coproducts, then there is a natural family ofmorphisms Y + Z α Y,Z / / Y × Z induced by the identities on Y and Z and the zero morphisms 0 : Y → Z and 0 : Z → Y . The categoryis semi-additive when these α Y,Z are invertible [6, VII.2]. A semi-additive category admits a canonicalenrichment over commutative monoids; conversely, any category enriched over commutative monoids whichhas either finite products or finite coproducts is semi-additive. Our result for semi-additive categories assertsonce again that the existence of any natural isomorphism Y + Z ∼ = Y × Z implies that the category is semi-additive. ∗ The support of the Australian Research Council and DETYA is gratefully acknowledged.
1e also show that the common part of the two arguments follows from a general result about monoidalfunctors; since the individual results are so easy to prove, however, we give them first, in Sections 1 and 2respectively, before turning to the general result in Section 3.
This section involves, as in the introduction, a category D with finite products and coproducts and a naturalfamily of isomorphisms X × Y + X × Z ∼ = X × ( Y + Z ). First we show, in the following Lemma, that sucha D will be distributive if X × ∼ = 0. Later on, we shall see that this Lemma follows from a more generalresult about coproduct-preserving functors due to Caccamo and Winskel; and that this in turn is a specialcase of a still more general result about monoidal functors: this is our Theorem 6 below. Lemma 1
Suppose that as above that we have natural isomorphisms X × Y + X × Z ψ / / X × ( Y + Z ) and that X × ∼ = 0 . Then the category D is distributive. Proof: If X × ∼ = 0, then ϕ X,Y, gives an isomorphism X × Y + X × ∼ = X × ( Y + 0), which we can regardas simply being an isomorphism X × Y ∼ = X × Y . By naturality, the diagram X × Y ψ X,Y, / / i (cid:15) (cid:15) X × Y X × i (cid:15) (cid:15) X × Y + X × Z ψ X,Y,Z / / X × ( Y + Z )commutes, and similarly we have a commutative diagram X × Z ψ X, ,Z / / i (cid:15) (cid:15) X × Y X × i (cid:15) (cid:15) X × Y + X × Z ψ X,Y,Z / / X × ( Y + Z )and now combining these we get a commutative diagram X × Y + X × Z ψ X,Y, + ψ X, ,Z / / X × Y + X × Z. δ X,Y,Z (cid:15) (cid:15) X × Y + X × Z ψ X,Y,Z / / X × ( Y + Z )In this last diagram, the ψ ’,s are all invertible, hence so is δ . (cid:3) Recall that an object T is called subterminal if for any object X there is at most one morphism from X to T . (If, as here, a terminal object exists, this is equivalent to saying that the unique map T → X × ∼ = 0 made in the Lemma is unnecessary. The remainder of this section will be devotedto doing so. 2 roposition 2 The product × is initial, and so is subterminal. Proof:
For the first part, observe that ψ , , gives an isomorphism 0 × × ∼ = 0 × (0 + 1), and that0 + 1 ∼ = 1 and 0 × ∼ = 0. For the second, we have an isomorphism 0 ∼ = 0 ×
0, and since 0 is initial, this canonly be the diagonal ∆ : 0 → ×
0. Thus any morphism X → × X → (cid:3) Next we consider the special case where D is pointed (0 = 1); ultimately we shall reduce the general caseto this. In a pointed category, every object has a (unique) morphism into 0; but in a distributive category,any morphism into 0 is invertible [2, Proposition 3.4]. It follows that any category which is pointed anddistributive is equivalent to the terminal category 1. Our next result shows that the same conclusion holdsunder the assumption of pointedness and a non-canonical distributivity isomorphism. Proposition 3 If D is pointed then D is equivalent to the terminal category 1. Proof:
Taking Y = Z = 1 gives a natural family θ X = ψ X, , : X + X ∼ = X . By naturality, the diagram X + X + X + X θ X + θ X / / θ X + X (cid:15) (cid:15) X + X θ X (cid:15) (cid:15) X + X θ X / / X commutes, and now since θ X is invertible θ X + θ X = θ X + X . The diagram X + X ∇ (cid:15) (cid:15) θ X / / i X + X ( ( PPPPPPPPPPPP X i GGGGGGGGG X + X + X + X θ X + θ X θ X + X / / ∇ + ∇ (cid:15) (cid:15) X + X ∇ (cid:15) (cid:15) X i ( ( PPPPPPPPPPPPPP X + X θ X / / X commutes, where i X + X denotes the injection of the first two copies of X into X + X + X + X . But now θ X = ∇ iθ X = θ X i ∇ is invertible, so ∇ : X + X → X is a monomorphism; since it also has a section i (and j ) it is invertible. This proves that any two maps X → Y must be equal. On the other hand, there is alwaysat least one such map, since D is pointed; thus there is exactly one, and so X ∼ = 0. Since X was arbitrary,the result follows. (cid:3) Theorem 4 If D is a category with finite products and coproducts, and with a natural 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. Since we have the projection X × → → X × → e : X × → → X × D /
0. So if D / e will be the identity, and X × D / → D is fully faithful, and preserves finite products as wellas coproducts. Thus the isomorphisms ψ X,Y,Z restrict to D /
0, thus equipping D / D / D is distributive. (cid:3) Non-canonical semi-additivity isomorphisms
We now give an analogous result for semi-additivity. An interesting feature is that this does not require usto assume that the category is pointed, although that will of course be a consequence.
Theorem 5 If A is a category with finite products and coproducts and with a natural family ψ Y,Z : Y + Z ∼ = Y × Z of isomorphisms, then A is semi-additive. Proof:
Taking Y = 1 and Z = 0 gives an isomorphism ψ , : 1 ∼ = 1 ×
0; composing with the projection1 × → →
0. By uniqueness of morphisms into 1 and out of 0, this is inverse to theunique map 0 →
1, and so A is pointed.Taking one of Y and Z to be 0 gives natural isomorphisms ψ Y, : Y ∼ = Y and ψ ,Z : Z ∼ = Z . By naturalityof the ψ Y,Z , the diagrams Y ψ Y, / / i (cid:15) (cid:15) Y ( Y ) (cid:15) (cid:15) Z ψ ,Z / / j (cid:15) (cid:15) Z ( Z ) (cid:15) (cid:15) Y + Z ψ Y,Z / / Y × Z Y + Z ψ Y,Z / / Y × Z commute, and so also Y + Z ψ Y, + ψ ,Z / / Y + Z α Y,Z (cid:15) (cid:15) Y + Z ψ Y,Z / / Y × Z commutes. Just as in the proof of the lemma, ψ Y, + ψ ,Z and ψ Y,Z are invertible, hence so is α Y,Z . (cid:3) In this section we prove a general result on monoidal functors, which could be used in the proof of bothof the other theorems. Recall that if A and B be monoidal categories, a monoidal functor F : A → B consists of a functor (also called F ) equipped with maps ϕ Y,Z : F Y ⊗ F Z → F ( Y ⊗ Z ) and ϕ : I → F I which need not be invertible, but which are natural and coherent [4]. The monoidal functor is said to be strong if ϕ Y,Z and ϕ are invertible, and normal if ϕ is invertible. Given such an F and another monoidalfunctor G : A → B with structure maps ψ X,Y and ψ , a natural transformation α : F → G is monoidal ifthe diagrams F Y ⊗ F Z α X ⊗ α Y / / ϕ X,Y (cid:15) (cid:15) GY ⊗ GZ ψ X,Y (cid:15) (cid:15) I ϕ / / ψ (cid:31) (cid:31) ???????? F I αI (cid:15) (cid:15) 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 strongmonoidal, with structure maps W ⊗ X ⊗ Y ⊗ Z W ⊗ γ ⊗ D / / W ⊗ Y ⊗ X ⊗ Z I λ / / I ⊗ I where γ denotes the braiding and λ the canonical isomorphism.4 heorem 6 Let A and B be braided monoidal categories, and F = ( F, ϕ, ϕ ) : A → B a normal monoidalfunctor (so that ϕ is invertible). Suppose further that we have a monoidal isomorphism A × A F × F / / ⊗ (cid:15) (cid:15) B × B ⊗ (cid:15) (cid:15) (cid:31)(cid:31) (cid:31)(cid:31) (cid:11) (cid:19) ψ A F / / B Then ϕ is invertible, and so F is strong monoidal. Proof:
The fact that ψ is monoidal means in particular that the diagram F W ⊗ F X ⊗ F Y ⊗ F Z ψ W,X ⊗ ψ Y,Z / / ⊗ γ ⊗ (cid:15) (cid:15) F ( W ⊗ X ) ⊗ F ( Y ⊗ Z ) ϕ W ⊗ X,Y ⊗ Z (cid:15) (cid:15) F W ⊗ F Y ⊗ F X ⊗ F Z ϕ W,Y ⊗ ϕ X,Z (cid:15) (cid:15) F ( W ⊗ X ⊗ Y ⊗ Z ) F (1 ⊗ γ ⊗ (cid:15) (cid:15) F ( W ⊗ Y ) ⊗ F ( X ⊗ Z ) ψ W ⊗ Y,X ⊗ Z / / F ( W ⊗ Y ⊗ X ⊗ Z )commutes. Taking X = Y = I and twice using the isomorphism ϕ gives commutativity of F W ⊗ F Z ⊗ ϕ ⊗ ϕ ⊗ (cid:15) (cid:15) SSSSSSSSSSSSSS SSSSSSSSSSSSSS
F W ⊗ F I ⊗ F I ⊗ F Z ψ W,I ⊗ ψ I,Z / / ϕ W,I ⊗ ϕ I,Z (cid:15) (cid:15)
F W ⊗ F Z ϕ W,Z (cid:15) (cid:15)
F W ⊗ F Z ψ W,Z / / F ( W ⊗ Z )in which all arrows except ϕ W,Z are invertible; thus ϕ W,Z too is invertible. (cid:3)
Remark 7
In the proof of Theorem 6, we have used rather less than was assumed in the statement. Forexample, we do not use the nullary part of the assumption that the natural transformation is monoidal.The following corollary appeared (in dual form) as [1, Theorem 3.3]:
Corollary 8 (Caccamo-Winskel)
Let A and B be categories with finite coproducts, and F : A → B afunctor which preserves the initial object. If there is a natural family of isomorphisms F X + F Y ψ X,Y / / F ( X + Y ) then F preserves finite coproducts. Proof:
In this case F has a unique monoidal structure, and ψ is always monoidal. (cid:3) In particular if D has finite products and coproducts, we may apply the Corollary to the functor X × − : D → D and recover Lemma 1.Section 2 involves the case where the categories A and B are the same, but the monoidal structure on A is cartesian and that on B is cocartesian. The functor F is the identity. One proves 0 → A → A has a unique normal monoidal structure, withbinary part precisely the canonical morphism α : Y + Z → Y × Z . Furthermore, any natural isomorphism ψ Y,Z : Y + Z ∼ = Y × Z is monoidal. 5 eferences [1] Mario Caccamo and Glynn Winskel. Limit preservation from naturality. In Proceedings of the 10thConference on Category Theory in Computer Science (CTCS 2004) , volume 122 of
Electron. NotesTheor. Comput. Sci. , pages 3–22, Amsterdam, 2005. Elsevier.[2] Aurelio Carboni, Stephen Lack, and R. F. C. Walters. Introduction to extensive and distributive cate-gories.
J. Pure Appl. Algebra , 84(2):145–158, 1993.[3] J. R. B. Cockett. Introduction to distributive categories.
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Proc. Conf. Categorical Algebra (La Jolla,Calif., 1965) , pages 421–562. Springer, New York, 1966.[5] Andr´e Joyal and Ross Street. Braided tensor categories.
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