MacLane s coherence theorem expressed as a word problem Paul-André Melliès Preuves, Programmes, Systèmes CNRS UMR-7126, Université Paris 7 ÑÐÐ ÔÔ ºÙ ÙºÖ DRAFT In this draft, we reduce the coherence theorem for braided monoidal categories to the resolution of a word problem, and the construction of a category of fractions. The technique explicates the combinatorial nature of that particular coherence theorem. 1. Introduction Let denote the category of braids and Å any braided monoidal category. Let Ö Åµ denote the category of strong braided monoidal functors from to Å and monoidal natural transformations between them. All definitions are recalled in section 5. The coherence theorem for braided monoidal categories is usually stated as follows: Theorem 1. The categories Ö Åµ and Å are equivalent. We prove the coherence theorem in four independent steps. To that purpose, we introduce the category whose objects are binary trees of Å, with leaves ¼ and ½, whose morphisms are sequences of rewriting steps «µ, µ, µ, µ, «¹½ µ, ¹½ µ, ¹½ µ, ¹½ µ, quotiented by the laws of braided monoidal categories. 1.1. First step Consider a braided monoidal category Å, and the category ËÖ Åµ of strict braided monoidal functors and monoidal natural transformations from to Å. We prove that Å and ËÖ Åµ are isomorphic. More precisely, we prove that the functor ¹ ËÖ Åµ Å which associates to a strict natural transformation the morphism ¾ ¼ µ ¾ ¼ µ ½ ½µ ½µ in Å, is reversible. To every object in the category Å, we associate a strict braided monoidal functor defined as follows: ¼µ ½µ Å µ µ Å µ 1
and «µ «µ µ µ This defines a functor because the image of a commutative diagrams in is always commutative in Å. The functor is strict braided monoidal by construction. To every morphism in the category Å, we associate a family of morphisms indexed by objects of, as follows: ¼ id ½ Å Å The family defines a natural transformation because the maps «,,, and are supposed natural in Å. The natural transformation is monoidal by construction. The map ¹ is functorial from Å to ËÖ Åµ and it is easy to see that it defines the inverse of the evaluation functor ¹ ËÖ Åµ Å. We conclude. 2 1.2. Second step We introduce the notion of contractible category. A category is contractible when: there exists at most one morphism between two objects of, every morphism of is reversible. In other words, a category is contractible when it is a preorder category and a groupoid. Consider a contractible subcategory of a category, full on objects. Write ³ Ó for two objects and of, when there exists a morphism in. Write ³ ÑÔ for two morphisms and when there exists two morphisms and in making the following diagram commute: The relations ³ Ó and ³ ÑÔ are equivalence relations on objects and morphisms of, respectively. We call orbit of an object or morphism its equivalence class wrt. ³ Ó or ³ ÑÔ. The quotient category is defined as follows: its objects are the orbits of objects, its morphisms Å Æ are the orbits of maps, its identities and composition are induced from. The two categories and are equivalent. Indeed, there exists a projection functor which maps every morphism Å Æ to its orbit, and an embedding functor depending on the choice, for every orbit wrt. ³ Ó, of an object in that class. Clearly, id, and there exists a natural transformation id.
3 1.3. Third step Consider a braided monoidal category Å, and the hom-category Ö Åµ of strong braided monoidal functors and monoidal natural transformations from to Å. We prove that the categories Ö Åµ and ËÖ Åµ are equivalent. Define as the subcategory of Ö Åµ containing all reversible maps ¾ ¼ µ ¾ ¼ µ such that ½ id ½µ. Every identity of Ö Åµ is element of. The main observation is that the category is contractible. Indeed, once the equality ½ id ½µ is fixed, the component morphisms ¼ and Å of the natural transformation are uniquely determined by the commutative diagrams µ Å µ ¾ µ Å µ ¼ ¼µ Å µ Å µ ¾ µ Å Å µ ¼ ¼ ¼µ Moreover, every identity of Ö Åµ is element of. We may therefore consider the category Ö Åµ, which we know is equivalent to Ö Åµ. We prove that the categories Ö Åµ and ËÖ Åµ are isomorphic. We need to prove that for every strong monoidal functor ¾ ¼ µ from to Å, there exists a strict monoidal functor, and a map ¾ ¼ µ in. Consider a family of isomorphisms ¼ ¼ ½ id Å ¾ µ Æ Å µ indexed by objects of. Then, associate to every morphism in, the morphism µ ¹½ Æ µ Æ in Å. A close look at the diagram shows that this defines a strict braided monoidal functor Å and a monoidal natural transformation in. We conclude. 1.4. Fourth step After steps 1. 2. and 3. the proof of coherence reduces to the comparison of two free categories : the formal braided monoidal category of Å-trees and rewriting paths, quotiented by the commutativities of braided monoidal categories, the geometric braided monoidal category of braids. This is the most interesting and difficult part of the proof, in fact the first and only time combinatorics plays a role. We prove 1. that the subcategory of consisting only of «, and maps, and their inverses, is contractible, 2. that the quotient category is isomorphic to the braid category, and equivalent to through strong braided monoidal functors. The first and second assertions are established in sections 3 and 4 respectively.
4 1.5. Conclusion The categories Ö Åµ and Å are equivalent by steps 1. 2. and 3. and the categories Ö Åµ and Ö Åµ are equivalent by step 4. This proves theorem 1. 2. A word problem Let be the category whose objects are Å-trees with leaves ¼ and ½, whose morphisms are sequences of «µ, µ and µ, quotiented by the monoidal law (1) the commutativity laws (5) (6) and the additional commutativity triangles (2). Å Å Å ¼ Å ¼ Å ¼ Å Å ¼ ¼ Å ¼ for ¼ (1) ¼ Å Å µ Å «Å µ Å Å Å Å Å µ Å Å Theorem 2. The category verifies the three following properties: «Å µ Å Å the category has pushouts, every morphism is epi, from every object, there exists a morphism to a normal object. (2) Proof. The proof appears in (Melliès, 2002). It is motivated by works of rewriters like Lévy (Lévy, 1978; Huet and Lévy, 1979) and algebraists like Dehornoy (Dehornoy, 1998). Definition 1. An object in a category is called normal when id is the only morphism outgoing from. 3. Calculus of fractions 3.1. Definition For every category and class of morphisms in, there exists a universal solution to the problem of reversing the morphisms in. More explicitly, there exists a category ¹½ and a functor such that: È ¹½ the functor È maps every morphism of to reversible morphism, if a functor Å makes every morphism of reversible, then factors as Æ È for a unique functor ¹½ Å.
The class is a calculus of left fraction in the category, see (Gabriel and Zisman, 1967), when contains the identities of, is closed under composition, each diagram where ¾ may be completed into a commutative diagram each commutative diagram Ø where Ø ¾ ¼ ¼ where ¾ may be completed into a commutative diagram 5 Ø ¼ where Ø ¾ The property of left fraction calculus enables an elegant definition of the category ¹½ and functor È ¹½. The category ¹½ is defined as follows. Its objects are the objects of, and its morphisms are the equivalence classes of pairs µ of morphisms of where ¾ and ¼ Ø and two morphisms ¼ ¼¼ ¼ under the equivalence relation which identifies two pairs when there exists a pair ¼¼ Ù a commutative diagram ¼ ¼¼ Ù where Ù ¾ Ø ¼ ¼ ¼ forming 3.2. Application By theorem 2, the class is a calculus of fraction in the category. But we can prove a bit more. A consequence of the pushout property, and definition of a normal object, is that any two morphisms and to a normal object are equal in. So, consider two morphisms in ¹½, represented as pairs in Ø
Any two morphisms and to a normal object in, make the following diagram commute in : We obtain ¼ Ù Ø ¼ Theorem 3. The category ¹½ is contractible. 6 4. From word problem to coherence theorem In section 3 we have proved that the category ¹½ consisting of «µ, µ µ steps and their inverse, is contractible. In step 1. we prove that ¹½ is a subcategory of, full on objects. In step 2. we prove that the category ¹½ is braided monoidal, and embeds in through strong braided monoidal functors, in step 3. we prove that the category ¹½ is isomorphic to the usual braid category. 4.1. First step The two triangles (2) are commutative in, as in every monoidal category, see exercise VII.Ü1.1. in (Mac Lane, 1991). This ensures the existence of a functor. By definition, this functor induces a functor ¹½. This functor is injective on objects. It is also faithful for the obvious reason that ¹½ is contractible. Thus, ¹½ defines an embedding of the contractible category ¹½ into. The embedding is full on objects. This enables to consider the quotient category ¹½. 4.2. Second step The category ¹½ becomes strict monoidal with the following definition. Let and be two morphisms of ¹½ represented by two morphisms ¼ ¼ ¼ and ¼ ¼ ¼ in the category. Define the tensor product Å Å Å as the projection of the tensor product ¼ Å ¼ ¼ Å ¼ ¼ Å ¼ and the unit object as the orbit of the unit in. Correctness follows from monoidality of Å in. Monoidality of Å is not difficult to prove, and diagrams (5) (6) are obvious. The category ¹½ is braided monoidal. Let and be two objects of the quotient category ¹½ represented by two objects ¼ and ¼ in the category. Define the braiding µ Å Å in the quotient category as the orbit of the braiding
¼ ¼ µ ¼ Å ¼. Correctness of the definition and naturality of follow from naturality of in. Diagrams (7) (8) (9) follow from the definition of in ¹½. Moreover, the projection functor ¹½ is strict braided monoidal. Diagrams (10) (11) are trivial, and diagram (12) follows from the definition of the braiding in ¹½. the embedding functor ¾ ¼ µ is strong braided monoidal, with the morphisms ¼ ¼ µ and ¾ µ µ Å µ Å µ determined as morphisms of. Diagrams (10) (11) hold because every maps are in ¹½, and diagram (12) holds because of the definition of the braiding in ¹½. 7 4.3. Third step We prove that ¹½ and are isomorphic categories. The objects of ¹½ are the natural numbers. The morphisms of ¹½ are sequences of Ñ Òµ and ¹½ Ñ Òµ steps, modulo monoidality, and commutativity of the triangles induced by (8) (9): Ñ Å Ò Å Ô ÑÅ ÒÔµ ÑÅÒÔµ Ô Å Ñ Å Ò Ñ Å Ò Å Ô ÑÒµÅÔ ÑÒÅÔµ Ò Å Ô Å Ñ Ñ Å Ô Å Ò ÑÔµÅÒ Ô Å Ñ Å Ò Ò Å Ñ Å Ô ÒÅ ÑÔµ Ò Å Ô Å Ñ (3) We construct a strict monoidal functor from ¹½ to by interpreting each Ñ Òµ by the braiding Ñ Òµ Ñ Ò Ò Ñ in consisting in permuting Ò braids over Ñ braids. The functor is full. The only difficult point to prove is that it is faithfull. This reduces to proving that the hexagonal diagram generating equality of braids in commutes (already) in the category ¹½ : Ò Å Ñ Å Ô ÒÅ ÑÔµ Ò Å Ô Å Ñ ÑÒµÅÔ ÒÔµÅÑ Ñ Å Ò Å Ô Ô Å Ò Å Ñ ÑÅ ÒÔµ ÔÅ ÑÒµ Ñ Å Ô Å Ò ÑÔµÅÒ Ô Å Ñ Å Ò The diagram is a nice geometric consequence of the commutative triangles 3, and monoidality of ¹½. We conclude that ¹½ and the category of braids are isomorphic. (4) References P. Dehornoy. On completeness of word reversing. Discrete mathematics, to appear.
8 P. Gabriel and M. Zisman. Calculus of Fractions and Homotopy Theory. Ergebnisse des Mathematik. Springer-Verlag, 1967. G. Huet, J.-J. Lévy. Call by Need Computations in Non-Ambiguous Linear Term Rewriting Systems. Rapport de recherche INRIA 359, 1979. Reprinted as: Computations in orthogonal rewriting systems. In J.-L. Lassez and G. D. Plotkin, editors, Computational Logic; Essays in Honor of Alan Robinson, pages 394 443. MIT Press, 1991. A. Joyal and R. Street. Braided tensor categories. Adv. Math. 102, 20-78, 1993. C. Kassel. Quantum groups. Graduate Texts in Mathematics, Springer-Verlag, 1991. J.-J. Lévy. Réductions correctes et optimales dans le -calcul. Thèse de Doctorat d Etat, Université Paris VII, 1978. S. Mac Lane. Categories for the working mathematician. Second edition. Graduate Texts in Mathematics, Springer-Verlag, 1991. P-A. Melliès. Residual Theory Revisited. Manuscript, january 2002. 5. Appendix: definitions of braided monoidal categories, functors and natural transformations 5.1. Monoidal category A monoidal category Å is a category with a bifunctor Å Å Å Å which is associative up to a natural isomorphism «Å Å µ Å µ Å equipped with an element, which is a unit up to natural isomorphisms Å Å These maps must make Mac Lane s famous pentagon commute Å Å Å µµ Å«Å Å µ Å µ as well as the triangle: «Å Å µµ Å Å Å µ Å Å «««Å Å µ Å Å Å A monoidal category is strict when all «, and are identities. Å µ Å Å µ «Å µ Å µ Å (5) (6) 5.2. Braided monoidal category A braiding for a monoidal category Å consists of a family of isomorphisms Å Å
9 natural in and, which satisfy the commutativity Å and makes both following hexagons commute: Å Å µ Å Å Å µ «Å µ Å «Å µ Å Å Å Å Å µ «Å µ Å (7) (8) Å µ Å «¹½ Å Å µ Å µ Å Å Å µ Å «¹½ Å Å µ Å «¹½ Å Å µ (9) 5.3. Monoidal functor A monoidal functor ¾ ¼ µ Å Å ¼ between monoidal categories Å and Å ¼ : an ordinary functor Å Å ¼, for objects in Å morphisms in Å ¼ : which are natural in and, for the units and ¼, a morphism in Å ¼ : making the diagrams commute: ¾ µ µ Å µ Å µ ¼ ¼ µ Å µ Å µµ µå ¾ µ µ Å Å µ «¼ µ Å µµ Å µ ¾ µå µ Å µ Å µ (10) ¾ ŵ Å Å µµ «µ ¾ ŵ Å µ Å µ µ Å ¼ µå ¼ µ Å µ ¾ µ µ µ Å µ ¼ Å µ ¼ Å µ µ Å µ ¾ µ µ µ Å µ (11)
A monoidal functor is said to be strong when ¼ and all ¾ µ are isomorphisms, strict when ¼ and all ¾ µ are identities. 10 5.4. Braided monoidal functors If Å and Å ¼ are braided monoidal categories, a braided monoidal functor is a monoidal functor ¾ ¼ µ Å Å ¼ which commutes with the braidings and ¼ in the following sense: µ Å µ ¼ µ Å µ (12) ¾ µ Å µ µ ¾ µ Å µ 5.5. Monoidal natural transformations ¾ ¼ µ Å Å ¼ between A monoidal natural transformation ¾ ¼ µ two monoidal functors is a natural transformation between the underlying ordinary functors making the diagrams µ Å µ Å commute in Å ¼. µ Å µ ¾ µ ¾ µ Å µ Å Å µ ¼ ¼ ¼ ¼ (13)