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Braided Compact Closed Categories with Applications to Low Dimensional Topology

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Braided Compact Closed Categories with Applications to Low Dimensional Topology CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector A,,VANCEi IN MATHEMATICS 77. 156-182 (1989) Braided Compact Closed Categories with Applications to Low Dimensional Topology PETER J. FREYD Departmcnr of Mathematics, Universir~ q/ Penns.~lvania, Philadelphia. Penns?;lvania 19104 AND DAVID N. YETTER Department qf Maihematics and Statisfics, McGii/ University. Montreal, Quebec. Canada H3A 2K6 INTRODUCTION The discovery by Jones [4] of a new isotopy invariant of classical links, and its subsequent generalizations [3, 7a, 7b, 7c], has lead to renewed interest in algebraic representations of knotting and linking. Prompted by this, Yetter [16] analysed the structure of “categories of tangles” to give a combinatorial description of composition of generalized tangles. In that work a great simplification was achieved by using a monoidal structure on the categories considered. As we shall see, the assumption in [ 161 that links (or more generally “Markov classes”) arise by the imposition of extrinsic structure (i.e., Markov moves) is unnecessary when the full categorical structure (including the correct generalization of compact closedness to the non-symmetric case) is understood, an observa- tion first made by Freyd [a]. This observation leads to a clear representa- tion-theoretic view of the new link invariants and, when coupled with the encoding of framings by a modified diagrammatic “isotopy” rather than the labels used in [ 161, to an “algebraicized” Kirby calculus from which it is actually possible to calculate invariants of 3-manifolds from their surgery presentation. In a parallel development, Joyal and Street [IS, 141 have considered “braidings” in the context of monoidal category theory: a “braiding” is a natural family of maps a/,,:A@B+B@A satisfying all the usual properties for a symmetry in monoidal category 156 cml-8708/89 $7.50 CopyrIght r!, 1989 by Academic Press. Inc. All rights of reproduction in any form reserved BRAIDED COMPACTCLOSED CATEGORIES 157 theory including the invertibility of the components, but excluding the usual requirement g.4.B”B.A --1 ,408. The first connection between braided categories and topology is given by Joyal and Street’s characterization of the free (strict) braided monoidal category as the category of geometric braids. We are indebted to them for the influence of their work and numerous helpful suggestions. 1. DEFINITIONS AND LEMMAS FROM CATEGORY THEORY We begin by recalling some notions from the theory of monoidal categories: DEFINITION 1.1. A nzonoidal category’ V = (V, 0, Z, CI,p, i) consists of a category V, a functor @ : V x V + V (written in infix notation), and natural isomorphisms c(~,~,~.:(A@B)@C+A@(B@C), p,,:A@Z-+A, and A,,: I@ A + A such that Ml M2 (A@Z)@BA A@(Z@B) A@B V is strict if all components of ~1,p. and 2 are identity maps. Much of the following will be simplified by the fact that the monoidal categories arising from topological considerations are strict, and by the coherence theorem of MacLane [ 121, by which any monoidal category is equivalent to a strict monoidal category. That is, one may safely assume that all monoidal categories are strict. 158 FREYD AND YETTER Following Joyal and Street [S] we make DEFINITION 1.2. A braiding in a monoidal category is a natural isomorphism satisfying Bl AO(BOC) A (B@C)OA % 2 / \ (AOB)OC BO(COA) O@C B@O \ / (BOA)@C--, B@(AOC) B2 (A@B)@CA CO(AOB) A@(C@B)- (AOC)OB 3-1 For the “correct” generalization of compact closed categories to the non-symmetric case, we take the following partially restricted version of the notion of pivotal category (see Joyal and Street [.5] for the relaxed version 1. DEFINITION 1.3. A (strict) pivotal category is a monoidal category, V, equipped with a (strict) anti-involution of monoidal categories (-)* (i.e., a contravariant functor, satisfying moreover (fog)* =g* of*, I= I*, and (()** = Id,), and a family of maps satisfying BRAIDED COMPACT CLOSED CATEGORIES 159 Pl (AOB)O(B*OA*) x .4@(B@(B*@A*))3=+ AO((BOB*)OA*) P2 and, letting qA = (Ed*)*, P3 Ll* PA* i / B*OI I@A* Epc3.4’ / B*@(A@A*)w B*@“@A*) B*@(B@A*) * ’ + (B*@B)OA* We then have LEMMA 1.4. A pivotal category is closed in the sense that for all objects B, the finctor - @ B has a right adjoint Hom( B, --). 160 FREYD AND YETTER Proof: Let Hom( B, -) = ~ @ B*. The component of the unit (resp. counit) for the adjunction at an object A is then A @qe (resp. A@E~). All necessary verifications follow routinely from P3 and are left to the reader. The following definition and lemma (found in Kelly and Laplaza [9] for the symmetric compact closed case) are crucial to the applications of pivotal categories in low dimensional topology: DEFINITION 1.5. The trace of an endomorphism f: A + A in a pivotal category is the endomorphism of Z given by the formula LEMMA 1.6. In any pivotal category, the trace satisfies Wg) = trkf) whenever f: A -P B and g : B -+ A for anq’ objects A and B. The reader will note that this trace is preserved by monoidal functors which preserve (-)* and E (henceforth called piootal.functors), and that if the pivotal category is additive, then this is a trace in the ordinary sense, taking values in the ring End(I). 2. DEFINITIONS AND LEMMAS FROM CLASSICAL KNOT THEORY Throughout, we work in the piecewise-linear category, except where it is necessary to pass to the smooth category to discuss framings of the normal bundle, and we assume that S3 is endowed with a fixed orientation. We recall the standard reduction of ambient isotopy of piecewise-linear links in S3 to combinatorial moves on regular projections with specified crossing and orientation data (i.e., Knot diagrams) given in Reidemeister [ 133 (see also Burde and Zieschang [ 1 ] for a good exposition): DEFINITION 2.1. A link diagram is a projection of a link in [w3 onto a plane (if we wish we may choose a copy of S’ in S” and project stereographically from any point on it but not on the link) such that the resulting curve has only isolated doubie points with transverse intersection as singularities, together with a choice at each double point of which line is the overcrossing (and, if the link is oriented, the orientation inherited from the link). BRAIDED COMPACT CLOSED CATEGORIES 161 THEOREM 2. (Reidemeister). TLVOlinks are ambient isotopic !f and only ij the}2 have knot diagrams ,rJlich are equivalent under Q.0 Isotopies of the plane qf projection Q.l -\ / i \ f) > ‘Ib / 52.3 Furthermore, isotopies of the projection can he replaced with moves of the f orms A.nl 162 FREYD AND YETTER A.n.2 A-A\ / By abuse of language we will say that two diagrams related by the moves of the previous proposition are isotopic. It is also necessary, following Kauffman, to make DEFINITION 2.3. Two knot diagrams are regulurlv isotopic if they are equivalent under moves 52.2, 0.3, Axl, and A.7c.2 (no Q.l!). Finally, we make DEFINITION 2.4. A tangle is a portion of a knot diagram contained in a rectangle, and incident with the boundary only on the top and bottom edges, where it intersects transversely. We say that two tangles are equal if there is an isotopy of the plane carrying one to the other in such a way that corresponding edges of the rectangle are preserved setwise. 3. CATEGORIES OF TANGLES We begin with an omnibus definition: DEFINITION 3.1. The category of oriented tangles up to regular isotopy (resp. oriented tangles up to isotopy; unoriented tangles up to regular isotopy; unoriented tangles up to isotopy; S-colored oriented tangles up to regular isotopy; . ..). denoted IwOUanrmg (resp. OUarmg; RUaung; Tauug; S-ROTaung; . ..). is the category whose objects are words on the symbols ( 1, 7 } (resp. words on the symbols ( 1, r }; natural numbers; natural numbers; words on the symbols ( 1, 7 ) x S; . ..) and whose maps are named by oriented tangles (resp. oriented tangles; unoriented tangles; BRAIDED COMPACT CLOSED CATEGORIES 163 unoriented tangles; oriented tangles each arc of which is assigned a “color” from the set S; . ..) with two tangles naming the same map if they are regularly isotopic (resp. isotopic; regularly isotopic; isotopic; regularly isotopic with no color changes permitted; . ..). The source/target of a tangle is the word of local orientations for (resp. word of local orientations for; number of points in; number of points in; word of local orientations and colors for; . ..) the intersection with the top/bottom of the bounding rec- tangle. In all cases composition is made by matching top and bottom of tangles. Moreover, each of the categories has a strict monoidal structure in which the map f@g is obtained by matching the right edge of a repre- sentative off‘with the left edge of a representative of g, the identity object being the empty word (resp. the empty word: 0; 0; the empty word: ...). The composition and monoidal structure are illustrated in the following scheme: 1 164 FREYD AND YETTER THEOREM 3.2. ROTmng (resp. OTaung; IRTaong; Taq; S-Rolfarm~; ...) has a braiding, o, in which (T,~,~is given by passing the strands of A over the strands of B. Proof. The proof is essentially the same as that given in the case of the braid category, EL, in Joyal and Street [5]; as was dune there, we give pictorial proof: For naturality: X Y ‘Y 7 ‘X = Y X 9 f A W z and for BI: X Y z X Y 2 7‘Y ‘X 1, \v ‘x/ Y X 2 \f!J ‘Y 1, lb r lyj/4jf( \ h\Y 2 X Y z X (this latter is idential to the case of B), B2 being similar.
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