LOWÐDIMENSIONAL TOPOLOGY AND HIGHERÐORDER CATEGORIES

Transcription

1 LOWÐDIMENSIONL TOPOLOGY ND HIGHERÐORDER TEGORIES ollaborators: Ross Street Iain itchison now at University o Melborne ndrž Joyal UniversitŽ d QŽbec ˆ MontrŽal Dominic Verity Macqarie University Todd Trimble Macqarie University John Power University o Edinbrgh (Scotland) Robert Gordon Temple University (Philadelphia) thors o some contribting works: Roger Penrose (diagrammatic tensor calcls) G. Max Kelly and M.L. Laplaza (combinatorial string diagrams) Peter Freyd and David Yetter (category o tangles verss dality) N.Y. Reshetikhin and V.G. Traev (invariants via categories) Mei hee Shm (Macqarie PhD thesis on tangles o ribbons) J.S. arter and M. Saito (movies) J. E. Fischer Jr (the monoidal bicategory o 2-tangles) Good textbook or backgrond applications: hristian Kassel Qantm Grops Grad Texts in Math 55 (Springer-Verlag, 995) Page

2 n arrow : a D in a monoidal category is depicted as ollows. D omposition g : a o arrows : a, g : a is perormed vertically p the plane (electronics term: in series): g g Page 2

3 Tensoring : a, Õ : Õ a Õ to get Õ : Õ a Õ is depicted horizontally rom let to right (electronics term: in parallel): ' ' The nit or the tensor prodct is denoted by I. n arrow : I a wold be depicted by: Page 3

4 D E H h g F G The vale o the above diagram is a certain arrow F G aaa D E H. Theorem (Joyal-Street) The vale o a progressive plane string diagram in a monoidal category is deormation invariant. Page 4

5 Example o a monoidal category Let n be the rtin n string braid grop. Here is an element o 5. t = t = presentation or n is given by the generators s,..., s n- and the relations () s i s i+ s i = s i+ s i s i+ or i n-2, (2) s i s j = s j s i or i < j- n-2. 2 i i + n Ð n s i 2 i i + n Ð n Page 5

6 The braid category is the disjoint nion o the n. More explicitly, the objects o are the natral nmbers 0,, 2,..., the homsets are given by (m, n) = n when m = n otherwise, and composition is the mltiplication o the braid grops. The category is eqipped with a strictly associative tensor strctre deined by addition o braids : m n aaa m+n which is algebraically described by s i s j = s i s m+j. = α β α β Page 6

7 Model category or cbical set cointerval in a monoidal category V is a diagram s J i t I s i = = t i I where I is the nit or the tensor prodct. an we ind a model or the ree monoidal category containing a generic cointerval? This will be a monoidal category generated by a single object J and three arrows depicted diagrammatically by s t J J J i sbject to the two relations s t J Empty = diagram = J i i Objects will be tensor powers J n = J J... J (n terms) o J. typical arrow J 5 aa J is depicted below. Page 7

8 i i i s s t s t t s This diagram can be interpreted as a nction ξ : < 5 > aa < > where < k> = { Ð, +,, 2,..., k} as ollows. Ð Ð Page 8

9 So or model category I has objects the bi-pointed sets < k> and arrows ξ : < m> aa < n> those nctions which preserve Ð, + and have i < j i ξ (i) < ξ(j) whenever ξ (i), ξ (j) {, 2,..., n}. The tensor prodct is given by The cointerval in I is < m> < n> = < m + n> (ξ ζ)(i) = ξ(i) or 0 < i m ζ(i) or m < i m + n s < > i < 0> t which is generic in the sense that the tensor-preserving nctors T rom I into any monoidal category V are in natral bijection with cointervals in V. The bijection takes T to the image o the generic cointerval nder T. cbical set, as sed in algebraic topology, is precisely a nctor : I aa Set. Page 9

10 raided monoidal categories braiding or a monoidal category is a natral amily c, : aka o isomorphisms compatible with the tensor prodct in the sense that the ollowing two diagrams commte. c, c, c, c, c, c, braided monoidal category selected braiding. is a monoidal category with a Example The braid category is braided monoidal. by the elements c = c m, n : m + n aa n + m braiding is given illstrated by the ollowing igre. m n Theorem [JS] category generated by a single object. The braid category is the ree braided monoidal Page 0

11 c, c, c, c, c, natrality c, c, c, c, c c c = c c c Page

12 Enter 3 Dimensions raid relation, Yang-axter eqation, or Reidemeister move III = Theorem (Joyal-Street) The vale o a progressive 3D string diagram in a braided monoidal category is deormation invariant. Page 2

13 Dality in monoidal categories let dal or an object o a monoidal category consists o an object together with arrows : aa I, : I aa sch that = =, = =, For monoidal categories with dality on both sides, this leads to string diagrams in the plane which have winding, and, or braided monoidal categories with dality, this leads to tangles (these inclde both braids and links). gain, each sch diagram has a deormation invariant vale. Page 3

14 String diagrams or monoidal categories are in act appropriate or bicategories in the sense o Žnabo. monoidal category is a bicategory with one object (in the same way as a monoid is a category with one object). What are called arrows o the monoidal category are called 2-cells in the bicategory; what are called objects o the monoidal category are called -cells in the bicategory; the one object (or 0-cell) o the bicategory never rates a mention in the monoidal category. However, in the string diagram, we shold really think o this single 0-cell as labelling the plane regions between the strings. The more sal diagrams or bicategories have been called pasting diagrams. The passage rom pasting diagrams to string diagrams is via planar PoincarŽ dality. For example, consider the pasting diagram below. d a e h φ ψ D a b E c θ b E g The corresponding string diagram is obtained by replacing 2-cells by nodes, -cells by edges, and 0-cells by plane regions, while preserving the incidence relations. d a e h φ c b ψ D a b E θ E g Page 4

15 String diagrams have an advantage over pasting diagrams especially when identity -cells are involved. Identity arrows occr in some o the basic concepts in bicategories. s an example, consider a pair o adjoint arrows : a, : a in a bicategory. This means that there are 2-cells, (called the conit composites and nit) satisying the two conditions that the pasting are eqal to the identity 2-cells o,, respectively. In terms o string diagrams, these conditions become the ollowing two eqations between vales. = = djoints in bicategories generalize dals in monoidal categories and lead to diagrams with winding as beore; bt now 2D regions are labelled by objects. We shall later consider diagrams or higher adjoints. Page 5

16 lternative view o braidings ommtativity can be expressed by saying the operation is a homomorphism. n abelian monoid ÒisÓ a monoidal category with one object; that is, a bicategory with one object and only an identity arrow. braided monoidal category is a monoidal category or which the tensor prodct preserves the tensor prodct p to coherent natral isomorphism. That is, it is a monoidal bicategory with one object. That is, it is a tricategory with one object (= 0-cell) and only an identity arrow (= -cell). Diagrams or n-th order categories belong in n- dimensional Eclidean space. This is the explanation o why diagrams or monoidal categories are 2D and those or braided monoidal categories are 3D. Symmetric monoidal categories are one object, one arrow, one 2-cell tetracategories. Diagrams or tetracategories belong in 4D. In act, diagrams or symmetric monoidal categories belong in 4 and all higher dimensions: they are combinatorial. Page 6

17 Sraces in 3D and tricategories The starting point is a 3-dimensional generalization o the Penrose notation. 3-cell in a tricategory g α a β transormations via 3D PoincarŽ dality to α g a β onsider the case where both α and g are identities; the pictre has one single 3D region and no speciic distingished plane. a β Page 7

18 be Example Pasting version a α β γ δ ζ g h v w p r q x y z g h v w Movie version a α γ ζ δ h p r v z h g g h β g w q v w v w x y Page 8

19 3D version Take the ollowing three planes in xyt-space: Λ :x+ y + t = 0 Π : x y = 0 Σ :x+ y t = 0. Then the -cells, r, x, h label parts o the plane Λ, the -cells v, p, z, g label parts o the plane Π, and the -cells w, q, y, h label parts o the plane Σ. The 2-cells α, ζ label parts o the line Π Σ, the 2-cells β, label parts o the line Σ Λ, and the 2-cells γ, δ label parts o the line Λ Π. O corse, the 3-cell a labels the point Λ Π Σ. y = Ð y = + t y = 0 Π Λ Σ Λ Σ Σ Λ Π Π y x This relates to the Zamolodchikov tetrahedra eqations. Page 9

20 Lax adjnctions in tricategories lax adjnction in a tricategory consists o objects,, arrows : a, : a, 2-cells, and 3-cells n, e n e satisying the ollowing two conditions: Page 20

21 n e n e Page 2

22 Movie or n n Movie or e e Page 22

23 Srace or n n Srace or e e Page 23

24 Movie or lax adjnction axiom n e Srace diagram or axiom n e Page 24

25 Example rom lte-ockett-seely-trimble They expressed the logic o their Òweakly distribtive categoriesó in terms o string diagrams and then sed rewrite rles on them to ind normal orms. data rewrites r ; s axiom s r is the identity Page 25

26 ÒThe ockett PocketÓ (Verity) r s = Page 26