Tangled Circuits

T angled circuits R. Rosebrugh Department of Mathemati cs and Computer Science Moun t A l lison Univ ersity Sac kvill e, N. B. E4L 1 E6 C anada rros ebrug h@mt a.ca N. Sabadini Diparti men to di Infor m atica e Comunicazione Universit` a dell’ Insubria via Carl o ni, 78, Com o, Italy nico letta .sab adini@uninsubria.it R. F. C. W a lters Diparti men to di Infor m atica e Comunicazione Universit` a dell’ Insubria via Carl o ni, 78, Com o, Italy robe rt.wa lter s@uninsubria.it No v em b er 1 2, 20 18 1 1 In tro duc t ion The theme of the pap er is the use of c ommutative F r o b enius algebr as in br aide d strict monoidal c ate gori e s in the study of v arieties of circuits and comm unicating systems whic h o ccur in Computer Science, including circuits in w hic h the wires are tangled. W e indicate a lso some p ossible nov el geomet- ric in t erest in suc h algebras. The con tribution of the pap er is the intro duction a nd application of sev- eral new suc h categories, and a ppro priate functors b etw een them. The au- thors and collab ora t o rs ha ve previously studied similar systems using sym- metric monoidal categories ([8, 9, 10, 1 1, 16, 17, 18, 5 ]), with separable al- gebras instead of F robenius algebras. These earlier w orks did not tak e in to consideration an y ta ng ling of the wires. F urther we will see in section ?? the imp ortance of considering F rob enius a lgebras r a ther than the mor e sp ecial separable algebras ev en in the symmetric monoidal case (no tangling). 1.1 T angled circuit diagrams W e prop ose a definition for a category of tangle d cir cuit diagr ams , in whic h it is p ossible to distinguish, fo r example, the first and second of the f ollo wing circuit diagrams, while the second a nd third are equal. R R R The notion of tang led circuit diagram is pa r a metrized b y a m ultigraph (or tensor sche me) of comp onen ts (suc h as the comp onen t R in the exam- ple ab ov e). Giv en suc h a multigraph M , a tangled circuit dia g ram (or more briefly , a circuit diagra m) is a n arrow in the free braided strict monoidal cate- gory on M in whic h ob jects of the m ultigraph M are equipp ed with symmet- ric F rob enius algebra structures; w e denote this category b y TCircD M . The ob jects of the multigraph M may b e thought of as typ es of wir es . Give n any ob ject A of M it is straig h tfo rw ard to see that there is an appropriate functor from F reyd, Y etter’s category T angle ([6]) to TCircD M since a symmetric F rob enius structure on A induces a tangle algebra structure on A . As a result an y inv arian ts of tangled circuit diagrams provide also in v a rian t s for tangles 2 and knots. W e conjecture that suc h functors T angle / / TCircD M are faith- ful. W e also conjecture that there is a top olog ical description o f TCircD M related to F reyd, Y etter’s description of T angle and to cob ordisms. 1.2 Relations The category Rel whose ob jects are sets, a nd whose arrow s are relations is symmetric monoidal with the tensor of sets b eing the cartesian pro duct, and eac h ob j ect has a symmetric F rob enius (ev en separable) algebra struc- ture pro vided b y the diagonal functions and their reve rse relat io ns. I n fact this w as the mo t iv ating ex ample for the intro duction in [3] of the F rob enius equations (equiv alent a xioms had b een giv en earlier b y Law v ere in [14]). W e describe here a mo dification of Rel whic h we call TRel G , whic h dep ends on a group G , and whic h is braided rather tha n symmetric. W e further describe a comm utativ e F rob enius algebra in TR el G whic h hence yields a represen tation of TCircD M , and this represen tation enables us, fo r exam- ple, to distinguish the t w o differen t circuits ab ov e. W e discuss distinguishing closed circuits, a problem analog o us to classifying knots, using TRel G . 1.3 Spans and cospans The principal category we ha ve using in the earlier w ork on circuits and comm unicating-pa rallel algebras o f pro cesses has b een the category Span ( Graph ) of spans of graphs (and fo r sequen tial systems Cospan ( Graph )). Already in the original pap er [8] the separable algebra structure on each ob ject play ed a crucial ro le. The relatio n b etw een another mo del of circuits, namely Mealy auto mata and Span ( Graph ) w as discusse d in [9]. One of the motiv atio ns of the presen t w o rk is to pro duce an seman tic algebra in whic h the tw isting of wires is also (at least partially) expressible. T o this end we in tro duce first a simple bra ided mo dification TSpan G of Span ( Set ), dep ending on a group G , with a comm utative F rob enius algebra. It is clear that a similar construction TSpan G ( C ) could b e made f or a group ob ject G in a category C with limits in the place of Set . Again, there is a represen tation of T angle (via a represen tation of TCircD ) whic h takes a tangle to the span of colourings of the tangle (in tro duced b y John Armstrong in [2]). Applied to knots the set of c olourings is one of the simplest in v ariants for dis tinguishing knots (as a first ex ample it allo ws one 3 to sho w that a trefoil is not an unknot). The extended notion of colourings of tangled circuit diagrams gives further aid in distinguishing circuit diagrams. The categor y Group op , the dual of the category of groups has finite lim- its. F urther F , the free group on o ne generator is a group ob ject in Group op . The catego ry TSpan F ( Group op ) is braided monoidal with F equipp ed with a comm utative F rob enius structure. The in duced represen tatio n T angle / / TSpan F ( Group op ) asso ciates the cospan of groups in t ro duced b y John Armstrong in [1] to a tangle, and the knot group t o a knot. 1.4 Linear analogue circuits This example comes from the pap er [9] where it is discussed in detail. How- ev er the F rob enius algebra structure was not noticed in that pap er. The category is analogous to TSpan G ( Graph ) where the gr o up G is the real n umbers under addition. The F rob enius a lgebra structure arises from the Kirc hhoff law for curren ts. Since the group is a b elian there is no information ab out the tangling of wires . W e describ e, as an e xample, circuits comp osed of resistors, capacitors and inductors. 1.5 Remaining questions Pro ving that t wo e xpressions in TCircD y ield differen t circuit seems to b e a difficult question even in apparen tly simple cases some of whic h w e note b elo w. If a s w e suspect knots are f a ithfully represen ted in TC ir cD this is not surprising, though for knots there a r e k no wn though non-trivial a lgorithms. 1.6 References There is a h uge literature now relating monoidal categories and geometry b eginning with [15, 1 2 , 7]. W e m en tio n just t w o further items of an exp osi- tory na ture useful to reading this paper (apar t from o ur o wn w o rk men tioned ab ov e): the first [19] is a surv ey for computer scien tists and others whic h dis- cusses man y additional structures but strangely no t F rob enius algebras, and ignores our w ork on separable algebras; the second [13] is an in tro ductory b o ok on the relation b et w een F rob enius algebras and 2-dimensional cob or- dism. 4 1.7 Ac kno wledgemen t W e would like to tha nk Aurelio Carb o ni fo r helpful suggestions. 2 Braided mon o idal categor ies and F rob enius algebras W e review immediately the notions fundamental for the pap er. 2.1 Braided monoidal categories Definition 2.1 A braided strict monoidal category ([7]) i s a c ate gory C with a f unc tor, c al le d tensor, ⊗ : C × C / / C and a “ unit” obje ct I to gether with a natur a l family of i s o morphisms τ A,B : A ⊗ B / / B ⊗ A c al le d twist satisfying 1) ⊗ is as so ciative and unitary on obje cts and arr ows, 2) the fol lowing dia gr am s c omm ute for obje cts A, B , C : B 1 : A ⊗ B ⊗ C B ⊗ A ⊗ C τ ⊗ 1 $ $ J J J J J J J J J J J J A ⊗ B ⊗ C B ⊗ C ⊗ A τ / / B ⊗ C ⊗ A B ⊗ A ⊗ C : : 1 ⊗ τ t t t t t t t t t t t t and B 2 : A ⊗ B ⊗ C A ⊗ C ⊗ B 1 ⊗ τ $ $ J J J J J J J J J J J J A ⊗ B ⊗ C C ⊗ A ⊗ B τ / / C ⊗ A ⊗ B A ⊗ C ⊗ B : : τ ⊗ 1 t t t t t t t t t t t t Among the consequenc es of the definition is the Y ang-Baxter equation whic h reads : (1 ⊗ τ )( τ ⊗ 1)(1 ⊗ τ ) = ( τ ⊗ 1)(1 ⊗ τ )( τ ⊗ 1) : A ⊗ B ⊗ C / / C ⊗ B ⊗ A A compact and comprehensible f o rm ulatio n of suc h prop erties is provided b y circuit or“wire” diagrams like the follo wing. Comp osition is read from left to righ t and ⊗ is vertical j uxtap osition. The t wist is expressed by the “p ositiv e crossing” (top wire ov er bo ttom) and it s inv erse by the negativ e crossin g. 5 = Another conse quence of the axioms ab ov e is that τ A,I = τ I ,A = 1 A : A / / A . The natura lit y of the t wist τ leads to the following kind of equalit y of diagrams: R = R In the case when the co domain of the comp onen t is I naturalit y is dra wn, for example, as: R = R 2.1.1 F rob enius algebras Definition 2.1 A comm utativ e F rob enius algebra in a br aide d monoida l c a t- e gory c o n sists of a n obje ct G and four arr ows ∇ : G ⊗ G / / G , ∆ : G / / G ⊗ G , n : I / / G and e : G / / I mak ing ( G, ∇ , e ) a mono id, ( G, ∆ , n ) a c omono id and s atisfying the e q uations (1 G ⊗ ∇ )(∆ ⊗ 1 G ) = ∆ ∇ = ( ∇ ⊗ 1 G )(1 G ⊗ ∆) : G ⊗ G / / G ⊗ G (D) ∇ τ = ∇ : G ⊗ G / / G τ ∆ = ∆ : G / / G ⊗ G Definition 2.2 A multigr aph M c onsists of two sets M 0 (vertic es or wir es) and M 1 (e dges or c omp on e nts) and two functions dom : M 1 / / M ∗ 0 and cod : M 1 / / M ∗ 0 wher e M ∗ 0 is the fr e e monoid on M 0 . 6 Definition 2.3 Given a multigr aph M the fr e e br aide d strict monoidal c at- e gory in which the ob j e cts of M ar e e quipp e d with c om mutative F r ob enius algebr a structur es is c al le d TCir cD M . Its arr ows ar e c al le d tangle d cir cuit diagr ams, or mor e briefly cir cuit diagr ams. In the c as e that M has one vertex and n o arr ows we wil l denote TCircD M simply as TCircD . 2.1.2 T angle algebras Definition 2.2 An obje ct X in a br aide d strict monoidal c ate gory (with twist τ )is c al le d a tangle a lgebra when it is e quipp e d with a rr ow s η : I / / X ⊗ X and ǫ : X ⊗ X / / I that sa tisfy the e quations (wher e w e write 1 for al l identities): (i) ( ǫ ⊗ 1)(1 ⊗ η ) = 1 = (1 ⊗ ǫ )( η ⊗ 1) (ii) ǫτ = ǫ and τ η = η . Axiom (i) says that X is a self-dual ob ject . The r e a d er c an tr ansl a te these in to wir e diagr ams . An example is the wir e diagr a m for (i): = = Theorem 2.1 I f G is a c om m utative F r o b e n ius a l g ebr a in a br aide d m o noidal c ate g ory, then the arr ow s ǫ = e ∇ , η = ∆ n , τ satisfy the ax ioms of the gener ating obje ct of the c ate gory of tangles, and so G is a tangle alge br a . Pro of. Let G b e a comm utativ e F rob enius algebra in a braided monoidal category . It is stra ig h tfo rw ard to give algebraic pro ofs for the tangle algebra axioms, but w e r emind the reader that these can b e more easily found using wire diagrams. T o se e that ( ǫ ⊗ 1)(1 ⊗ η ) = 1 notice that ( e ⊗ 1)( ∇ ⊗ 1)(1 ⊗ ∆)(1 ⊗ n ) = ( e ⊗ 1 )∆ ∇ (1 ⊗ n ) = 1 · 1 = 1 . 7 Definition 2.3 (F r eyd-Y etter) Th e c a te go ry T angle is the fr e e strict monoidal c ate g ory gener ate d by one obje c t X , e quipp e d with a tang l e algebr a structur e. The category T angl e has a geometric description [20] consonant with its name. In t ha t description the arrows fro m I to I are knots and links. Corollary 2.1 Given an ob je c t A of m ultigr aph M ther e is a uniq ue br aide d strict monoi d a l functor T angle / / TCircD M taking the gener ating obje ct to A and the structur e maps of T angle to the c orr esp o nding s tructur e maps of A in TCircD M . 2.1.3 Example equations W e now give some examples o f equations b et w een circuit diagrams. Prop osition 2.1 I f R : X × Y / / I is an arr ow in t he multigr aph M then Rτ − 1 Y ,X = ( ǫ X )(1 X ⊗ R ⊗ 1 X )( η ⊗ 1 Y ⊗ 1 X ) = Rτ X,Y . Pro of. First a picture of the equations: R = R R = It is clearly sufficien t to prov e the first e quation. 8 ( commutativ ity ) R = = R ( natur al ity ) R R = R = ( natur al ity ) ( dual ity ) Prop osition 2.2 I f R : I / / X ⊗ X and S : X ⊗ X / / I then S τ 2 n R = S R ; that is R and S joine d by an even n umb er of twists is e qual to R and S joine d dir e ctly. A couple more equations prov able in TCircD ; 2.1.4 Example = 9 Remark 2.1 T h e ge ometric intuition i s that the wir es a r e thic k and so c an b e deforme d c ontr acting se gmen ts. Notic e h owever that it is not true i n gen- er al in TCircD that the sep ar abl e axio m ∇ ∆ = 1 holds . That is, cycles c annot b e c ontr acte d to a p oint. 2.1.5 Example = H 1 H 2 U 1 U 2 H 1 H 2 U 1 U 2 2.1.6 Example If R : I / / X ⊗ X and S : X ⊗ X / / I then S ( ǫ ⊗ ǫ ⊗ 1 ⊗ 1)(1 ⊗ τ − 1 ⊗ τ − 1 ⊗ 1)(1 ⊗ 1 ⊗ η ⊗ η ) R = S τ τ R . Diagrammatically: R S = R S Pro of. W e will giv e a diagrammatic pro of. A more explicit picture of the left hand expression is 10 R S By naturalit y this is equal to R S and hence to R S This simplifies by duality t o R S 11 It is now clear that rep eating the a rgumen t using natura lit y and dualit y w e obta in the result. 3 A braided catego r y of relations 3.1 The definition of TRel G W e will describ e a braided mo dification of the catego ry Rel with a comm u- tativ e F rob enius ob ject. Definition 3.1 L et G b e a gr oup. The obje cts of TRel G ar e the formal p owers o f G , and the arr ows fr om G m to G n ar e r elations R fr om the set G m to t he set G n satisfying: 1) if ( x 1 , ..., x m ) R ( y 1 , ...y n ) then also for al l g in G ( g − 1 x 1 g , ..., g − 1 x m g ) R ( g − 1 y 1 g , ..., g − 1 y m g ) , 2) if ( x 1 , ..., x m ) R ( y 1 , ...y n ) then x 1 ...x m ( y 1 ...y n ) − 1 ∈ Z ( G ) (the c enter of G ). Comp osition and identities ar e define d to b e c omp osition and identity of r e- lations. It is straigh tfo rw ard to v erify that TRel G is a category . W e introduce some useful nota tion. W rite x = ( x 1 , ..., x m ), y = ( y 1 , ..., y n ), and so on. W rite x = x 1 x 2 ...x m and for g , h in G , as g h = hg h − 1 . F or g in G write x g = ( x g 1 , x g 2 , ..., x g m ). Th us, ( x ) g = x g , and of course for an y x, y in G m × G n , x g y g = ( xy ) g where w e write xy for ( x 1 , ..., x m , y 1 , ..., y n ). Theorem 3.1 TR el G is a br aide d strict monoidal c ate gory with tensor de- fine d on obje cts by G m ⊗ G n = G m + n and o n arr ows by pr o duct of r elations. The t wist τ m,n : G m ⊗ G n / / G n ⊗ G m is the functional r elation ( x, y ) ∼ ( y x , x ) 12 Pro of. As noted abov e it is easy to sho w that iden tit ies and comp os- ites of r elat io ns satisfying 1) and 2) also satisfy 1) and 2), so TRel G is a category . The monoidal structure of Rel also restricts to TRel G since if R : G m / / G t and S : G n / / G u satisfy 1) and 2) then so also do es R × S . T o see t ha t R × S satisfies 1) notice that if xRy and z S w then for and g ∈ G , x g Ry g and z g S w g and hence ( xz ) g ( R × S )( y w ) g . T o see tha t R × S satisfies 2) notice that, if x ( y ) − 1 ∈ Z ( G ) and z ( w ) − 1 ∈ Z ( G ), then xz ( y w ) − 1 = ( x )( z )(( y )( w )) − 1 = ( x )( z ) ( w ) − 1 ( y ) − 1 . Bu t z ( w ) − 1 ∈ Z ( G ), so ( x )( z )( w ) − 1 ( y ) − 1 = ( x )( y ) − 1 ( z )( w ) − 1 and the latter is in Z ( G ). W e show that B 1 holds for τ as defined. B 2 is similar. First note that τ m,n + p ( xy z ) = ( y z ) x x . F urther ( τ m,n ⊗ 1 G p )( xy z ) = y x xz while (1 G n ⊗ τ n,p )( xy z ) = xz y y } . Th us (1 G n ⊗ τ n,p )( τ m,n ⊗ 1 G p )( xy z ) = (1 G n ⊗ τ n,p )(( y x ) xz ) = ( y x )( z x ) x = τ m,n + p ( xy z ) . Lastly w e need to sho w that τ m,n : G m × G n / / G n × G m is natural. This amoun t s to t wo conditions. Consider R : G p / / G m and S : G q / / G n in TRel G . The first conditio n for naturality is that τ m,n ( R ⊗ 1 G n ) = (1 G n ⊗ R ) τ p,n : G p + n / / G n + m . But xy z w ( x ∈ G p , y ∈ G n , z ∈ G n , w ∈ G m ) belongs to the left-hand side iff xRw and z = y w , whereas xy z w b elongs to the righ t- ha nd side iff xRw and z = y x . But condition 2) implie s that if xRw then for an y y it follows that y x = y w , and hence the result. The second condition fo r natura lit y is t ha t ( τ m,n )(1 G m ⊗ S ) = ( S ⊗ 1 G m )( τ m,q ) : G m + q / / G n + m . But xy z w ( x ∈ G m , y ∈ G q , z ∈ G n , w ∈ G m ) b elongs to the left-hand side iff x = w and y S ( z x − 1 ), whereas xy z w b elongs to the righ t-hand side iff x = w and y x S z . Condition 1) implies the result. Remark 3.1 Notic e that a r elation in TR el G fr om I to G × G is just a subset of G × G close d under c onjugation by elements of G and whose elements ( x, y ) satisfy xy ∈ centr e ( G ) . F urther a r ela tion fr om I to I is either the empty set or the one-p oint set. Notic e also that if the gr oup G is ab elian the c onditions (1) and (2) of the definition 3.1 ar e trivial ly true. 13 3.1.1 The comm utativ e F rob enius structure on G The comm utativ e F rob enius structure on the ob ject G of T R el G men tioned ab ov e is as follo ws: ∇ is a function, namely the m ultiplication of the group G , n : I / / G is also a function, the identit y of t he gro up; ∆ is the opp osite relation of ∇ , e is the opp osite relatio n of n . Notice that η is the relation ∗ ∼ ( x, x − 1 ), and ǫ is the opp osite relation of η . It is straigh tfo r w ard to chec k that these relations belong to TRel G . W e will just c heck one of the F rob enius equations, namely that (1 G ⊗ ∇ )(∆ ⊗ 1 G ) = ∆ ∇ : G × G / / G × G. If g , h , p , q are in G then ( g , h, p, q ) b elongs to the left- ha nd r elation if there is a r ∈ G suc h that g = pr and r h = q . But this is the same a s sa ying that p − 1 g = q h − 1 or g h = pq whic h is exactly the condition for ( g , h, p, q ) to b e in the righ t-hand relation. 3.2 Pro ving circuits d istinct in TRel In this section w e discuss the p o ssibility o f distinguishing v arious tangled circuits, including the analogue of knots, closed circuits, that is, circuits from the one-p oint set I to I , by lo oking in TRel G . 3.2.1 Example First an example where t wo circuits may b e distinguished in T R el S 3 , where S 3 is the symmetric group on three letters. The circuits are: R S R S Pro of. Let eac h of R a nd S b e the set of conjugates of u = (12 , 13 , 23 , 13) under the action of G (not G × G × G × G ). Notice that (12)(13)(23)( 1 3) is the iden tity . The second circuit e v a luates as the one p oint set. 14 The first circuit ev aluates instead as the empt y set since the braid in the first c ircuit relates (12 , 13 , 23 , 13 ) in R to (13 , 23 , 23 , 13) whic h is not in the conjugacy class of u since the second and third elemen ts are equated by the braid. Notice that a sim ilar argumen t using the sy mmetric group S 3 w orks for t wo comp onen ts joined by n > 3 wires, the first t w o of which a re ta ngled. 3.2.2 Example W e will see that the first tw o circuits in section 1.1 can also b e shown distinct in TRel S 3 . It is clearly s ufficien t to show the follo wing circuits distinct: R R T a k e R to b e the follo wing subset of ( S 3 ) 2 × ( S 3 ) 2 : the conjugacy class of the elemen t ((12 , 13) , (12 , 13)). Then the first circuit ev aluates as ∅ and the second as the one-p oint set. 3.2.3 Example Next an example of t w o circuits whic h w e believ e are distinct in TCircD M but are a lw ays equal in TRel G . F or an y group G , TRel c annot dis tinguish them. R S R S 15 Pro of. Suppose ( x, y , z ) is an elemen t of comp o nent R . Notice t ha t since xy z is in the cen tre xy z = y z x = z xy . The bra id b et we en the t wo components in the first circuit relates ( x, y , z ) to u = ( xy x − 1 z xy − 1 x − 1 , xy x − 1 , z − 1 xz ) = ( xy x − 1 z xy − 1 x − 1 , z − 1 y z , z − 1 xz ) since y z x = z xy . Instead the bra id in the second circuit relates ( x, y , z ) to v = ( z , z − 1 y z , z − 1 y − 1 xy z ) = ( z , xy x − 1 , x ) since z − 1 y − 1 xy z = z − 1 y − 1 y z x = x and z xy = y z x . But x z uz − 1 x − 1 = v since xz xy x − 1 z xy − 1 x − 1 z − 1 x − 1 = xy z xx − 1 z xx − 1 z − 1 y − 1 x − 1 = xy z y − 1 x − 1 = z and hence u and v are conjugate. Since S is closed under conjugacy , the elemen t ( x, y , z ) give s rise to an elemen t of the first circuit if and only if it do es for the second circuit. Since this is true for any ( x, y , z ) the t w o circuits are equal in TRel G . 3.2.4 Example In f act the last example is general for three wires. The cir cuit ob taine d by c omp osing in TRel G any two two c omp o nents R : I / / G 3 and S : G 3 / / I with a br aiding in b etwe en dep ends o nly on the p ermutation, not the br aidin g . Pro of. Supp ose ( x, y , z ) ∈ R t hen xy z ∈ centr e ( G ) and hence xy x − 1 = z − 1 y z , y z y − 1 = x − 1 z x and z xz − 1 = y − 1 xy . Consider t wo comp osites R comp osed with τ ⊗ 1 and R composed with τ − 1 ⊗ 1. Consider ( x, y , z ) ∈ R . W e will sho w that these t w o comp osites asso ciate ( x, y , z ) with conjugate triples. Rep eating this w e see that the argumen t give n in the ab o v e example can b e applied, sho wing that in a comp o site τ and τ − 1 are in terch angeable. In the first comp osite ( x, y , z ) is related to u = ( xy x − 1 , x, z ) = ( z − 1 y z , y , z ). In the second composite ( x, y , z ) is related to ( y , y − 1 xy , z ). It is immediate that z uz − 1 = v . Of course differen t p erm utations can be distinguished e v en in R el . 3.2.5 Example Another t w o c ircuits w e can distinguish in TRel S 3 : 16 H 1 H 2 U 1 U 2 U 3 U 4 H 1 H 2 U 1 U 2 U 3 U 4 Pro of. Replace each of the four comp onen ts U 1 , U 2 , U 3 , U 4 b y ǫ . Let R b e the conjugacy class of ( 12 , 1 3 , 2 3 , 13 ). T he wires of the first circuit re- late this elemen t to u = ( 1 2 , 2 3 , 12 , 13 ), and of the second circuit to v = (13 , 12 , 12 , 13).Clearly u and v are not conjugate, and hence we can c ho ose S so that the tw o circuits ev aluate differen tly in T R el S 3 . 3.2.6 Example The follo wing t w o circuits can b e distinguished in TRel S 3 . R S R S Pro of. T ake R to b e the conjugacy class of (12 , 13 , 23 , 13) and S the con- jugacy class of (() , 13 , () , 1 3). The firs t circuit ev aluates as the one-p oin t set and the second as ∅ . 4 A braided catego r y of spans In this section we b egin to extend the previous sections with a mo dification of the category Span of spans of sets with a bra iding f or some spans. 17 Definition 4.1 L et G b e a gr oup. The obje cts of TSpan G ar e the formal p owers of G , and an arr ow fr om G m to G n is an iso m orphism c lass of sp an s in sets, G m o o δ 0 S δ 1 / / G n , fr om the set G m to the set G n such that ther e exist a function G × S / / S of G written ( g , s ) 7→ g s yielding a bije ction for e ach g ∈ G , and satisfying : 1) if δ 0 ( s ) = ( x 1 , ..., x m ) and δ 1 ( s ) = ( y 1 , ..., y n ) then δ 0 ( g s ) = ( x g 1 , ..., x g m ) and δ 1 ( g s ) = ( y g 1 , ..., y g m ) for al l g in G , 2) if δ 0 ( s ) = ( x 1 , ..., x m ) and δ 1 ( s ) = ( y 1 , ..., y n ) then x 1 ...x m ( y 1 ...y n ) − 1 ∈ Z ( G ) . Comp osition an d iden tities ar e c omp osition and identity of sp ans. It is straigh tfo r w ard that TSpan G is a category . L ike TRel G it has the structure of a braided strict mono ida l catego ry . Theorem 4.1 TSpan G is br aide d strict monoidal with tensor de fi ne d by G m ⊗ G n = G m + n and twist τ m,n : G m ⊗ G n / / G n ⊗ G m is the sp an de termi ne d by t he function δ 1 wher e: δ 1 ( x 1 , ..., x m , y 1 , ..., y n ) = ( y x 1 , ..., y x n , x 1 , ..., x m ) wher e x = x 1 x 2 ...x m and y x = xy x − 1 . Pro of. This is similar to Theorem 3.1. W e use the same notatio n as ab ov e. As noted, it is easy to sho w that iden tities a nd comp osites of spans satisfying conditions 1) and 2) also satisfy 1 ) a nd 2), so TSpan G is a category . T o see t ha t ⊗ is a functor recall that pro duct of spans defines a tensor functor on the category Span of spans. It remains to sho w that TSpan G is closed under ⊗ . Supp o se R : G m / / G t and S : G n / / G u . If x = δ 0 ( r ) , y = δ 1 ( r ) and z = δ 0 ( s ) , w δ 1 ( s ), then for an y g , x g = δ 0 ( g r ) , y g = δ 1 ( g r ) a nd z g = δ 0 ( s ) , w g = δ 1 ( s ), whence ( xz ) g = δ 0 ( g r, g s ) , ( y w ) g = δ 1 ( g r, g s ), so taking g ( r, s ) to b e ( g r, g s ) conditio n 1) is satisfied. F or x, y , z , w as defined, condition 2) follow s exactly as in Theorem 3 .1. The ass o ciativ e and unita r y pro p erties for ⊗ in TSpan G are imme diate from the same prop erties in Span . W e show that B 1 holds for τ as defined. B 2 is similar. 18 Since the twis ts and iden tities are defined b y functions, the span comp o- sition is obtained b y comp osing func tions and w e calculate: (1 n ⊗ τ m,p )( τ m,n ⊗ 1 p )( xy z ) = (1 n ⊗ τ m,p )( y x xz ) (1) = y x z x x (2) = ( y z ) x x (3) = τ m,n + p ( xy z ) (4) (5) whic h pro v es B 1. As in t he case of TRel G the conditions 1) and 2 ) assure the naturality of τ . 4.0.7 A comm utativ e F rob enius structure on G As for TRel G and using the same functions view ed a s spans, G has the structure of a comm uta tiv e F robenius alg ebra in TSpan G . Conseq uen tly: Corollary 4.1 Ther e is a unique br aide d strict mon oidal func tor T angle / / TSpan G taking the gene r ating obje ct to G and the structur e m a ps of T angle to the c orr esp onding a rr ow s in TSpan G . 4.1 Knot colourings The description of T Span G mak es it cle ar that there is a fa ithful monoidal functor TSpan G / / Span ( Set ) . The follo wing comp osite of monoidal functors w e hav e describ ed we denote as col our ing s : col our ing s G : T angle / / TCircD / / TSpan G / / Span ( Set ) . col our ing s G tak es the generating ob ject X of T angle to the underlying se t of G , and tak es ǫ X to the span G × G ← { ( x, y ) : xy = 1 } / / I , η X to I ← { ( x, y ) : xy = 1 } / / G × G and τ X to ( x, y ) ← ( x, y ) 7→ ( xy x − 1 , x ). 19 Theorem 4.1 (J. Armstr ong [2]) I f K is a kno t then col our ing s G is the se t of c ol o urings of K in the gr oup G . Remark 4.1 B e c ause of the faithfulness of the functor TSpan G / / Span ( Set ) the c a lculation of the set of c olourings of a k n ot may b e done e qual ly in TSpan G or Span ( Set ) . The advantage of intr o ducing TSpan G as we do is that TSpan G has the sam e struct ur e as T angle (br aide d m o noidal with a tangle a l g e br a ) wher e as Span ( Set ) do es not. 4.1.1 Colourings of a trefoil W e will calculate the colourings of a trefoil in the dihedral group D 3 to allo w us to in tro duce notation and indic ate relations with other w o rk. One expression for a trefoil in T angle is ( ǫ ⊗ ǫ )(1 ⊗ τ ⊗ 1)(1 ⊗ 1 ⊗ τ − 1 (1 ⊗ τ ⊗ 1)( η ⊗ η ) . It is conv enien t t o represen t the arrows in this expression as comp onen ts as follow s: η ǫ τ τ − 1 Then the trefoil ma y b e written a s the circuit diagram: η η τ τ − 1 τ ǫ ǫ 20 The ev aluat ion of the expression for the trefoil in Span ( Set ) is a limit of the diag ram in Set f o rmed by taking for eac h wire in the diagram the set G and for eac h componen t the pair of arro ws constituting its span o f sets (see [18] for the relation b et w een limits in C and expressions in Span ( C )). An elemen t of this limit is a tuple of elemen ts of G one for eac h w ire, satisfying the conditions of the comp onents . E ac h of the comp onents η , ǫ , τ , τ − 1 is actually a relation from its domain to co domain, that is a subset of pr o ducts of groups giv en b y equ ational conditions. It is con v enien t to refine the pictures of the comp onent to include the conditions as follow s: xy =1 x y η xy =1 x y xy = z w x = w τ x y z w xy = z w y = w τ − 1 ǫ x y z w Then a colouring of the trefoil, that is, a n elemen t of the limit is a tuple of elemen ts of G on the wires s atisfying the conditions of the comp onen ts: ab =1 ej =1 ad =1 hk =1 a b c d e f g h j k b = f be = cf c = h cg = dh j = g f j = g k 21 When the group is D 3 there are 12 colourings, one for eac h of ( a, c ) = (1 , 1), (123 , 123), (132 , 132), (1 2 , 1 2 ), (1 3 , 1 3 ), (23 , 23), (12 , 13 ), (1 2 , 23 ), (13 , 12), (13 , 2 3), (23 , 12) , (2 3 , 1 2 ), whereas the unknot has 6 colourings. 4.2 Knot groups Consider no w the the group ob ject F , the free group on o ne generator, in the category Group op . As we hav e men tioned the construction TSpan w orks for any category with finite limits, not just Set , and hence there is a braided mo no idal category TSpan F ( Group op ), and a corresp onding rep- resen tation Gp : T angle / / TSpan F ( Group op ) / / Span ( Group op ) = Cospan ( Group ). Theorem 4.2 (J. A rmstr ong [1]) I f K is a knot then Gp ( K ) is the kn ot gr oup of K . 4.2.1 The knot group of a t refoil Remark 4.2 Lim its in Group op ar e c oli m its in Group . We c an c alculate the knot gr o up fr om the same pictur e we use d to c alculate the knot c olouring. In t he diagr am ab =1 ej =1 ad =1 hk =1 a b c d e f g h j k b = f be = cf c = h cg = dh j = g f j = g k 22 a letter r epr esents the fr e e gr oup F on that gener ator, letters on a p air of wir es r epr esents the fr e e gr oup on two gener ators F × F in Group op . The c omp onents ar e q uotients of the fr e e gr oup on the b o undary wir es by the e quations. The evaluation of the c i r cuit in TSpa n F ( Group op ) is a c olimit, namely t he fr e e gr oup on al l the wir es q uotiente d by al l the e quations. In t he c ase of the tr efo i l the k n ot g r oup is < a, b, c, d, e, f , g , j, k ; ab = 1 , b = f , be = cf , c = h, cg = dh, ad = 1 , ej = 1 , j = g , f j = g k , hk = 1 > . 5 Extending TRel G and TSpan G 5.1 TRel X ,G W e now describe an extension o f TRel G whic h dep ends not o nly on the group G but also on a set X , and w e denote it TRel X,G , and a similar extension of TSpan G denoted TSpan X,G . These will enable us to mo del circuits with state. Definition 5.1 Th e c ate gory TRel X,G has obje cts ( X × G ) n . A n arr ow of TRel X,G is a r elation S in Set fr om ( X × G ) m to ( Y × G ) n such that 1) if ( x, h ) S ( y , k ) then for any g ∈ G , ( x, h g ) S ( y , k g ) , an d 2) if ( x, h ) S ( y , k ) then ( h )( k ) − 1 ∈ Z ( G ) . Comp osition and identities ar e define d as in R el In T R el X,G w e define a tensor pro duct by ( X × G ) m ⊗ ( X × G ) n = ( X × G ) m + n . Prop osition 5.1 T R el X,G is a br aide d strict mon o idal c ate gory with τ ( X G ) m ⊗ ( Y G ) n define d to b e t he r elation (( x, g ) , ( y , h )) ∼ (( y , h g ) , ( x, g )) . The idea is that in TRel X,G the ob ject X × G is a single wire car- rying data X . As in TRel G and TSpan G , a “single wire” X × G in TRel X,G admits a comm utativ e F rob enius algebra structure, namely the com ultiplication is t he relation ( ( x, g ) , ( x, h )) ∼ ( x, g h ); the multiplic ation is ( x, g h ) ∼ (( x, g ) , ( x, h )), the counit is ( x, 1) ∼ ∗ and t he unit is ∗ ∼ ( x, 1). 23 5.2 Analogue resistiv e c ircuits in TRel R , R W e begin by describing circuits of resistors whic h may be describ ed in TRel X,G where X = R is the real n um b ers, a nd G = R as a group u nder addition. It is useful to use a gr a phical notatio n similar to that of section 4.1 to do cal- culations in TRel R , R . F or example, we dra w a relation S : R × R / / X × G as: i 1 , v 1 i 2 , v 2 ( i 1 , v 1 ) S ( i 2 , v 2 ) With this notation, where i denotes curren t and v denotes v oltage, a resistor of resistance r is: i 1 , v 1 i 2 , v 2 i 1 = i 2 v 2 = v 1 − ir The unit and counit, whic h sometimes w e dra w as forks, and whic h e m- b o dy Kirc hho ff ’s la w of curren ts: i 1 , v 1 i 2 , v 2 i 3 , v 3 i 1 , v 1 i 2 , v 2 i 3 , v 3 i 1 = i 2 + i 3 v 1 = v 2 = v 3 i 1 + i 2 = i 3 v 1 = v 2 = v 3 24 Using the op erations of TRel R , R one can no w ev aluate a net work of resistors. F or example the circuit with t w o parallel resistors with resistances r 1 , r 2 resp ectiv ely r 1 r 2 ev aluates as: i 1 , v 1 i 2 , v 2 i 1 = i 2 v 2 − v 1 = i 1 ( r 1 r 2 r 1 + r 2 ) 5.3 TSpan X ,G Definition 5.2 Th e c ate gory TSpan X,G has obj e cts ( X × G ) n . A n arr ow of TSpan X,G is an isom o rphism of sp ans S in Set : ( X × G ) m o o δ 0 S δ 1 / / ( Y × G ) n such that such that ther e exist a function G × S / / S of G on S w ritten ( g , s ) 7→ g s yielding a bije ction for e ach g ∈ G , and satisfying: 1) if δ 0 ( s ) = ( x 1 , h 1 , ..., x m , h m ) and δ 1 ( s ) = ( y 1 , k 1 , ..., y n , k n ) then δ 0 ( g s ) = ( x 1 , h g 1 , ..., x m , h g m ) and δ 1 ( g s ) = ( y 1 , k g 1 , ..., y n , k g n ) for al l g in G , 2) if δ 0 ( s ) = ( x 1 , h 1 , ..., x m , h m ) and δ 1 ( s ) = ( y 1 , k 1 , ...y n , k n ) then h 1 ...h m ( k 1 ..., k n ) − 1 ∈ Z ( G ) . Comp osition and identities and tensor ar e defin e d as in Sp an . The br aiding and F r ob enius st ructur e ar e as in TRel X,G . It is clear that this definition ma y b e made in any catego r y C with finite limits to giv e a category TSpan X,G ( C ). 25 5.4 RLC circuits in TSpan X ,G ( Graph ) The algebra of R LC circuits w e will describ e was in tro duced in [9] but without the conscious ness of F rob enius algebras. W e will giv e a brief rec apitulation without full details. W e need to s a y something first ab out the some what un usual in t erpreta- tion of a graph in this setting. If the graph consists of the tw o (domain a nd co domain) functions φ : X / / Y and ψ : X / / Y w e will in t erpret t his a s the formal differen tial equation φ ′ = ψ . F or further explanation of this in- terpretation see [9 ]. In the examples w e describe the interpretation will hav e a clear meaning. There is a notion of b eha viour for suc h a syste m, namely a function x : R / / X suc h that φ ′ ( x ( t )) = ψ ( x ( t )) (only meaningful with smo othness assumptions). W e will no w consider TSpan X,G ( Graph ) where b oth X and G are the graph with one v ertex, and set of arrows R ; w e will ide n tif y b oth X and G with the set R , the group structure b eing addition. Again it is useful to use a graphical notation similar to that of Section 4.1 to do calculations in TSpan R , R . F or example, w e draw the spans cor r e- sp onding to the constan ts τ , ∆, ∇ , η , ǫ , and the resis tors of the a lgebra (in whic h all of the graphs hav e one ve rtex) exactly a s in section 5.2. Instead the g raph of a capacitor with capacitance c is the pair f unctions φ, ψ : R 3 / / R defined by φ ( i, v , q ) = q a nd ψ ( i, v , q ) = i ; the in terpretatio n of this is tha t a capacitor has stat e i , v , and also state q , the c har g e of the capacitor, and that q ′ = i . The boundary conditions (the morphism of the span) are on the left v 1 = v , and i 1 = i and on the right v 2 = v − q c and i 2 = i . Hence we dra w the capa cito r as follows: q ′ = i i 1 , v 1 i 2 , v 2 i 1 = i = i 2 , v 1 = v v 2 − v 1 = q c Similarly an inductor with inductance l has an extra v ariable of state p with graph R 3 / / R , and pictures; 26 i ′ = p i 1 , v 1 i 2 , v 2 i 1 = i = i 2 , v 1 = v v 2 − v 1 = l p Using the op eratio ns of TSpan R , R ( Graph ) one can no w ev aluate a net- w ork of resistors, capacitors and inductors. F or example the circuit of an inductance and a capacitance l c ev aluates as i, v 1 , v 2 , p, q ( − i ) ′ = p q ′ = i q c = v 1 − v 2 = l p A b eha viour consists o f fiv e functions from R to R , namely i ( t ), v 1 ( t ), v 2 ( t ), q ( t ), p ( t ) suc h that i ′ = − p , q ′ = i and q c = v 2 − v 1 = l p . 27 6 Dirac’s b elt tric k The claim is that the follo wing tw o circuits are equal in TCirc D , that is t hat a rotation through 2 π of a comp onent I / / X 3 is equal t o the iden tity . 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