A Guided Tour in the Topos of Graphs

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1 A Guided Tour in he Topo of Graph Sebaiano Vigna Abrac In hi paper we urvey he fundamenal conrucion of a preheaf opo in he cae of he elemenary opo of graph. We prove ha he raniion graph of nondeerminiic auomaa (a.k.a. labelled raniion yem) are he eparaed preheave for he double negaion opology, and obain a an applicaion ha heir caegory i a quaiopo. Keyword: opo heory, graph heory, auomaa heory, raniion yem 1 Inroducion Sheaf opoi are uually aociaed o coninuou mahemaic, uch a differenial or algebraic geomery. Nonehele, everal (pre)heaf opoi wih imple (even finie!) bae caegorie decribe fundamenal mahemaical objec uch a ree and graph. I ha been uggeed by Lawvere [5] ha hee imple opoi have a rich combinaorial rucure. Nonehele, very few reul are known abou hem, and heir inernal working eem o be moly unexplored. Thi noe ake a brief our hrough he opo of graph, a preheaf opo having a bae he caegory wih wo parallel arrow [5, 12, 6] (here and in he following we aume ha our graph are direced, and ha hey may poe parallel arc and elf-loop; noe ha elewhere hee graph have been called irreflexive, a hey do no poe an ideniy loop a each verex). We inroduce he baic definiion of opo heory and how heir meaning in he opo of graph, alway rying o decribe inuiively he eence of each conrucion. We aume ome knowledge of caegory heory, in paricular he noion of careian cloedne. Good reference are [7], [2] and [8]. Moreover, an elemenary inroducion o opoi can be found in [6]. Our purpoe i wofold: on one ide, our urvey on he opo of graph provide a deailed, complee and elf-conained example of combinaorial opo, which can be exremely ueful o anyone waning o ge acquained wih opo-heoreical definiion; on he oher ide, we how ha everal conrucion of he opo correpond o known conrucion in graph heory (uch a he induced ubgraph conrucion), and by analyzing he Lawvere opologie of he opo we obain a characerizaion of he caegory of auomaa a a caegory of eparaed objec of a opo; hi allow u o prove immediaely ha auomaa (i.e., labelled raniion yem) and oher mahemaical objec, uch a imple finie undireced graph and olerance pace, form a quaiopo. Diparimeno di Scienze dell Informazione, Univerià di Milano, Via Comelico 39/41, I Milano MI, Ialy, Fax: e_mail: vigna@di.unimi.i 1

2 Our reamen i le elemenary han he one given in [6] (ye we do have ome minor overlap, in paricular in he deailed decripion of given in Secion 3), a i aume a cerain knowledge of caegory heory; on he oher hand, we rive o minimize he number of noion inroduced, o o enlarge he audience. Thi omeime keep u from making ome inereing remark ha, however, will be eviden o he peciali, by whom we hope o be forgiven. 2 Baic definiion We begin wih he fundamenal definiion of elemenary opo: Definiion 1 An elemenary opo i a careian cloed caegory wih a ubobjec claifier, i.e., an objec wih an arrow : 1 uch ha for every mono m : S X here i a unique arrow χ m : X which make he following diagram a pullback: S 1 m X χm he arrow χ m i he claifying, or characeriic arrow for m. χ m generalize he well-known characeriic map from e heory (in he opo of e, = {0, 1} and he pullback exhibi he well-known correpondence beween ube and characeriic funcion). The ubobjec claifier can alo be een a an objec of ruh value: he arrow i called he rue map, while he fale map : 1 i he claifier of he unique map 0 1 (wih 0 and 1 we denoe he iniial and erminal objec, repecively; i can be hown ha every opo ha all finie colimi). Here we hall deal wih a paricular cla of elemenary opoi, i.e., hoe conruced by mean of preheaf caegorie: given a mall caegory C, he caegory Se Cop of funcor C op Se and naural ranformaion urn ou o be a opo. In fac, hi aemen i rue for any generalized funcor caegory E Cop, a long a E i a opo and C i a caegory in E (for he precie definiion, ee [8]); hi echnique give a way of building new opoi from given one. For inance, preheave of finie e wih a finie bae caegory form a opo. Le now conider he caegory Ɣ repreened in he following picure: N A The preheaf caegory G = Se Ɣop i he opo of graph. Each preheaf G in G i given by a e G(N), he e of node, and a e G(A) of arc. The arrow, are mapped o funcion G(), G( : G(A) G(N) which aign o each arc i ource and arge. The reader can eaily check ha naural ranformaion are he claical graph morphim (i.e., map of node and arc which preerve he ource/arge aignmen). Analogouly, FinSe Ɣop i he opo of finie graph (our reul are rue boh in he finie and in he general cae). 2

3 3 Repreenable, limi, exponenial and he claifier Preheaf caegorie poe andard conrucion for (co)limi, for and for he exponenial. We are going o dicu hee conrucion, giving he general definiion and applying hem o G. We do no give proof, for which he reader can refer o he abovemenioned ex. The bae of our conrucion i given by he repreenable funcor. There i one repreenable C(, X) for each objec X of C, which aociae o X he e of morphim ino X. The definiion on arrow i given by compoiion. In G, he repreenable funcor are N = Ɣ(, N) and A = Ɣ(, A). Since Ɣ(N, N) = {N} and Ɣ(A, N) =, he repreenable N ha ju one node, called N (noe ha, a uual, when no confuion i poible we wrie X for 1 X ). The repreenable A i he one-arrow graph picured below: A, becaue Ɣ(N, A) = {, } and Ɣ(A, A) = {A}. One can hink of repreenable funcor a a repreenaion in he real world (he opo) of concep (objec of he bae caegory): in our cae, he repreenable aociaed o he objec N, whoe image i he e of node, i a ingle node, while he repreenable aociaed o he objec A, whoe image i he e of arc, i a ingle arc. I can be hown ha he repreenable provide a e of generaor for he opo (i.e., for every f, g : X Y we have f = g iff f h = g h for each repreenable R and each arrow h : R X). Thu, wo arrow are differen iff hey are differen over ome repreenable. Moreover, every objec X of a opo i buil, in a (precie) ene, by glueing a uiable e of repreenable, becaue X i he colimi of he diagram r : R X, where R range over he repreenable and r range over he morphim R X. In a preheaf opo, um and produc are compued locally, i.e., he (co)limi L of any funcor F : D Se Cop i given by L(X) = (co) lim F( )(X), where F( )(X) : D Se i he funcor obained by fixing he econd coordinae of F (which can be een, by careian cloedne, a a funcor D C op Se) 1. Uing hi fac, i i eay o check ha 0 i he empy graph (i.e., 0(N) = 0(A) = ), while 1 i a elf-loop (becaue 1(N) = 1(A) = { }). Moreover, he produc G H of wo graph ha node e G(N) H(N) and arc e G(A) H(A); in oher word, we pu an arc beween x, y and x, y for each pair of arc beween from x o x (in G) and from y o y (in H). In general, he exponenial Y X in he caegory Se Cop i defined by Y X (Z) = Na(C(, Z) X, Y ) = Se Cop (C(, Z) X, Y ), and he definiion on arrow follow by nauraliy (Na(, ) i, of coure, he e of naural ranformaion beween wo funcor). In our cae, we have H G (N) = G(N G, H). 1 More generally, he forgeful funcor U : Se Cop Se/ C creae (co)limi. 3

4 Noe ha he effec of muliplying G by N i ju o rip away all of i arc. Thu, he node of H G are he funcion G(N) H(N). For wha maer arc, we have H G (A) = G(A G, H). The graph A G i buil by aking wo copie of G(N), and aaching each arc in G(A) in uch a way ha i ource i in he fir copy, while i arge i in he econd copy. Equivalenly, we can hink of aking G, adding a copy of G(N), and hen moving he arge of each arc o he econd copy. Each morphim A G H i an arc of H G, having a ource (arge he map induced by he fir (econd) copy of G(N) in A G. Inuiively, an arc of H G i a relaxed morphim G H which map arc wih a common ource (arge o arc wih a common ource (arge. Of coure, uch a morphim doe no correpond, in general, o a morphim G H: in fac, i i a farher from being uch a morphim a i ource and arge are differen : when hey are equal, he arc i a elf-loop, and i repreen a morphim G H (a i hould, ince he global elemen 1 H G are in bijecion wih he morphim G H). In he nex picure, we how a worked ou example. The econd copy of G(N) i denoed by overline, and we decribed he map {a, b, c} {0, 1} (or {ā, b, c} {0, 1}) by liing in order he number aigned o each elemen. G a b c H 0 1 A G a b c H G ā b c Finally, he claifier can be buil by conidering ubobjec of he repreenable. In general, for Se Cop we have (X) = {S S i a ubobjec of C(, X)}, and he morphim : 1 i given by (X) = X (i.e., i value on X i he repreenable on X). In our cae, he repreenable N ha ju wo ubobjec, 0 N and N, which provide he node e of, while he repreenable A ha five ubobjec, namely 0 A, A,, and (, which provide 4

5 he arc e 2. Thi give u he following graph: 0 A 0 N where he rue arrow : 1 end he only node and arc of 1 o N and A, repecively. Given a ubgraph m : S G, he claifying map χ m : G work a follow: node which are no in S are mapped o 0 N ; node which are in S are mapped o N; if an arc i no in S, we have four poibiliie: arc whoe ource and arge are no in S are mapped o 0 A ; arc whoe ource i in S, bu he arge i no, are mapped o ; arc whoe arge i in S, bu he ource i no, are mapped o ; arc having boh ource and arge in S are mapped o ( ; finally, an arc in S i mapped o A. In oher word, he poibiliie for node are ju wo (a node may, or may no, be in a cerain ubgraph); however, he iuaion i much more varied for an arc, o which five differen arc of can be aigned, depending alo on he aignmen of i ource and arge. A we already remarked, i he objec of ruh value of he opo. Logical operaor on hee ruh value can be defined a follow: he conjuncion : i he characeriic map of he ubobjec, : 1 ; he negaion : i he characeriic map of : 1. I i eay o check ha on he node and work exacly like in Se (a we already menioned, node have only wo ruh value). On he arc he iuaion i differen: reduce he ruh value of every pair of arc of o he minimum (he order being 0 A <, < ( < A); for inance, = 0 A, while ( =. The negaion map, on he oher hand, aifie 0 A = A and A = ( = 0A (o ha i forge wheher an arc wa in a ubgraph or no, a long a i ource and arge were). 4 Topologie on Anoher fundamenal ource of new opoi i given by opologie: a opology allow o e ou cerain objec, called heave, which form a new opo (in fac, heave were he moivaing example for opo heory). 2 We are uing a number of common abbreviaion: for every graph G, 0 G i he unique morphim 0 G. The map, : N A are defined in he obviou way, i.e., (N) =. The map ( ) : N + N A i induced by he univeral propery of he um. Ofen we do no diinguih explicily he node and arc componen of a graph morphim. ( N A 5

6 There are many way o define a opology on a opo. Here we will follow Lawvere idea of an elemenary decripion, given by a map j : which aifie a imple e of axiom. Definiion 2 A opology on a opo i a morphim j : uch ha j = (1) j j = j (2) j = ( j j). (3) In oher word, j preerve conjuncion and ruh, and i i idempoen. The original idea behind a Lawvere opology i an (inernal) generalizaion of a Grohendieck opology (which in urn i a caegorical generalizaion of a claical opology); nonehele, he definiion alo applie in iuaion where coninuiy i apparenly aben. In hi cae, j become a combinaorial, raher han a opological objec. In order o ee hi fac more clearly, we claify he nonrivial opologie on G. We remark ha he map : i alway a opology, called he double negaion opology, and ha wo rivial opologie are alway available: he ideniy on and he morphim! 1. Theorem 1 There i exacly one nonrivial opology on G beide he double negaion opology. Proof. The proof i elemenary (in fac, a cae-by-cae analyi). Fir of all noe ha j (N) = N and j (A) = A necearily by (1). If j (0 N ) = 0 N, he only choice we have i o end ( o ielf (which implie j = 1 ) or o A (he image of 0 A, and i forced by our choice for he node); in he la cae, we obain he double negaion opology, becaue : fixe all arc (and node), excep for (, which i en o A. If j (0 N ) = N, hen j (0 A ) = A implie j =! ; indeed, by (3) we have (( ) ) (( )) (( )) (( )) A = j (0 A ) = j 0 A = j j (0 A ) = j A = j, and ince by (2) j ( and j () mu be arc fixed by j, we have j () = j ( = A. Thu, we mu aume j (0 A ) = ( ( (which implie by (3) j () = j ( = ). There are no oher remaining cae. Anoher poible aemen of he la heorem i ha he laice of opologie on i he fourelemen boolean algebra. The inereed reader can check ha he only nonrivial opology beide double negaion i he o-called cloed opology on he global elemen ( : 1, which i defined a ( [8]. Since hi implie ha he double negaion opology i open, we hall call he only nonrivial opology differen from he cloed opology. 5 Cloure and deniy Definiion 3 Given a ubobjec m : S G, wih characeriic map χ m : G, he cloure of S in G (wih repec o he opology j) i he ubobjec S claified by j χ m ; S i aid o be dene if i cloure i equal o G. 6

7 Inuiively, compoiion wih j increae he ruh level of χ m, hu decribing a bigger objec. The axiom on j guaranee ha he behaviour of hi cloure operaion i reaonable, i.e., S S, S = S. The nex wo heorem how ha he opologie on generae cloure operaion and dene ubobjec which correpond o well-known conrucion from graph heory. Recall ha a panning ubgraph (or parial graph) of G i a ubgraph conaining all he node of G, while an induced ubgraph i a ubgraph H conaining all he arc of G connecing node of H. Theorem 2 The cloure aociaed o he cloed opology add o a ubgraph S all he node of he graph, i.e., S i he panning ubgraph generaed by S. The dene ubobjec of G are hoe ubgraph which include all he arc of G. In paricular, here i a minimum dene ubobjec, which i he arc e of G. Proof. Le m : S G be a ubgraph of G, and χ m i characeriic map. The compoiion wih he cloed opology ha he effec of adding all he node of G o S, becaue node previouly mapped o 0 N will be mapped o N; moreover, no arc will be added o S, becaue no arc of i en o A by he cloed opology, excep for A ielf. Thi mean ha a ubgraph S need exacly o include all he arc of G in order o be dene. Theorem 3 The cloure aociaed o he double negaion opology add o a ubgraph H all he arrow which have ource and arge in S, i.e., S i he induced ubgraph generaed by S. The dene ubobjec of G are hoe ubgraph which include all he node of G. In paricular, here i a minimum dene ubobjec, which i he node e of G. Proof. Le m : S G be a ubgraph of G, and χ m i characeriic map. The compoiion wih ha he effec of adding o S all arc of G lying beween node of S; in fac, uch arc are en by χ m o (, and hi arc i in urn en o A by. Thi mean ha a ubgraph S need exacly o include all he node of G in order o be dene. 6 Sheave and eparaed objec Definiion 4 An objec X of a opo wih opology j i aid o be j-eparaed if for every objec Y, every j-dene ubobjec m : S Y, and every morphim f : S X here i a mo one facorizaion g : Y X of f hrough m: m S f Y g X An objec i j-complee uch a facorizaion alway exi. A j-heaf i an objec which i j-eparaed and j-complee 3. 3 We hall uually underand he reference o j if i i clear from he conex. 7

8 In oher word, a morphim ino a eparaed objec X i compleely deermined by i rericion o a dene ubobjec of he domain. If he objec X i alo complee (i.e., a heaf), every morphim defined on a dene ubobjec S Y ino X exend o a unique morphim Y X. Sheave are imporan becaue heir full ubcaegory i again a opo. Geing back o G, i i eay o check ha eparaed graph and heave for he cloed opology are raher rivial: heave are graph wih ju one node, and heir opo i equivalen o Se (via he funcor ( )(A) : G Se); he only eparaed, noncomplee objec i he empy graph. Bu in he cae of he double negaion opology, omehing radically differen happen: Theorem 4 The objec of G which are eparaed for he double negaion opology are exacly he graph wihou parallel arc. The heave are exacly he complee graph (wih elf-loop), and hey form a opo equivalen o Se. Proof. Conider he monomorphim ( : N + N A, and a eparaed graph G wih wo arc a and b beween he node x and y. The morphim a, b : A G defined in he obviou way are boh exenion of ( x y) : N + N G; by eparaene, hi implie a = b. On he oher hand, if G i complee for any pair of node x and y an exenion a : A G of ( x y) : N + N G deermine an arc beween x and y. Thu eparaed (complee) graph have a mo (lea one arc beween each pair of node. On he oher hand, if m : S H i dene, by Theorem 3 he incluion m ha o be a bijecion on he node; hu, every exenion of a map f : S G o H ha o be defined on node a f N ( N ) 1. If G ha no parallel arc, hen he exenion (if i exi) i alo uniquely deermined on he arc, and if G ha an arc beween each pair of node an exenion alway exi. The la aemen i immediae, he equivalence being induced by he rericion of he funcor ( )(N) : G Se. The la heorem ugge ha a reaonable name for graph wihou parallel arc hould be eparaed (he eparaed objec for he cloed opology being uninereing). Noe ha, by one of he myeriou coincidence of mahemaic, claical complee graph are complee in he opo-heoreical ene. 7 Arc labelling and raniion yem We now wan o apply our eparaene reul. Conider an alphabe and he graph (we are lighly abuing he noaion) defined by (N) = { }, (A) = (he arrow are forced by he univeral propery of { }). The lice caegory G/ i formed in he uual way, by aking a objec he morphim g : G, and a arrow he commuing riangle G a H g The fundamenal heorem of opo heory ay ha every lice of a opo i again a opo, and hi fac of coure applie o G/. Indeed, licing i a mo imporan ource of new opoi. h 8

9 Since a morphim g : G mu end all node of G o, and can chooe freely an elemen of for each arc of G, G/ conain he graph arc-labelled on. The ubobjec claifier i eenially he ame, bu we mu add a copy of each arrow for each elemen of (becaue morphim in G/ have o preerve label). Technically, he new ubobjec claifier i given by he fir projecion [8]. Our inere i in he raniion graph of (nondeerminiic) auomaa 4, alo known a labelled raniion yem, whoe caegory ha been widely udied in compuer cience, a i i ued for he emanic of proce algebra (ee, for inance, [9] and [4]). No all objec of G/ are raniion yem; indeed, raniion yem are exacly hoe graph aifying he following condiion: for each pair of node (ae) x, y here i a mo one arc (raniion) labelled by α wih x a ource and y a arge. Thu, he following graph α α i no a raniion yem. I i eay o check ha he proof of Theorem 4 work alo for he lice opo G/ ; only, hi ime we mu require ha here are no parallel arc wih he ame label. Indeed, he double negaion opology i idenical, excep ha i work labelwie. Thi yield he following Theorem 5 Traniion yem labelled on are exacly he objec of G/ which are eparaed for he double negaion opology. Moreover, we can now eaily prove ha Theorem 6 Traniion yem labelled on form a quaiopo. The reul i immediae by Theorem 5 and by [3], where i i hown ha eparaed objec form a quaiopo. Since he caegory of auomaa and he caegory of heir raniion graph (wih graph morphim) are equivalen, we have alo he following Corollary 1 Auomaa on form a quaiopo. A deep analyi of quaiopoi can be found in [12], where everal of heir properie, uch a local careian cloedne, finie compleene, repreenabiliy of relaion and o on, are proved. We ju noe ha he eenial difference beween a opo and a quaiopo i ha ha doe no claify all ubobjec, bu raher hoe ubobjec defined by rong monomorphim. A monomorphim m : X Y i aid o be rong iff every commuaive quare f X d e Y g X m Y wih e an epimorphim ha a diagonal d a indicaed (hi mean ha he riangle commue, o d i unique). In he cae of raniion yem, we have our la 4 By an auomaon on an alphabe we mean a funcion δ : X 2 X ; a morphim beween auomaa δ : X 2 X and δ : X 2 X i a funcion f : X X uch ha f (δ(α, x)) δ (α, f (x)). 9

10 Theorem 7 The rong monomorphim are hoe defining induced ubgraph, i.e., a rong ubobjec of a raniion yem i defined by he removal of a ube of ae. Proof. The neceiy of hi condiion can be een in he following diagram, where X ha an arc a beween x and y and m : S X conain x and y, bu no a: N + N ( x y) ( A S m X I i clear ha no diagonal exi. Sufficiency can be proved by uing he fac ha (he monomorphim of) an induced ubgraph i alway an iomorphim on i image. a 8 Concluion We hope ha he reader appreciaed he rich inernal rucure of (he opologie on) G. Of coure, oher opoi of graph are alo of inere: in paricular, by adding an arrow ε : A N o Ɣ, aifying he imple equaion ε = ε = 1, we obain he opo of reflexive graph [5], in which every node ha an aigned elf-loop which i preerved by morphim (in fac, node hould be more correcly idenified wih uch elf-loop). Thi doe no aler in an eenial way he combinaorial rucure of a ingle graph, bu now morphim can collape arc (he almo andard name in he graph-heoreical lieraure i degenerae map), o all conrucion inide he opo are influenced one can eaily check ha he produc i now anoher claical operaion on graph. Shrimpon [11] purue a mo inereing udy of he rucure of he group-graph of auomorphim in hi new opo. Boh (reflexive and irreflexive) opoi can be made undireced by augmening Ɣ wih a ymmery arrow, i.e., an involuion on A aifying obviou equaion w.r.. and (and poibly ε). The reuling graph are very imilar o claic undireced graph, bu wih a diinguihed feaure: here are elf-loop ha are fixed by he ymmery and elf-loop ha are no, omehing which i no expreible in erm of undireced graph. Thi fac ha a rong impac on he combinaorial rucure of covering, a dicued in [1]. However, in all cae graph ha are eparaed for he double negaion opology are graph wihou parallel arc, which reinforce he idea ha eparaed hould be he righ word for uch graph (no maer wheher hey are alo undireced or reflexive). In oher word, Theorem 4 applie alo o he opoi dicued in hi ecion, o everal caegorie of andard mahemaical rucure (wih heir andard morphim) urn ou auomaically o be quaiopoi: ju o name a few, imple finie undireced (chlich graph (bu wih elf-loop allowed), olerance pace [10] and (of coure) binary endorelaion, a.k.a. digraph wihou parallel arc. Reference [1] Paolo Boldi and Sebaiano Vigna. Fibraion of graph. Dicree Mah., 243:21 66,

11 [2] Franci Borceux. Handbook of Caegorical Algebra 1, volume 50 of Encyclopedia of Mahemaic and I Applicaion. Cambridge Univeriy Pre, [3] Peer Johnone. On a opological opo. Proc. London Mah. Soc., 38: , [4] Sefano Kaangian and Sebaiano Vigna. The opo of labelled ree: A caegorical emanic for SCCS. Fund. Inform., 32:27 45, [5] F. William Lawvere. Qualiaive diincion beween ome opoe of generalized graph. In Caegorie in compuer cience and logic (Boulder, CO, 1987), volume 92 of Conemp. Mah., page American Mahemaical Sociey, [6] F. William Lawvere and Sephen H. Schanuel. Concepual Mahemaic. Cambridge Univeriy Pre, [7] Saunder Mac Lane. Caegorie for he Working Mahemaician. Springer Verlag, [8] Colin McLary. Elemenary Caegorie, Elemenary Topoe. Number 21 in Oxford Logic Guide. Oxford Univeriy Pre, [9] Robin Milner. Communicaion and Concurrency. Inernaional Serie in Compuer Science. Prenice Hall, [10] Yuri A. Shreyder. Tolerance pace. J. Cyberne., 1(2): , [11] John Shrimpon. Some group relaed o he ymmery of a direced graph. J. Pure Appl. Algebra, 72(3), [12] Owald Wyler. Lecure Noe on Topoi and Quaiopoi. World Scienific,