Morphisms of Coloured Petri Nets

Transcription

1 orphsms o Coloured Petr ets Joachm Wehler Ludwg-axmlans-Unverstät ünchen, Deutschland joachmwehler@gmxnet Classcaton: Structure and behavour o nets Abstract We ntroduce the concept o a morphsm between coloured nets Our denton generalzes Petrs denton or ordnary nets A morphsm o coloured nets maps the topologcal space o the underlyng undrected net as well as the kernel and cokernel o the ncdence map The kernel are lows along the transtonbordered bres o the morphsm, the cokernel are classes o markngs o the place-bordered bres The attachment o bndngs, colours, lows and markng classes to a subnet s ormalzed by usng concepts rom shea theory A coloured net s a shea-coshea par over a Petr space and a morphsm between coloured nets s a morphsm between such pars Coloured nets and ther morphsms orm a category We prove the exstence o a product n the subcategory o sort-respectng morphsms Ater ntroducng markngs our concepts generalze to coloured Petr nets Keywords Petr topology, coloured net, shea-coshea par, morphsm, product Introducton The concept o a morphsm between ordnary nets s well-dened: A morphsm maps the nodes and respects adjacency and orentaton The present paper generalzes ths denton to coloured nets For a morphsm between ordnary nets the nverse mage o a place (resp transton) s a place-bordered (resp transton-bordered) subnet Ths property has a smple translaton nto the language o topology The bpartte structure o an undrected net ntroduces two Petr topologes on the set o nodes One o them, the P-topology, has open sets the place-bordered subnets The other has open sets the transtonbordered subnets Each Petr topology expresses all graph theoretcal propertes o an undrected net In partcular, a map respects the adjacency, t s contnous wth respect to one and hence to both Petr topologes Chapter 3 The two Petr topologes o a net surveys some o the topologcal propertes o Petr spaces and ther contnous maps Whle adjacency s a topologcal property, the orentaton o a net supersedes topology Besdes ther orentaton, arcs o a coloured net have weghts An arc weght maps bndng-elements to token-elements These addtonal propertes o a coloured net express a second, algebrac structure It conssts o monods and modules and ther correspondng morphsms These algebrac objects are attached to the open resp closed subsets o the underlyng undrected net They arse rom transtons, beng closed sngletons, and places, beng open sngletons Shea and coshea are sutable mathematcal concepts, to endow a topologcal space wth a amly o algebrac objects Chapter 4 Coloured nets as shea-coshea par ntroduces these concepts It ormalzes a coloured net as a par o a shea and a coshea together wth two morphsm between them (Denton 49) From the kernel o the ncdence morphsm derves the shea o lows It attaches to every closed subset o the Petr space the lows o ths transton-bordered subnet Analogously, rom the cokernel o the ncdence morphsm derves the coshea o markng classes Two markngs, whch are potentally reachable rom each other, are consdered equvalent The coshea o markng classes attaches to every open subset the markng classes o ths place-bordered subnet One essental property o a low s emboded nto the shea denton: A low o a net restrcts to a low on every transton-bordered subnet Smlarly, a markng class o a place-bordered subnet extends to a markng class on every embeddng net Ths property s emboded nto the coshea denton Chapter 4 requres the wllngness o the reader to consder some concepts rom mathematcs, whch are new n computer scence Chapter 5 orphsms o coloured nets denes such morphsms as a morphsm between the two sheacoshea pars (Denton 5) The denton s nspred by the work o Lakos The morphsm has three components: A contnous map between the underlyng Petr spaces, a map between the two sheaves o lows and a map between the two cosheaves o markng classes All three maps must be compatble wth

2 Introducton respect to the ncdence morphsms A morphsm between coloured nets s not necessarly sort-respectng Specal types o morphsms are embeddngs and abstractons Ater addng ntal markngs to the coloured nets we dene morphsms between two Petr nets as morphsms o the underlyng coloured nets, whch map the ntal markng as well as certan actvated occurrence sequences (Denton 58) We prove that a morphsm wth open mage always extends to a morphsm o Petr nets (Proposton 57) orphsms n the sense o Wnskel can be represented - ater a modcaton o ther mage - by our morphsms (Proposton 53) Ths modcaton keeps the behavour For morphsms o ordnary nets no categorcal product s known We prove n Chapter 6 Products o coloured nets the exstence o a product n the category o coloured nets wth ratonal coecents and sortrespectng morphsms (Proposton 64) The product serves as the base or the categorcal product o coloured Petr nets and sort-respectng morphsms (Proposton 66) Applyng the dagonal constructon we derve the exstence o bre products (Proposton 6) All nets n ths paper are assumed to be nte - concernng the number o transtons and places as well as concernng the number o ther colour-elements evertheless, the jont methods o shea theory and the theory o topologcal vector spaces allow to deal wth nnte nets, too Ths wll be detaled elsewhere Runnng example We wll llustrate the new concepts o the paper at the example rom Fgure It shows a morphsm between two nets The source s an ordnary net, the target s a coloured net The coloured net has a sngle transton a wth two bndng-elements b, b and a sngle place u wth one token-element c Each bndng-element consumes and creates two token-elements The topologcal part o the morphsm s a contnous map : between the underlyng Petr spaces, whch maps the nodes o onto the nodes o We want to drect the attenton to the bre structure o, whch results rom Hence we dstngush between T-bres as nverse mages o trans- : = a, a transton-borde- tons and P-bres over places The set { p, p, t, t, t 3, t 4 } spans,a red T-bre The set { p 3, p 4, t 5, t 6 } spans the place-bordered P-bre,u : ( u) = The map renes a transton (resp a place) o by a transton-bordered (resp place-bordered) subnet o (, F, ): > t p p a t t 3 t 4 w w - p 3 t 5 p 4 u t 6 Fgure orphsm onto a coloured net The transton-bordered bre,a has two lows τ := t t 3 and τ := t t t 4 Followng the proposal o Lakos ([Lak997]) we consder them as the bndng-elements o the whole T-bre We consder a markng as a lnear unctonal on the places Then the place-bordered bre,u has one markng class

3 Runnng example 3 [ p ] = [ ] π = 3 p 4 wth respect to potental reachablty We attach t to the whole P-bre as ts token-element In order to extend the contnous map on the level o undrected nets to a morphsm o coloured nets (, F, ):, we wll ntroduce to other maps: The map F wll map bndngs-elements o T-bres to bndng-elements o the target, and the map wll map token-elements o P-bres to tokenelements o the target Both maps have to be compatble wth the ncdence morphsms The nets rom rom Fgure wll serve as our runnng example n Chapter The two Petr topologes o a net Graph-theoretcal propertes o an undrected net can be characterzed by topologcal methods, too Ths nsght marked already the early perod o Petr net theory ([Fer975], [GLT980]) 3 Denton (Petr space) A topologcal space s a Petr space, t satses the ollowng two condtons: Arbtrary ntersectons o open subsets are open, e or any not necessarly nte - amly (U ) I o open sets U the ntersecton I U s open agan Every pont o s ether open or closed The concept o a Petr space expresses the undamental dualty o the bpartton as a topologcal dualty 3 Proposton (The two Petr topologes o a net) For an undrected net = (, ad ) the adjacency relaton ad x denes two Petr topologes on the set o nodes: A subset U s open wth respect to the P-topology, [ p and (p, t) ad or t U ] p U A subset A s open wth respect to the T-topology, [ t and (p, t) ad or p A ] t A Open sets o the P-topology are the place-bordered subnets, whle transton-bordered subnets are open wth respect to the T-topology I a set s open wth respect to one o both topologes, then t s closed wth respect to the other and vce versa Topologcal statements n ths paper wll always reer to the P-topology We denote by j P : P the embeddng wth respect to the P-topology o the dscrete subspace o open ponts, called places Analogously we denote by j T : T the embeddng wth respect to the T-topology o the dscrete subspace o closed ponts, called transtons Correspondngly, or a subset Q we set T : = Q T und P : = Q P Q Q 33 Proposton (Canoncal bass o a Petr space) For a Petr space the amly o open sets p P t T p ~ t s a bass o the P-topology, ts canon- ~ = denotes the smallest neghbourhoud o the closed pont t wth respect to the P-topology The canoncal bass o the T-topology has as members ether a sngle transton p = { p } pre p post p, a place p together wth ts pre- and postset cal bass Here t : {t } pre( t) post( t) or the set 34 Denton (Fbres o a morphsm) Consder a contnous map between two Petr spaces :

4 The two Petr topologes o a net 4 ) The nverse mage o a pont y, consdered as a subspace o, s called the bre o over y and s denoted by y := - (y) It s called a T-bre, y T s a transton, and a P-bre, y P s a place ore generally, we denote by Q := - (Q) the nverse mage o an arbtrary subset Q ) The map s called dscrete, all ts bres equpped wth the subspace topology nherted rom - carry the dscrete topology A contnous map between two Petr spaces : s dscrete, t respects the sorts, e (P ) P and (T ) T Some authors call such maps a oldng 35 Runnng example (Open resp closed sets) Typcal subsets o, whch are open wth respect to the P-topology, are places and the P-bre,u Typcal closed subsets are transtons and the T-bre,a The map : s contnous, but not dscrete 4 Coloured nets as shea-coshea pars Shea theory s the subject, n whch you do topology horzontally and algebra vertcally attrbuted to Auslander A coloured net attaches a set o bndng-elements to every transton and a set o token-elements to each o ts places Varaton o transtons and places as a parameter establshes the common amles o colours These two amles are a specal case o the mathematcal concept o a coshea (resp a shea), whch consders all closed (resp all open) subnets as parameters Frst we dene a preshea and a precoshea makng use o the elegant language o category theory (Denton 4) Then we comment on these concepts more down to earth usng a notaton wth amles Presheaves and precosheaves are only a transton stage, the nal objects are sheaves and cosheaves As a general reerence we recommend ([Bre997], [994]) We denote by Ab the category o Abelan groups or -modules It s an Abelan category, and the concept o an exact sequence s well-dened 4 Denton (Preshea and precoshea) Consder a xed topologcal space as a category P(): Objects are the open subsets o For morphsms between two open subsets there exst two possbltes: I one set contans the other, then there s one sngle morphsm, the embeddng Otherwse the set o morphsms s empty A preshea (resp precoshea) o Abelan groups on s a contravarant (resp covarant) unctor F: P() Ab A morphsm between two presheaves (resp precosheaves) s a natural transormaton between the two unctors The essental property o the unctor s a set o restrctons (resp extensons) between the Abelan groups belongng to derent open subsets These morphsms have to respect composton and denttes 4 Remark (Preshea and precoshea) ) A preshea C o Abelan groups on a topologcal space s a amly o Abelan groups (C(U)) U open and a amly o morphsms n Ab, called restrctons, r V,U : C(U) C(V), V U open, wth r U,U = d and r W,V o r V,U = r W,V or W V U open

5 Coloured nets as shea-coshea par 5 ) A precoshea B o Abelan groups on s a amly o Abelan groups (B(U)) U open and a amly o morphsms n Ab, called extensons, e U,V : B(V) B(U), V U open, wth e U,U = d and e U,V o e V,W = e U,W or W V U open ) The elements o the Abelan groups C(U) (resp B(U)) are called the sectons over U o the preshea C Γ U, C : = C U (resp the precoshea B) A common notaton s 43 Remark (Change o coecents and postvty) Besdes the category Ab we consder presheaves and precosheaves wth values n derent categores They arse rom extendng the coecents o the sectons wth respect to a dstngushed bass along the chan o nclusons Q The algebrac operaton or the extenson Q s the tensor product It extends a -module B to the Q-vector space B Q := B Q and smlar presheaves and precosheaves The cone o non-negatve sectons s ormed by sectons wth coecents rom resp Q A morphsm, whch maps non-negatve sectons to non-negatve ones, wll be called sgned morphsm A shea adds to the propertes o a preshea the possblty to glue global sectons rom local ones A secton o a shea s characterzed by a product o local sectons, whch concde ater restrcton on ther common doman o denton Analogously, a secton o a coshea s represented by a sum o local sectons It s an equvalence class o a coproduct o local sectons The equvalence relaton s generated by sectons, whch extend a common element on the ntersectons 44 Denton (Sheaves and cosheaves) Denote by a topologcal space ) A preshea C o Abelan groups on s a shea, or every open set U and every open coverng (U ) I o U the ollowng sequence n Ab s exact 0 C(U) r Π I C(U ) s Π j,k I C( U j U k ) Both morphsms to a product are determned by the correspondng morphsms to the actors r := r U,U : C(U) C(U ) s jk : Π I C(U ) C( U j U k ), s jk (( ) I ) := r Uj Uk,Uj ( j ) - r Uj Uk,Uk ( k ) ) A precoshea B o Abelan groups on s a coshea, or every open set U and every open coverng (U ) I o U the ollowng sequence n Ab s exact C j,k I B( U j U k ) d C I B(U ) e B(U) 0 Both morphsms rom the coproduct are determned by correspondng morphsms rom the summands d jk : B( U j U k ) C I B(U ), d jk := e Uj,Uj Uk - e Uk,Uj Uk e := e U,U : B(U ) B(U) ) A morphsm between two sheaves (resp cosheaves) s a morphsm between the correspondng presheaves (resp precosheaves) 45 Remark (Shea and coshea as lmt resp colmt) For readers nterested n the categorcal background: Denton 44 expresses the characterstc glueng

6 Coloured nets as shea-coshea par 6 property o a shea (resp coshea) as a lmt (resp colmt) Global sectons C(U) resp B(U) are the equal- zer o C U C( U U j ) resp the coequalzer o C B ( U U j ) CB ( U ) akng use o these types o lmts one can dene sheaves and cosheaves wth values n arbtrary categores I,j I,j I I C I each object o C has an underlyng set, then the orgetul unctor transorms a preshea wth values n C nto a preshea wth values n Set, the category o sets Because the unctor respects lmts, t respects the shea property The orgetul unctor does not respect colmts Thereore a coshea wth values n C has no underlyng coshea o sets n general Instead one has to use the adjont unctor Eg the unctor, whch attaches to a set the correspondng ree algebrac object rom C (Abelan group, Abelan monod, etc), transorms a coshea o sets nto a coshea wth values n C ([Ber99]) To determne a shea or a coshea, t s not necessary to consder n Remark 4 or Denton 44 all open sets It suces to x the sectons over the elements o a bass For a Petr space we wll always take the canoncal bass rom Proposton 33 Usng a contnous map between topologcal spaces one can map (pre)sheaves and (pre)cosheaves n a covarant way rom one topologcal space to the other Sectons o the drect mage shea over an open set are sectons o the orgnal shea over the nverse mage o ths set The nverse mage o an open set under a contnous map s by denton open agan 46 Proposton (Drect mage o sheaves and cosheaves) Consder a contnous map : between two topologcal spaces Let F be a precoshea and G a preshea on The covarant unctor U a F ( U ) on open sets U, s a precoshea F on, the drect mage o F I F s a coshea, then also F s a coshea The contravarant unctor U a G ( U ), on open sets U, s a preshea G on, the drect mage o G I G s a shea, then also G s a shea The drect mage s a natural transormaton between precosheaves: A morphsm between precosheaves on nduces a morphsm between the correspondng drect mages on Ths attachment s compatble wth composton An analogous statement holds or presheaves For a Petr space one can dene the concept o a morphsm between a (pre)coshea wth respect to the T-topology and a (pre)shea wth respect to the P-topology 47 Denton (orphsm between a precoshea and a preshea) Denote by a Petr space A morphsm w: B C between a precoshea B o Abelan groups on the closed sets o and a preshea C o Abelan groups on the open sets o s a amly o morphsms o Abelan groups w U,A : B (A) C (U), U open, A closed, such that or each par o nclusons A A and U U the ollowng dagram commutes B (A ) w U,A C (U ) e r A, A U, U B (A ) w U,A C (U ) Here e denotes the extenson o B and r the restrcton o C A morphsm between a coshea and A, A a shea s a morphsm between the correspondng precoshea and preshea U, U

7 Coloured nets as shea-coshea par 7 In ths paper we take the ollowng denton o a coloured net as our startng pont ([Jen98]): 48 Denton (Tuple-notaton o a coloured net) A coloured net n tuple-notaton (T, P, (B(t)) t T, (C(p)) p P, (w / (t, p)) (t,p) TxP ) comprses: Two non-empty, nte dsjont sets T o transtons and P o places a amly o non-empty, nte sets (B(t)) t T (bndng-elements) a amly o non-empty, nte sets (C(p)) p P (token-elements) and two amles w = (w (t, p)) (t,p) TxP, w = (w (t, p)) (t,p) TxP o -lnear maps between the ree -modules ( bags ) B(t) and C(t), the negatve resp postve ncdence unctons, w (t, p), w (t, p) Hom (B(t), C(p) ) ow we are ready to translate the tuple-denton o a coloured net (T, P, (B(t)) t T, (C(p)) p P, (w / (t, p)) (t,p) TxP ) nto the language o shea theory We consder the underlyng undrected net :=T P as a Petr space We attach to every subset A T o transtons the coproduct o bndng-elements CB(t) Ths attachment s a coshea o sets CB(t) t T t A on the subspace T Smlarly, by attachng to every subset U P o places the product o token-elements C (p) we obtan a shea o sets C(p) on the subspace P In p U p P order to lnk both unctors by the ncdence morphsms, we map them along the correspondng embeddng nto the common Petr space and extend ther values rom sets to ree Abelan monods 49 Denton (Coloured net as shea-coshea par) A coloured net n tuple-notaton has the ollowng representaton as shea-coshea par (, B, C, w / ): Transtons and places orm the Petr space := T P, bndng-elements orm the coshea o bndngs B : = j T CB(t) on the closed subsets o, t T token-elements orm the shea o tokens C : = j P C(p) on the open subsets o p P and the ncdence unctons orm the two negatve resp postve ncdence morphsms w / : B C wth w / U,A: B(A) C(U), U open, A closed, whch are nduced by the unversal property o coproduct and product rom the ncdence unctons w / (t, p): B(t) C(p), (t, p) A x U 40 Runnng Example (Shea-coshea par) The net has two non-empty closed subsets, the set A = { a } and the whole set A = and two nonempty open subsets U = { u } and U = Sectons o the coshea B are the two bndng-elements o the transton: B (A ) = B (A ) = B (a) = {b, b } The only non-trval secton o the shea C s the tokenelement: C (U ) = C (U ) = C (u) = {c} The amles, whch consttute the two ncdence morphsms w / : ( B ) ( C / ), contan the morphsms w ;U,A = ( ): B ( A ) C ( U ) the ree Abelan monods o rank and between

8 Coloured nets as shea-coshea par 8 The state equaton o a coloured net ntroduces the derence w := w - w o the ncdence morphsms To ths end one has to embed the ree Abelan monods o sectons nto ther Abelan group extenson Ths task s perormed by extendng coecents rom to The resultng coshea B has sectons B (A) =CB(t) and the resultng shea C has sectons C (U) = C(p) The state equaton lnks t T A bndng and token-elements The kernel o w are local lows, whch glue to a shea F o Abelan groups on the closed sets o The cokernel o w are classes o markngs wth respect to potental reachablty They glue to a coshea o Abelan groups on the open subsets o It suces to dene F and over the members o the canoncal bass 4 Denton (Flows and markng classes) Consder a coloured net = (, B, C, w / ) wth ncdence morphsm w := w - w : B C ) The shea F o lows on the closed subsets o s dened or elements o the canoncal bass as F(A) := ker [ w p,p : B B (p) C (t) (p) ] p P U A = A = p, p { t } P, t T wth the restrcton F( p ) F(t) = B(t) or t p nduced rom the ncluson C u p B( u) u p B( u) composed wth the canoncal projecton rom the product Sectons rom F(A) are lows o the restrcton o the subnet nduced by A ) The coshea o markng classes on the open subsets o s dened or elements o the canoncal bass as (U) := coker [ w ~ t,t : B C (t) C (p) ~ ( t )] ~ U = t, t T U = { p }, p P wth the extenson (p) = C(p) ( ~ t ) or p ~ t nduced rom the canoncal njecton nto the coproduct composed wth the ncluson C ~ q t C( q) ~ q t C( q) Sectons rom (U) are markng classes o the subnet nduced by U The shea o lows and the coshea o markng classes generalze token-elements and bndng-elements Sectons o F over a closed sngleton orm the ree Abelan group wth bass the bndng-elements o the F t = B t Sectons o over an open sngleton orm the ree Abelan group wth bass the transton token-elements o the place n queston ( p) C( p) = 4 Proposton (Flows and markng classes) Consder a coloured net = (, B, C, w / ) over a Petr space From Denton 4 derves the ollowng orm o lows and markng classes over general closed resp open sets ) A low τ F(A) over an arbtrary closed set A s a amly o bndng-elements τ = b ( a ) w a a T A B, whch satses or every place p P A the low condton ( b ) 0 a T A t p p,t t = ) Every markng class µ (U) over an arbtrary open subset U can be represented by a amly o

9 Coloured nets as shea-coshea par 9 token-elements c c u C ( u ) Another amly c' c' u ( u ) = C C u P U u P U u P U u P U = C CC represents the same markng class, there exsts a step b B ( U) wth c c' = w ( b) ) The ncdence morphsms w / groups : F( A) ( U) w / U, A : B 43 Denton (arkng and step) C PU, T U nduce on the level o sectons a morphsm o Abelan or each closed subset A and open subset U An nteger-valued markng o a coloured net = (, B, C, w / ) s a secton µ Γ( P, ) o the coshea o markng classes over the open set P o all places An nteger-valued step s a secton τ Γ( T, F ) o the shea o lows over the closed set T o all transtons The subcone o non-negatve sectons denes markngs and steps wth coecents rom 44 Remark (arkng and markng class, step and low) For a coloured net = (, B, C, w / ) one has a surjectve morphsm e : ( P, ) (, ) Γ,,P Γ whch projects markngs onto global markng classes The map annhlates those markngs, whch result r : Γ, F Γ T,F, whch rom the rng o a step Smlarly one has an njectve morphsm embeds global lows nto steps 45 Runnng Example (Flows and markng classes) Over the closed subsets A =,a resp A = o the net the shea o lows F has as sectons the ree Abelan groups F ( A ) span < τ τ > resp F ( A ) span < τ, τ τ > wth τ 3 = t 5 t 6 These =, T, =, 3 groups have resp rank and 3 The ncluson A A nduces the canoncal restrcton F (A ) F (A ) Over the open subset U =,u the coshea o markng classes has as sectons the ree Abelan group ( U) = p / span < (, ) > and over U = the ree Abelan group = 3,4 ( U ) = p / span < (,,, ), (,0,,0 ), (,,, ), ( 0,,0, ), ( 0,0,, ), ( 0,0,, ) > =,,4 have rank, the class [ ] 3 Both p serves as a base n both cases The ncluson U U nduces an extenson o markng classes (U ) (U ) by extendng markngs by the zero-markng 46 Remark (Shea-coshea par) ) What s the benet to represent even uncoloured p/t nets as shea-coshea par? In a p/t net every transton has only a sngle bndng-element, every place has only a sngle tokenelement Hence, nether B nor C add to the topologcal propertes o the net any algebrac normaton Qute derent s the sgncance o F and Global sectons Γ (, F ) o F are the T-lows o the net But beng a shea, F provdes much more normaton: It attaches to every transton-bordered subnet ts T-lows, and t compares two subnets - one contanng the other - by ther restrcton o lows The shea F emphaszes the mportance o the T-lows o all transton-bordered subnets as a structural nvarant o the net Γ are equvalence classes o markngs wth respect to potental reachablty The coshea attaches n a unctoral way to every place-bordered subnet all o ts markng classes I one subnet contans the other, then both are compared by the contnuaton o markngs The coshea emphaszes the mportance o markng classes Whch structural net propertes does the coshea express? Global sectons (, )

10 Coloured nets as shea-coshea par 0 wth respect to potental reachablty o all place-bordered subnets as a structural nvarant o the net The shea-coshea approach takes nto account also P-lows A lnear unctonal on markng classes maps two markngs to the same value, one s potentally reachable rom the other Hence the lnear unctonals on the sectons o are P-lows: The dual o the coshea are the P-lows o a net and o all ts place-bordered subnets ) The dentty between P-lows o a net and the dual o holds or arbtrary coloured nets The rst step n the study o dualty s to dene the dual o a coloured net A rst result: The lnear unctonals on the markng classes o a net are the lows o the dual net, the markng classes o the dual net are the lnear unctonals on the lows o the orgnal net Hence P-lows o a net are T-lows on ts dual and vce versa There are two undamental concepts n lnear net theory: Flow n the sense o T-low and markng class They are ormalzed by the shea T on the closed subnets and the coshea on the open subnets Here rom the concept o P-lows derves by dualty 5 orphsms o coloured nets A morphsm between coloured nets s a morphsms between two shea-coshea pars Hence t has as rst component a contnous map between the two Petr spaces nvolved We dene over ths map two other morphsms One maps the shea o lows o both nets, the other maps the coshea o markng classes Both maps are lnked by the ncdence morphsms o the nets 5 Denton (orphsm between coloured nets) A morphsm (, F, ): between two coloured nets = (, B, C, w / ) and = (, B, C, w / ) s ormed by: A contnous map : o the underlyng Petr spaces, a sgned morphsm F : F F between the sheaves o lows and a sgned morphsm : between the cosheaves o markng classes, whch render commutatve the ollowng two dagrams - or w - and w - wth T := T () and P := P () / ( w ) F P,T F;T ( T) F ( T) w ;P ( P) ( P) The morphsm s dscrete, the contnous map s dscrete 5 Proposton (Composton o morphsms) / ;P,T The composton o two morphsms o coloured nets s a morphsm I both are dscrete, then the composton s dscrete, too Proo The proposton s true, because the drect mage o sheaves and cosheaves s a unctor wth respect to contnous maps, QED Two mportant types o morphsms are abstractons and embeddngs T-bres o an abstracton are transton-renements, P-bres are place-renements An abstracton maps every low o the rened net onto a bndng-element o the abstracted net Analogously, every token-element o the abstracted net arses as the mage o a markng class o the rened net An embeddng extends njectvely every bndng resp token-element o the subnet to a bndng resp token-element o the ambent net The subnet contans wth two nodes rom the ambent net also all ncdent arcs

11 orphsms o coloured nets 53 Denton (Abstracton, embeddng) A morphsm o coloured nets (, F, ): s named abstracton,, F and are surjectve, embeddng, s an embeddng o topologcal spaces, whle F and are njectve 54 Runnng Example (Abstracton) Wth the notaton o Example 40 and 45 we dene the morphsm (, F, ): on the level o F a = F A, F = F We dene lows The drect mage has sectons A b =, F ;a : ( F )( a) F ( a), τ a b, =,, and F; : ( F ) F ( ), τ a On the 0 = 3 u = U, = We dene level o markng classes we have ;u : U ( )( u) ( u), [ p ] a c and : ( ), [ p ] a c 3 ; compatblty we have to very the commutatvty o / ( w ) u,a F F;a ( a) F ( a) w ;u ( u) ( u) / ;u,a 3 To prove the 0 Wth respect to the bases ntroduced above ths reduces to the matrx-equaton ( ) = ( ) 0 All components o the morphsm are surjectve, hence t s an abstracton o coloured nets 55 Remark (Relaton to the work o Lakos) Lakos dentes three orms o ncremental changes or coloured nets He ormalzes each one by a certan type o morphsm In our nterpretaton they are specal cases o our denton 53: A morphsm, whch captures a subnet renement ([Lak999], De 49), s an embeddng wth F and beng bjectve A morphsm, whch captures a node renement ([Lak999], De 47), s an abstracton; we consder ([Lak999], De 3) to be a msprnt A morphsm, whch captures a type-renement ([Lak999], De 4), s an embeddng wth beng the dentty So ar we have dealt wth the structural aspect o coloured nets A morphsm o coloured nets maps the structural propertes o a net, expressed by ts topology, ts lows and ts markng classes We now add a dstngushed ntal markng and consder the behavoural aspect o Petr nets We requre that a morphsm o coloured Petr nets compares also the ntal markngs As a consequence t maps even the behavour, there are no topologcal obstructons These obstructons are transtons n the mage wth a pre- and postset, whch s not ully contaned n the mage Such transtons cannot appear, the map s surjectve or - at least - has open mage In ths paper we dene the behavour o Petr nets by occurrence sequences A morphsm maps saturated occurrence sequences and the change o markng, whch results rom ther rng Saturated occurrence sequences lp between two types o components: Occurrence sequences o one type occur wthn a T-bre and have a low as Parkh-vector Occurrence sequences o the other type occur n a P-bre It s not excluded, that some components are zero A saturated occurrence sequence does not change the markng wthn T-bres, but t may change the markng across ts border Ths change s mapped to the mage net 56 Denton (Saturated occurrence sequence) Denote by (, F, ): a morphsm o coloured nets An occurrence sequence σ o s satu-

12 orphsms o coloured nets rated, t s a catenaton o ntely many occurrence sequences o σ = σ T, σ P, σ T, σ P, σ T,n σ P,n,, a T, s a low Every Parkh- wth the ollowng property: Every Parkh-vector τ ( σ T, ) F ( a ) vector τ ( σ P, ) B ( u ) vector o a low maps to a bndng-element ( τ ) B ( a ), u P, belongs to a P-bre Due to the non-negatvty o F, each Parkh- F ; a σ T, o all these bndng-elements denes an occurrence sequence, named ( σ), =,,n, o and the catenaton F;T 57 Proposton (appng o the behavour) A morphsm o coloured nets (, F, ):, whch has open mage (), maps an actvated, saturated occurrence sequence o to an actvated occurrence sequence o : I µ pre s a markng and σ a saturated occurrence sequence o, then µ,pre σ µ,post ;P F;T σ µ µ,pre Proo It suces to prove the proposton or occurrence sequences o length wthn a T-bre Consder a transton a T, a transton t - (a) and a bndng-element b B (t), whch s actvated at the mark- ng µ,pre For all places p holds µ,pre ( p) = w ;p, t ( b) µ ( p) wth a token-element ( p) C ( p) Because () s open, also ~ a For arbtrary, but xed place u ~ a holds ;u ;P,post ( µ ) = ( w ( b) ) ( µ ) = w ( ( b) ) ( µ ),pre Varaton o the place a ~ u ;u ;u,t ;u u ;u,a F;a ;u u as parameter shows ( µ ) = w ( ( b) ) ( µ ) ;P,pre ;P,a F;a ; P µ u Hence the bndng-element F;a (b), whch s the mage o the orgnal occurrence sequence, s actvated at the mark- µ, whch s the mage o µ pre under the sgned map The nal statement about the result- ng ;P,pre ng markng ollows rom the state equaton = µ w ( b) ;P µ by applyng,post,pre ;P, t ;P : ( µ ) = ( µ ) ( w ( b) ) = ( µ ) w ( ( b) ),post ;P,pre ;P ;P,t 58 Denton (orphsm o coloured Petr nets) ;P,pre ;P,a F; a, QED A morphsm o coloured Petr-nets (, F, ): (, µ ) (, µ ) s a morphsm o the correspondng coloured nets (, F, ):, whch satses the two ollowng condtons: The composton wth the extenson map o the coshea and Γ e ( P ),P ( P, ) Γ ( P ) ;P (, ) Γ( P ), maps the ntal markng o (, µ ) to the ntal markng o (, µ ): e ( µ ) ;P ( ) = µ F;T maps every actvated, saturated occurrence sequence σ n (, µ ) to an actvated occur- σ σ rence sequence o (, µ ): [µ > σ [µ > F;T F;T P,P 59 Remark (orphsm o coloured Petr nets) ) The second requrement o Denton 58 s always satsed, the contnous map s surjectve or has an open mage (Proposton 57) ) A dscrete morphsm (, F, ): satses ( P ) = P and ( T ) = T In ths case the

13 orphsms o coloured nets 3 extenson map n Denton 58 s the dentty and every occurrence sequence s saturated The denton smples as ollows: A dscrete morphsm denes a morphsm (, F, ): (, µ ) (, µ ) o coloured Petr nets, maps the ntal markngs and maps actvated occurrence sequences ;P F;T 50 Denton (odcaton) A surjectve, dscrete morphsm (, F, ): between two coloured net s a modcaton, both o the maps F : F F and : are sgned somorphsm A modcaton wll be called a place-modcaton (resp transton-modcaton), all ts T-bres (resp ts P-bres) are sngletons A well known example s the unoldng o a coloured net by an uncoloured p/t net The uncoloured net maps as a modcaton onto the coloured net The map does not the change the behavour o the net 5 Proposton (Invarance o the behavour under modcaton) between two coloured nets extends or arbtrary ntal mark- A modcaton (, F, ): ngs µ o to a morphsm o Petr nets (,, ): (, ), ( µ ) nduces a bjecton o reachable markngs F µ Ths morphsm Proo A modcaton extends to a morphsm o Petr nets by Proposton 57 It nduces a bjectve map ( ) µ and consder an occurrence sequence µ µ, post Wthout re- ;P o markngs Γ( P, ) = Γ ( P ), = Γ P, Γ P, o reachablty Set : = ;P ( µ ) strcton σ s a sngle bndng-element b B ( a) o a transton T phsm, there s a unque non-negatve low τ wth b µ τ µ,post,p, whch respects the relaton, then = ( µ ),post ;P,post T a µ, QED 5 Remark (Relaton to the work o Wnskel) σ a Because F; a s a sgned somor- τ It s actvated at µ I F ; a = A Wnskel-morphsm = (β, η): between two p/t nets n tupel-notaton s a par (β, η): One component s a multrelaton β: P P between the places, the other a partal uncton η: T T between the transtons and both satsy β( pre (t) ) = pre ( η(t) ) and β( post (t) ) = post ( η(t) ) or all t T ([W995]) The ollowng derences between Wnskel-morphsms and morphsms n the sense o Denton 5 attract notce: A Wnskel-morphsm s not necessarly globally dened The doman o ts multrelaton and ts partal uncton may be a proper, closed subset o the source net A Wnskel-morphsm s not necessarly contnous wth respect to the Petr-topology: A gven place o the source may be related to many derent places o the target net A Wnskel-morphsm has open codoman Hence there are no obstructons aganst the mappng o the behavour (Proposton 57) It wll turn out, that a Wnskel-morphsm can be represented by a morphsm n the sense o Denton 5: The restrcton o the Wnskel-morphsm on ts doman and a subsequent modcaton o ts codoman s a dscrete morphsm between coloured nets The modcaton contracts those places o the codoman, whch are related to the same place o the doman The modcaton smoothes the dscontnutes, but keeps the behavour by colourng the contracton 53 Proposton (orphsm and Wnskel-morphsm) A Wnskel-morphsm = ( β, η ) : ~ ~ has closed doman : dom( β) dom( η) ~ = and open

14 orphsms o coloured nets 4 codoman ( ): cod( β) cod( η) ~ = There exsts a place-modcaton π : ~ onto a coloured net, such that the composton o relatons g : π ( β η ) : g, g, g = o s a contnous uncton and extends to a dscrete morphsm o coloured nets ( F ) Proo (Scetch): ad ) From the denton β post ( t) = post ( t ) ~ η and ( t) : ( pre ) = pre ( t) ~ η ~ β ~ ollow both statements, the closedness o the doman and the openess o the codoman, by drect vercaton / ad ) Denote by = (,, C, w ) / We dene the coloured net = (,, C, w ) whch s generated by the relaton ( y ~ y : x P : y ( x) and y ( x) ) ~ β β on the set o equvalence classes : ( ~ / ~ ) B the p/t net, whch s generated by the closed subnet o ~ B as ollows: Consder on ~ the equvalence relaton, -/ -/ ; w ~ ;p,a U The quotent topology = s a Petr topology and the projecton π : ~ s contnous, open and dscrete We dene B : = π B ~, C : = π C ~ and w U,A : = π( p) the composton o the two relatons g : π ( β η ) : By constructon, = o s a map Its contnuty ollows agan by drect vercaton orm the denton o a Wnskel-morphsm ) By part ) and ) the map g : s contnous and dscrete The closed Petr space ~ generates a subnet, whch s a p/t net The target net, whch belongs to the Petr space, has -dmensonal modules o bndngs B ( a ) But possbly the modules o token-elements C ( u) are hgher-dmensonal and the ncdence morphsms w -/ are represented by matrces We dene the morphsm o sheaves g F : g F F We have g F ( a) = F ( a ) = B ( t) = C B ( t) or a closed g( t) = a g( t) = a sngleton a = g(t) The map g : B ( t) F;a g coproduct rom the denttes C B a s nduced by the unversal property o the ( t ) = a d : B t B a For a closed basc set A = u wth a A = F A = F U τ = n t g F A, n g( p) = u place u holds ( g F ) We map a low p onto the element g ( τ ) n g( t ) B ( A) In order to show, that g : ( g F )( A) F ( A) F;A = F;A s well-dened, we have to prove, that g F;A (τ) s a low The condtons on the multrelatons o source and target o mply by denton o the ncdence morphsm n β ( post ( t ) ) n post ( η ( t ) ) n w ( ) = w g ( τ) = Analogously n β( pre ( t ) ) = w g ( τ) ;,t ~ F;A = ;,t mples n β( post ( t ) ) = n β( pre ( t ) ) Hence w g F,A ( τ) ;,q w ( g ( τ ) ) 0 ;,q F, A = cosheaves g : g ;,t The low-condton n post ( t ) = n pre ( t ) F;A = w ( g ( τ) ) = 0 It s let to the reader, to gve an analogous denton or the morphsm o, QED 54 otatons (Categores o coloured nets) We denote by et the category o all coloured nets and morphsms accordng to Denton 49 and 5 Sheaves, cosheaves and morphsms rom ths category are dened over the monod and ts Abelan group extenson Sectons have natural or nteger numbers as coecents Due to the lnear structure o the category et one can easly extend the rng o coecents (Remark 43) and consder eg nets wth ratonal coecents We denote by et Q the category o ratonal coloured nets and ther morphsms We wll omt the subscrpt or sheaves and cosheaves, the context makes clear the rng o denton The ;,q F, A and

15 orphsms o coloured nets 5 subcategores wth only dscrete morphsms are denoted by et-ds resp et-ds Q A Petr net s a marked net We denote by Pet the category o coloured Petr nets and ther morphsms accordng to denton 58 Here rom derves the category Pet Q by rng extenson rom the rng to the eld Q In close analogy to the unmarked case we dene the categores Pet-ds resp Pet-ds Q As a relatvaton o the categores consdered so ar we ntroduce the comma categores et-ds() and et-ds() Q o relatve coloured nets Objects o these categores are dscrete morphsms to a xed coloured net as bass orphsms o these categores are dscrete morphsms, whch commute wth the bass morphsms o ther doman and codoman Products n these categores are called bre products 6 Products o coloured nets In the present chapter we show, that all o the categores et-ds Q, Pet-ds Q and et-ds() Q have products Categorcal products help to buld complex nets rom smpler one or to smply the study o a net by rst checkng, there exst some actors Frst we construct the Kronecker product o coloured nets I we restrct to dscrete morphsms, then the Kronecker product has the unversal property o the categorcal product The Kronecker product o two undrected nets s a subset o the topologcal product o ther nodes amely the subset o sort-respectng pars, ormed ether by two places or by two transtons The parng o a place wth a transton s not allowed as a node o the product net Due to the restrcton to sort-respectng pars also the morphsms o the category are restrcted to dscrete morphsms C (p,p ) = w - w - w w - w w - w - w w w B (t,t ) = w w w w - C (p ) = w B (t ) = w - w C (p ) = B (t ) = Fgure A coloured net, whch s the Kronecker product o two ordnary nets Products o nets and Petr-nets have been ntroduced by Dörler n the settng o p/t nets and sort-respectng maps ([Dör976]) They are named Kronecker product, because they are a close analogue to smlar constructons rom graph theory Coloured nets had not yet been nvented at the tme o Dörlers paper Our denton or the Kronecker product o coloured nets bulds on Dörlers denton or the underlyng undrected net and constructs on ths base a coloured net Fgure shows as an example the Kronecker product o two ordnary nets The product s a coloured net, each node has a -dmensonal module o colours A transton o the product has two types o bndng-elements: Ether each bndng-element o ts actors separately or both as a synchronzed mode Both bndng-elements keep ther pre- and postsets rom ther actor 6 Denton (Kronecker product) The Kronecker product o two coloured nets = (, B, C, w / ), =,, s the ollowng coloured net := := (, B, C, w / ): The Petr space : : = ( P x P ) ( T x ) x o the product, = carres the nduced topology as a subspace T

16 Products o coloured nets 6 sectons o the coshea B o bndngs over the product o closed sets are ( A A ): = ( B ( a ) B ( a ) ) B C, ( a,a ) TA xta sectons o the shea C o tokens over the product o open sets are ( U U ): ( C ( u ) C ( u ) ) C =, u,u PU xpu the extensons o the coshea B and restrctons o C are nduced by the product o the correspondng maps o both actors and the two ncdence morphsms w / : B C are nduced over product sets by the maps w / ;U U,A A ( B ( a ) B ( a ) ) : C C( u) C( u ), ( a,a ) ( u,u ) whch are sum and product o the ncdence maps o both actors w : ( a ) ( u ),, / ;u,a B C = We denote the two projectons o the topologcal product by p :, (x, x ) a x, =, 6 Remark (Flows and markng classes o the Kronecker product),f;a a T A a, Let = (, B, C, w / ) := denote the Kronecker product o two coloured nets Over the product o two closed sets A = A A a secton τ = C ( τa, τa, ) B ( A) s a low τ F ( A), ts a= ( a,a ) trace p ( τ) : = τ F ( A ), =,, s a low n both actor nets Varaton o A as a parameter denes two shea morphsms, the trace o lows, markng classes p, : p p,f : p F F, =, To dene the ratonal trace o Q one has to take nto account, that the product reproduces each place o the rst actor wth multplcty equal to the number o places o the second actor Ths amplcaton on the level o topology has to be corrected by a correspondng dvson on the level o markng classes Thereore we extend the doman o coecents rom to the eld Q Over the product U = U wth open set U a markng class π = C ( π π ) ( U ) = ( p ) ( U ) maps to p ( π) : = card ( P ) π,;u u, U u U Q p, between ratonal cosheaves Smlarly we dene the morphsm p 63 Proposton (Projectons o the Kronecker product) u u, u, Varaton o U as a parameter denes the morphsm, : p,f, = Both projectons o the Kronecker product ( p, p, p ):,, the category et-ds Q 64 Proposton (Categorcal product or dscrete morphsm) The Kronecker product wth ts two projectons s the product n the category et-ds Q Proo We show that the Kronecker product (,p, ) Q, are morphsms n p o two ratonal coloured nets together wth the two projectons rom Remark 6 has the unversal property o the product We consder a ratonal,, :, =, Due coloured net = (, B, C, w / ) and two dscrete morphsms (,F, ) to the unversal property o the topologcal product the two contnous maps nduce a contnous map nto

17 Products o coloured nets 7 the product : : x, y ( ( y), ( y) ) = a It s dscrete and ts mage s contaned n =, because both actor maps respect the sorts oreover t s unquely determned by the condton p o =,, In the next step we extend the contnous map to a morphsm o coloured nets = (, F, ): Frst we dene the morphsm o ratonal sheaves F : F F ; we drop the subscrpt Q For a gven transton a = (a, a ) actorzes ( F ) ( a) = F ( ) = F ( ) = B a a a t ( t ) = aund ( t) = a as a product o bndng-elements The canoncal embeddng o the ntersecton nto the product nduces a map We dene by composton F;a : = B ( t ) a und ( t ) ( t) B ( t) = a ( t) = = a ( t ) = a B ( t) ( ) F ( ) F ( ) F ( ) τa τ τ F a = a a a a,,f;a,f;a [ F ( a ) F ( a ) F ( a ) F ( a) F ( a ) = F ( a) ] Due to the nteness o all nets there s no derence between the product and the coproduct o sectons over closed sets A Because the bres are dscrete we have We dene the composton F;A F ( ) B ( ) = F ( ) and ( a) B ( A) A A a F = F;a : = F A F a F ( a) = B ( A) It remans to show, that the mage o ths map s a low F,A ( τ) F ( A) B ( A) or τ = ( ) F ( ) τ a F A Consder an arbtrary place p A o rom the open kernel o A Because both maps, =,, are morphsms, we have w o = o w We conclude w ;p,a,f;a,;p ( ( τ) ) = w ( τ ) ;p, a = ( w ( ( τ ) ) w ( ( τ ) ) ) ;p,a F;A ;p,a F;a a ;p,a,f;a a, ;p,a,f;a ( ( ( w )( τ ) ) ( ( w )( τ ) ) ) =,;p ;p,a a,,;p ;p,a ( ) w ( τ ) w ( τ ) ( ) =,;p,;p ;p,a a, ;p,a ( ) ( w w )( τ τ ) = ( ) ( w )( τ),;p,;p ;p,a ;p,a a, a, a, a, a a, = 0,;p,;p ;p, = A By constructon all local maps F,A are compatble wth restrctons Hence they dene a shea morphsm : F F It s unquely determned by the equatons p ( ) o p =,, We leave to the F reader the analogous denton : F,F, F = and the check, that (, ) F, : s

18 Products o coloured nets 8 a morphsm, whch completes a commutatve dagram, QED A unversal object lke the product s determned only up to a canoncal somorphsm In et-ds Q we obtan derent products by varaton o the scalng actor n the denton o the traces ow we ntroduce ntal markngs and consder Petr nets We wll select a dstngushed representaton by xng the scalng actor wth a condton on the ntal markngs 65 Denton (Kronecker product o coloured Petr nets) The Kronecker product o two coloured Petr nets (, µ ), =,, s the coloured Petr net (, µ ) (, µ ) := (, µ µ ) wth the product markng ( P, ), ( µ µ ) ( p, p ): = µ ( p ) µ ( p ) C ( p ) C µ µ Γ p A markng µ o the product s saturated, = ( µ ) ( µ ) µ, p, 66 Proposton (Categorcal product o Petr nets) p wth p,, =,, the trace morphsms The Kronecker product o Petr nets wth ts two projectons s the product n the category Pet-ds Q Even both coloured Petr nets belong to Pet-ds, ther product does not satsy the unversal property n ths category, because the trace maps are only dened over Q In the ollowng Proposton 67 we show, that a markng n the product s reachable, both o ts actor markngs are reachable n the correspondng actor nets I these actors belong to Pet-ds, then also the ntal markng and all reachable markngs n the product as well as ther traces have coecents rom 67 Proposton (Reachablty n the product o Petr nets) Reachable markngs n the product o two Petr nets rom Pet-ds Q correspond bjectvely to pars o reachable markngs n both actor nets For actor nets rom Pet-ds ths bjecton restrcts to a bjecton o nteger-valued markngs Proo Denote by (, ): = (, µ ) ( µ ) µ, the Kronecker product o two coloured Petr nets Both projectons (, µ) (, µ ), =,, are morphsms o Petr nets by Proposton 57 Hence the product ( P, ) Γ ( P, ) ( P ) p,;p p,;p : Γ Γ, restrcts to a map o reachable markngs p [ µ [ µ [,;P p,;p : µ The ntal markng µ s saturated In order to prove the njectvty o the map, we show that saturated occurrence sequences transorm saturated markngs nto saturated markngs Consder a saturated occurrence sequence σ wth Parkh-vector τ = τ(σ) We assume wthout loss o generalty that σ s the bndng-element τ = τ τ B ( t) = B( t) B( t ) o a sngle transton t = ( t, t ) Denote by µ pre a saturated markng, whch actvates τ, and denote by µ post the markng, whch results rom the rng o τ at µ pre We apply p, =,, to the state equaton post = µ pre w ;, ( τ) and obtan p, ( post ) = p, ( µ pre ) w ;,a p,t ( τ ) µ a µ Spellng out the components o the state equaton shows ( p ) = µ ( p ) µ ( p ), ( w ( τ ) w ( τ )) = µ post pre, pre, ;p,a ;p,a ( µ ( p ) w ( τ ) ) µ ( p ) w ( τ ) =, ;p,a pre, ;p,a pre Comparng the result under varaton o the place p shows = p ( µ ) p ( µ ) µ Hence the markng µ post s saturated In order to show the surjectvty o the map under consderaton we start wth post, post, post

19 Products o coloured nets 9 two occurrence sequences σ, =, rom the actors Wthout loss o generalty, each o them s a sngle τ B t Reversng the drecton o the above arguments shows, that n the product bndng-element ( ) the occurrence sequence σ wth the sngle bndng-element = τ ( t) = B ( t ) B τ τ B t s actvated at the product markng µ = µ µ, QED 68 Remark (Dagonal morphsm) Accordng to Proposton 64 each coloured net has a dagonal morphsm d d: wth the ollowng unversal property: For a coloured net and two morphsms :, =,, n et-ds Q, the product : actorzes over the dagonal as, = We prove, that the dagonal morphsm maps somorphc onto a subnet o the product, e t embeds nto the product 69 Proposton (Exstence o the dagonal) For a coloured net the dagonal morphsm nduces an somorphsm δ: onto a subnet (the dagonal) / / Proo We dene the dagonal o = (, B, C, w ) as the coloured net : = (,, C, w ) The Petr space : = {( x,x) : x } has the subspace topology Bndng-elements over closed product sets are B ( A A) : = B A B A, B ( A), the dagonal o the -module Token-elements over open product sets are C U U : = ( ) C ( U), the dagonal o the -module ( U) C ( U) C, d d B : The extensons and restrctons o the coshea B resp the shea C as well as the ncdence morphsms w result rom the correspondng morphsms o B resp C / The dagonal has the lows F ( A) and the markng classes ( U) = In order to consder the dagonal as a subnet B F ( a ) A = { t }, transton t = ( a,a) ( u ) A = p, place p = ( u,u) ( u) U = { p }, place p = ( u,u) ~ ( a ) = = ~ U t, transton t a,a C =, we have to extend the embeddng o topologcal spaces j: to a morphsm o coloured nets The morphsm o sheaves dened over a basc closed set A as the embeddng o ( A) F ( A) = ker F nto the -module j F : B ( a) B ( a) A = { t }, transton t = ( a,a) [ F ( p ) F ( p ) ] A = p place p = ( u,u) A smlar embeddng o modules denes the morphsm o cosheaves j : F F s It s let to the reader, to very that (j, j F, j ): s an embeddng o coloured nets and that the dagonal morphsm d d: restrcts to an somorphsm δ:, QED 60 Proposton (Inverse mage o dscrete morphsms) For a dscrete morphsm : o coloured nets and a subnet j : there exsts a subnet j : ( ), the nverse mage, such that the ollowng dagram s Cartesan

20 Products o coloured nets 0 ( ) j j / = V V CV V Proo We dene the coloured net ( ): = (V,,,w ) B as ollows: The set-theoretcal nverse mage V : t T V are dened as B V ( t) : F; ( t) ( jf; ( t) ( B ( ( t) )) ) B ( t) B ( A) : = B ( t) Token-elements o a place p P V are C ( p) : j ( B ( ( p) )) V C t T A V = nherts the subspace topology rom Bndng-elements o transton Sectons over arbtrary open sets are ( U) : = C ( p) = Sectons over arbtrary closed sets are / V ; p C ( p) = ; p C V V The ncdence morphsms w / V : B V C V p P U are dened as restrctons o w : B C Dagram chasng shows, that these dentons determne a subnet V, whch completes the above dagram as a Cartesan dagram, QED The exstence o bre products ollows by a standard constructon rom the exstence o products, the dagonal and the nverse mage A posteror the dagonal and the nverse mage turn out as specal cases o the bre product 6 Proposton (Fbre product o coloured nets) The category et-ds() Q has a product, t s a subnet o the product n et-ds Q Proo For two objects g, =,, rom et-ds() Q we dene the coloured net as nverse mage o the dagonal under the morphsm ( g g ) : = g g object o et-ds() Q wth ( g g ) (Proposton 60) It s an as bass morphsm The exstence o a morphsm = :, as requred or the unversal property, ollows rom the unversal property o and the actorzaton o the nduced morphsm over the dagonal, QED 7 Concluson and uture work The class o morphsms studed n the present paper s much more general than the class o sort-respectng morphsms The only restrcton or the underlyng maps s to be contnous, whle sort-respectng morphsms have dscrete bres n addton An mportant open queston concerns the exstence o products n the category et Q, whch has arbtrary contnous, but not necessarly dscrete morphsms An nterestng approach to morphsms between coloured nets s due to Keller [Kel000] He works n a strct categorcal settng and ormalzes a coloured net as the morphsm, whch maps ts unoldng onto ts underlyng uncoloured net It would be useul to compare Kellers representaton wth the shea-coshea approach Also the shea-unctor can be represented by a map over the underlyng topologcal space (espace étalé) Abstractng rom the orgn o the shea-coshea par n Chapter 4 one can try to generalze the topologcal and algebrac stuaton: A topologcal space merges two dsjont, dscrete subspaces, they are lnked by two morphsms between a coshea on one o ths spaces and a shea on the other Another type o generalzaton deals wth nnte nets, whch allow also nntely many colourelements or a gven node From a mathematcal pont o vew one perorms the eld extenson