Trees as operads. Lecture A formalism of trees

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1 Ltur 2 rs s oprs In this ltur, w introu onvnint tgoris o trs tht will us or th inition o nroil sts. hs tgoris r gnrliztions o th simpliil tgory us to in simpliil sts. First w onsir th s o plnr trs n thn th mor gnrl s o non-plnr trs. 2.1 A ormlism o trs A tr is non-mpty onnt init grph with no loops. A vrtx in grph is ll outr i it hs only on g tth to it. All th trs w will onsir r root trs, i.., quipp with istinguish outr vrtx ll th output n (possily mpty) st o outr vrtis (not ontining th output vrtx) ll th st o inputs. Whn rwing trs, w will lt th output n input vrtis rom th pitur. From now on, th trm vrtx in tr will lwys rr to rmining vrtx. Givn tr, w not y V ( ) th st o vrtis o n y E( ) th st o gs o. h gs tth to th lt input vrtis r ll input gs or lvs; th g tth to th lt output vrtx is ll output g or root. h rst o th gs r ll innr gs. h root inus n ovious irtion in th tr, rom th lvs towrs th root. I v is vrtx o init root tr, w not y out(v) th uniqu outgoing g n y in(v) th st o inoming gs (not tht in(v) n mpty). h rinlity o in(v) is ll th vln o v, th lmnt o out(v) is th output o v, n th lmnts o in(v) r th inputs o v. As n xmpl, onsir th ollowing pitur o tr: I. Morijk n B. oën, Simpliil Mthos or Oprs n Algri Gomtry, Avn Courss 11 in Mthmtis - CRM Brlon, DOI / _2, Springr Bsl AG 2010

2 12 Ltur 2. rs s oprs v w h output vrtx t th g n th input vrtis t, n hv n lt. his tr hs thr vrtis r, v n w o rsptiv vlns 3, 2, n 0. It lso hs thr input gs or lvs, nmly, n. h gs n r innr gs n th g is th root. A tr with no vrtis whos input g (whih w not y ) oinis with its output g will not y η, or simply y η. Dinition A plnr root tr is root tr togthr with linr orring o in(v) or h vrtx v o. h orring o in(v) or h vrtx is quivlnt to rwing th tr on th pln. Whn w rw tr w will lwys put th root t th ottom. On rwk o rwing tr on th pln is tht it immitly oms plnr tr; w thus my hv mny irnt piturs or th sm tr. For xmpl, th two trs r two irnt plnr rprsnttions o th sm tr. 2.2 Plnr trs Lt plnr root tr. Any suh tr gnrts non-σ opr, whih w not y Ω p ( ). h st o olours o Ω p ( ) is th st E( ) o gs o, n th oprtions r gnrt y th vrtis o th tr. Mor xpliitly, h vrtx v with input gs 1,..., n n output g ins n oprtion v Ω p ( )( 1,..., n ; ). h othr oprtions r th unit oprtions n th oprtions otin y ompositions. his opr hs th proprty tht, or

3 2.2. Plnr trs 13 ll 1,..., n,, th st o oprtions Ω p ( )( 1,..., n ; ) ontins t most on lmnt. For xmpl, onsir th sm tr pitur or: v h opr Ω p ( ) hs six olours,,,,, n. hn v Ω p ( )(, ; ), w Ω p ( )( ; ), n r Ω p (,, ; ) r th gnrtors, whil th othr oprtions r th units 1, 1,..., 1 n th oprtions otin y ompositions, nmly r 1 v Ω p ( )(,,, ; ), r 3 w Ω p ( )(, ; ), n r(v, 1, w) = (r 1 v) 4 w = (r 3 w) 1 v Ω p ( )(,, ; ). his is omplt sription o th opr Ω p ( ). Dinition h tgory o plnr root trs Ω p is th ull sutgory o th tgory o non-σ olour oprs whos ojts r Ω p ( ) or ny tr. W n viw Ω p s th tgory whos ojts r plnr root trs. h st o morphisms rom tr S to tr is givn y th st o non-σ olour opr mps rom Ω p (S) to Ω p ( ). Osrv tht ny morphism S in Ω p is ompltly trmin y its t on th olours (i.., gs). h tgory Ω p xtns th simpliil tgory. In, ny n 0 ins linr tr L n n 2 1 with n + 1 gs n n vrtis v 1,..., v n. W not this tr y [n] or L n. Any orr-prsrving mp {0,..., n} {0,..., m} ins n rrow [n] [m] in th tgory Ω p. In this wy, w otin n ming v n v 2 v 1 0 u Ω p. his ming is ully ithul. Morovr, it sris s siv (or il) in Ω p, in th sns tht or ny rrow S in Ω p, i is linr thn so is S. In th nxt stions w giv mor xpliit sription o th morphisms in Ω p. w

4 14 Ltur 2. rs s oprs F mps Lt plnr root tr n n innr g in. Lt us not y / th tr otin rom y ontrting. hn thr is nturl mp : / in Ω p, ll th innr mp ssoit with. his mp is th inlusion on oth th olours n th gnrting oprtions o Ω p (/), xpt or th oprtion u, whih is snt to r v. Hr r n v r th two vrtis in t th two ns o, n u is th orrsponing vrtx in /, s in th pitur: / u w v w Now lt plnr root tr n v vrtx o with xtly on innr g tth to it. Lt /v th tr otin rom y rmoving th vrtx v n ll th outr gs. hr is mp ssoit to this oprtion, not v : /v, whih is th inlusion oth on th olours n on th gnrting oprtions o Ω p (/v). hs typs o mps r ll th outr s o. h ollowing r two outr mps: /v w v v w w v /w Not tht th possiility o rmoving th root vrtx o is inlu in this inition. his sitution n hppn only i th root vrtx is tth to xtly on innr g, thus not vry tr hs n outr inu y its root. hr is nothr prtiulr sitution whih rquirs spil ttntion, nmly th inlusion o th tr with no vrtis η into tr with on vrtx, ll oroll. In this s w gt n + 1 mps i th oroll hs n lvs. h opr Ω p (η) onsists o only on olour n th intity oprtion on it. hn mp o oprs Ω p (η) Ω p ( ) is just hoi o n g o. W will us th trm mp to rr to n innr or outr mp Dgnry mps hr is on mor typ o mp tht n ssoit with vrtx v o vln on in s ollows. Lt \v th tr otin rom y rmoving th vrtx v n mrging th two gs inint to it into on g. hn thr is mp

5 2.2. Plnr trs 15 : \v in Ω p ll th gnry mp ssoit with v, whih sns th olours 1 n 2 o Ω p ( ) to, sns th gnrting oprtion v to i, n is th intity or th othr olours n oprtions. It n pitur lik this: 1 v 2 F mps n gnry mps gnrt th whol tgory Ω p. h ollowing lmm is th gnrliztion to Ω p o th wll-known t tht in h rrow n writtn s omposition o gnry mps ollow y mps. For th proo o this t w rr th rr to Lmm 2.3.2, whr w prov similr sttmnt in th tgory o non-plnr trs. Lmm Any rrow : A B in Ω p omposs (up to isomorphism) s A B σ whr σ : A C is omposition o gnry mps n δ : C B is omposition o mps Dnroil intitis In this stion w r going to mk xpliit th rltions twn th gnrting mps (s n gnris) o Ω p. h intitis tht w otin gnrliz th simpliil ons in th tgory. Elmntry rltions Lt : / n : / istint innr s o. It ollows tht th innr s : (/)/ / n : (/)/ / xist, w hv (/)/ = (/)/, n th ollowing igrm ommuts: C δ \v (/)/ / /.

6 16 Ltur 2. rs s oprs Lt v : /v n w : /w istint outr s o, n ssum tht hs t lst thr vrtis. hn th outr s w : (/v)/w /v n v : (/w)/v /w lso xist, (/v)/w = (/w)/v, n th ollowing igrm ommuts: (/v)/w w /v v /w w. v In s tht hs only two vrtis, thr is similr ommuttiv igrm involving th inlusion o η into th n-th oroll. h lst rmining s is whn w ompos n innr with n outr on in ny orr. hr r svrl possiilitis n in ll o thm w suppos tht v : /v is n outr n : / is n innr. I in th g is not jnt to th vrtx v, thn th outr v : (/)/v / n th innr : (/v)/ /v xist, (/)/v = (/v)/, n th ollowing igrm ommuts: (/v)/ /v v /. v Suppos tht in th innr g is jnt to th vrtx v n not th othr jnt vrtx to y w. Osrv tht v n w ontriut vrtx v w or w v to /. Lt us not this vrtx y z. hn th outr z : (/)/z / xists i n only i th outr w : (/v)/w /v xists, n in this s (/)/z = (/v)/w. Morovr, th ollowing igrm ommuts: z (/v)/w (/)/z / w /v v. It ollows tht w n writ v w = z, whr z = v w i v is losr to th root o or z = w v i w is losr to th root o. Elmntry gnry rltions Lt : \v n σ w : \w two gnris o. hn th gnris : \w ( \w)\v n σ w : \v ( \v)\w xist, w hv

7 2.3. Non-plnr trs 17 ( \v)\w = ( \w)\v, n th ollowing igrm ommuts: \v σ w σ w \w σv ( \v)\w. Comin rltions Lt : \v gnry n : mp suh tht : \v mks sns (i.., still ontins v n its two jnt gs s sutr). hn thr xists n inu mp : \v \v trmin y th sm vrtx or g s :. Morovr, th ollowing igrm ommuts: \v \v. Lt : \v gnry n : mp inu y on o th jnt gs to v or th rmovl o v, i tht is possil. It ollows tht = \v n th omposition \v \v is th intity mp i \v. 2.3 Non-plnr trs Any non-plnr tr gnrts (symmtri) olour opr Ω( ). Similrly s in th s o plnr trs, th st o olours o Ω( ) is th st o gs E( ) o. h oprtions r gnrt y th vrtis o th tr, n th symmtri group on n lttrs Σ n ts on h oprtion with n inputs y prmuting th orr o its inputs. Eh vrtx v o th tr with output g n numring o its input gs 1,..., n ins n oprtion v Ω( 1,..., n ; ). h othr oprtions r th unit oprtions n th oprtions otin y ompositions n th tion o th symmtri group. For xmpl, onsir th tr v w

8 18 Ltur 2. rs s oprs h opr Ω( ) hs six olours,,,,, n. h gnrting oprtions r th sm s th gnrting oprtions o Ω p ( ). All th oprtions o Ω p ( ) r oprtions o Ω( ), ut thr r mor oprtions in Ω( ) otin y th tion o th symmtri group. For xmpl i σ is th trnsposition o two lmnts o Σ 2, w hv n oprtion v σ Ω(, ; ). Similrly i σ is th trnsposition o Σ 3 tht intrhngs th irst n thir lmnts, thn thr is n oprtion r σ Ω(,, ; ). Mor ormlly, i is ny tr, thn Ω( ) = Σ(Ω p ( )), whr is plnr rprsnttiv o. In t, hoi o plnr strutur on is prisly hoi o gnrtors or Ω( ). Dinition h tgory o root trs Ω is th ull sutgory o th tgory o olour oprs whos ojts r Ω( ) or ny tr. W n viw Ω s th tgory whos ojts r root trs. h st o morphisms rom tr S to tr is givn y th st o olour opr mps rom Ω(S) to Ω( ). Not tht ny morphism S in Ω is ompltly trmin y its t on th olours (i.., gs). h morphisms in Ω r gnrt y s n gnris (s in th plnr s) n lso y (non-plnr) isomorphisms. Lmm Any rrow : S in Ω omposs s S σ S ϕ whr σ : S S is omposition o gnry mps, ϕ: S is n isomorphism, n δ : is omposition o mps. Proo. W pro y inution on th sum o th numr o vrtis o S n. I n S hv no vrtis, thn = S = η n is th intity. Not tht, without loss o gnrlity, w n ssum tht sns th root o S to th root o ; othrwis w n tor it s mp S tht prsrvs th root ollow y mp tht is omposition o outr s. Also, w n ssum tht is n pimorphism on th lvs sin, i this is not th s, tors s S /v v, whr v is th vrtx low th l in tht is not in th img o. I n r gs o S suh tht () = (), thn n must on th sm (linr) rnh o S n sns intrmit vrtis to intitis. Sin is mp o olour oprs, w n tor it in uniqu wy s surjtion ollow y n injtion on th olours. his orrspons to toriztion in Ω, S ψ S ξ, δ

9 2.3. Non-plnr trs 19 whr ψ is omposition o gnris n ξ is ijtiv on lvs, sns th root o S to th root o, n is injtiv on th olours (y th prvious osrvtions). I ξ is surjtiv on olours, thn ξ is n isomorphism. I ξ is not surjtiv, thn thr is n g in not in th img o ξ. Sin is n intrnl g (not l), ξ tors s S ξ /. Now w ontinu y inution on th mp ξ. In gnrl, limits n olimits o not xist in th tgory Ω; or xmpl, Ω lks sums n prouts. Howvr, rtin pushouts o xist in Ω, s xprss in th ollowing lmm: Lmm Lt : R S n g : R two surjtiv mps in Ω. hn th pushout R S g P xists in Ω. Proo. h mps n g n h writtn s omposition o n isomorphism n squn o gnry mps y Lmm Sin pushout squrs n pst togthr to gt lrgr pushout squrs, it thus suis to prov th lmm in th s whr n g r gnry mps givn y unry vrtis v n w in R, i.., : R S is : R R\v n g : R is σ w : R R\w. I v = w, thn th ollowing igrm is pushout: R R\v R\v R\v. I v w, thn th ommuttiv squr R R\v σ w σ w R\w σv (R\v)\w = (R\w)\v is lso pushout, s on sily hks.

10 20 Ltur 2. rs s oprs Dnroil intitis with isomorphisms h nroil intitis or th tgory Ω r th sm s or th tgory Ω p plus som mor rltions involving th isomorphisms in Ω. As n xmpl, w giv th ollowing rltion, tht involvs innr s n isomorphisms. Lt tr with n innr g n lt : (non-plnr) isomorphism. hn th trs / n / xist, whr = (), th mp rstrits to n isomorphism : / /, n th ollowing igrm ommuts: / /. Similr rltions hol or outr s n gnris Isomorphisms long s n gnris For ny tr in Ω, lt P ( ) th st o plnr struturs o. Not tht P ( ) or vry tr. hus, th tgory Ω is quivlnt to th tgory Ω whos ojts r plnr trs, i.., pirs (, p) whr is n ojt o Ω n p P ( ), n whos morphisms r givn y Ω ((, p), (, p )) = Ω(, ). A morphism ϕ: (, p) (, p ) in Ω is ll plnr i, whn w pull k th plnr strutur p on to on on long ϕ, thn it oinis with p. Using this quivlnt ormultion o Ω, th tgory Ω p is thn th sutgory o Ω onsisting o th sm ojts n plnr mps only, i.., ompositions o s n gnris. In Ω p, th only utomorphisms r intitis. I δ : S is omposition o s n α: S S is n isomorphism, thr is toriztion δ S α α δ S, whr δ is gin omposition o s n α is n isomorphism. his toriztion is uniqu i on ixs som onvntions,.g., on tks th ojts o Ω to plnr trs, n tks s n gnris to plnr mps. Similrly, isomorphisms n push orwr n pull k long omposition o gnris. Lt σ : S omposition o gnris n α: S S n

11 2.3. Non-plnr trs 21 β : two isomorphisms. hn thr r toriztions σ S σ S α α σ S β σ S β whr α n β r isomorphisms n σ n σ r ompositions o gnris. hus, ny rrow in Ω n writtn in th orm δσα or δασ with δ omposition o s, σ omposition o gnris, n α n isomorphism h prsh o plnr struturs Lt P : Ω op Sts th prsh on Ω tht sns h tr to its st o plnr struturs. Osrv tht P ( ) is torsor unr Aut( ) or vry tr, whr Aut( ) nots th st o utomorphisms o. Rll tht th tgory o lmnts Ω/P is th tgory whos ojts r pirs (, x) with x P ( ). A morphism twn two ojts (, x) n (S, y) is givn y morphism : S in Ω suh tht P ()(y) = x. Hn, Ω/P = Ω p n w hv projtion v : Ω p Ω. hr is ommuttiv tringl u Ω p i v Ω, whr i is th ully ithul ming o into Ω whih sns th ojt [n] in to th linr tr L n with n vrtis n n + 1 gs or vry n Rltion with th simpliil tgory W hv sn tht oth th tgoris Ω n Ω p xtn th tgory, y viwing th ojts o s linr trs. In t, it is possil to otin s omm tgory o Ω or o Ω p s ollows. Lt η th tr in Ω onsisting o no vrtis n on g, n lt η p th plnr rprsnttiv o η in Ω p. I is ny tr in Ω, thn Ω(, η) onsists o only on morphism i is linr tr, or it is th mpty st othrwis. h sm hppns or Ω p n η p. hus, Ω/η = Ω p /η p =.

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