Iterative Algebras for a Base

Transcription

1 Elctronic Nots in Thortical Computr Scinc 122 (2005) Itrativ Algbras for a Bas JiříAdámk 1,2 Stfan Milius 2 Institut of Thortical Computr Scinc, Tchnical Univrsity, Braunschwig, Grmany Jiří Vlbil 1,3 Faculty of Elctrical Enginring, Tchnical Univrsity, Pragu, Czch Rpublic Abstract For algbras A whos typ is givn by an ndofunctor, itrativity mans that vry flat quation morphism in A has a uniqu solution. In our prvious work w provd that vry objct gnrats a fr itrativ algbra, and w providd a coalgbraic construction of thos fr algbras. Itrativity w.r.t. an ndofunctor was gnralizd by Tarmo Uustalu to itrativity w.r.t. a bas, i.., a functor of two variabls yilding finitary monads in on variabl. In th currnt papr w introduc itrativ algbras in this gnral stting, and provid again a coalgbraic construction of fr itrativ algbras. Kywords: fr itrativ thory, rational monad, coalgbra 1 Introduction In our prvious work w introducd, for vry finitary 4 ndofunctor H, th concpt of an itrativ H-algbra. Th aim was to gnraliz and simplify th dscription of fr itrativ thoris of Calvin Elgot and his coauthors [10], [11]. In that w followd th footstps of Evlyn Nlson [16] who introducd itrativ Σ-algbras (in St) and simplifid th dscription of th fr itrativ 1 Th first and th third author acknowldg th support of th Grant Agncy of th Czch Rpublic undr th Grant No. 201/02/ {adamk,milius}@iti.cs.tu-bs.d 3 vlbil@math.fld.cvut.cz 4 finitary mans: prsrving filtrd colimits Elsvir B.V. Opn accss undr CC BY-NC-ND licns. doi: /j.ntcs

2 148 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) thory R Σ on Σ: that thory assigns to vry st Y th algbra R Σ Y of all rational Σ-trs on Y, i.., thos Σ-trs that hav finitly many subtrs up to isomorphism (a charactrization providd by Susanna Ginali [12]). Whras th original proof, using constructions on algbraic thoris, occupid most of th tchnical paprs [10], [8], [11], th proof du to Evlyn Nlson was concis and intuitiv. In [5] w introducd itrativity for H-algbras for vry finitary ndofunctor H of a locally finitly prsntabl catgory A this includs catgoris such as many-sortd sts A = St S and th catgory Fin[St, St] of all finitary st functors. W provd that vry objct Y of A gnrats a fr itrativ H-algbra, RY, and w providd a coalgbraic construction of RY : for th cas of Y = 0 (initial in A), R0 is a colimit of all finit H-coalgbras, and in th gnral cas w work with H( ) +Y instad of H. And, again, w provd that th monad R( ) of fr itrativ H-algbras is a fr itrativ monad on H. In th prsnt papr w work with a bas instad of an ndofunctor, and study itrativity of bas algbras. By a bas w undrstand a finitary ndofunctor from A to FM(A), th catgory of finitary monads on A. This was introducd by Tarmo Uustalu [18] undr th nam paramtrizd monad. Th motivating ida is to study itrativity of Σ-algbras, for a signatur Σ, whr ach opration symbol coms with th information in what placs itration is allowd to occur. Lt us illustrat this on th simpl xampl of a signatur consiting of a singl binary opration symbol. Cas 1: full itrativity. This is th concpt of itrativ algbra of Evlyn Nlson [16]: An algbra (A, ) is itrativ if vry systm x 1 t 1. (1.1) x m t m of quations in variabls X = { x 1,...,x m } and with right-hand sids t i = x j x k for x j,x k in X, ort i A, has a uniqu solution in A. Equivalntly vry systm (1.1) whracht i is a trm on X + A, t i X, has a uniqu solution. A fr itrativ algbra, RY, on a st Y is th algbra of all rational binary trs on Y. Cas 2: rstrictd itrativity. Hr w rquir that th fr variabls ar only allowd to occur on th lft-hand position of. Thus, an itrativ algbra is on in which vry systm (1.1) with right-hand sids t i = x a for x X and a A, ort i A has a uniqu solution. A fr itrativ algbra, for itrativity w. r. t. ths systms of quations, is th algbra of

3 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) all right-wllfoundd rational binary trs ovr Y, i.., thos which hav th right-most path from vry nod finit. Obsrv that for an itrativ algbra all right-hand sids t i = x i1 (x i2 ( (x in a)) ), x ij X, j =1,...,n (1.2) can b allowd: vry systm (1.1) with such right-hand sids has a uniqu solution, too. Cas 3: no itrativity. Hr no variabl is allowd to occur on righthand sids of systms (1.1), i.., w ar lft with th trivial systms in which all right-hand sids li in A. Evry algbra is thn itrativ. In ordr to formaliz such paramtrizd itrativity, w mov from finitary functors H : A A, usd for classical algbra, to finitary functors H : A A A, calld paramtrizd ndofunctors. In th cas of on binary opration th classical polynomial ndofunctor H : St St, HX = X X, is now substitutd by thr paramtrizd ndofunctors: H(X, Y ) = X X for Cas 1, H(X, Y )=X Y for Cas 2, and H(X, Y )=Y Y for Cas 3. Lt us dnot by X Y (rad X box Y ) afrh(x, )-algbra on Y (for all pairs of objcts X, Y A). Mor prcisly, for vry objct X w dnot by X th fr monad on th ndofunctor H(X, ) (which, as provd by Michal Barr [7] is just th monad of th fr algbras of H(X, )). This yilds a bas, i.., a functor A FM(A) in th obvious way. Exampl: for on binary opration with full itrativity, H(X, ) =X X is th constant ndofunctor whos fr algbra on Y is X Y = X X + Y. Th cas of rstrictd itrativity, H(X, ) = X corrsponds to unary opration symbols indxd by X th fr algbras ar X Y = X Y whr X is a fr monoid on X. Finally, th trivial cas of no itrativity yilds th bas indpndntly of X. X Y = fr algbra on Y

4 150 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) In gnral, w us th uncurrid form of a bas, i.., a functor : A A A finitary in both variabls and quippd with monad units A X A and monad multiplications X (X A) X A for all pairs of objcts X, A satisfying som obvious compatibility conditions (s Sction 2). A bas algbra is a monad algbra of th monad A on th objct A. Thatis,abas algbra is givn by a morphism α : A A A satisfying th Eilnbrg-Moor conditions in th scond variabl. Th bass X Y = X X + Y and X Y = X Y on St yild th usual algbras on on binary opration as bas algbras. Howvr, itrativity is diffrnt, as w dmonstrat blow. For a givn bas algbra A lt us call morphisms : X X A, X finitly prsntabl, quation morphisms. For a fully itrativ binary opration this is : X X X + A xprssing prcisly a systm (1.1), for th rstrictd itrativity w gt : X X A as in (1.2) abov. Dfinition 1.1 Abasalgbraα : A A A is itrativ providd that for vry quation morphism : X X A thr xists a uniqu solution, i.., a uniqu morphism : X A for which th squar X X A A A α A A commuts. Th main rsult of our papr is that (i) fr itrativ bas algbras always xist, and (ii) th monad thy prsnt in A is a fr itrativ monad on th givn bas. This mans that th full strngth of th rsults of [5], concrning (fully) itrativ H-algbras of a givn ndofunctor H, gnraliz to th cas of bas algbras. In that spcial cas on works with th bas X Y = HX + Y

5 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) whr itrativity of bas algbras is th full itrativity abov, and th monad of fr itrativ algbras was provd to b a fr itrativ monad on H. Th notion of a bas (undr th nam paramtrizd monad) has bn introducd by Tarmo Uustalu [18] who gnralizd som rsults of our papr [1]. In th prsnt papr w continu in th sam vin by gnralizing rsults of [5] from H-algbras to bas algbras. Although th structur of th prsnt papr follows that of [5] closly, it turns out that all th proofs ar substantially mor difficult. Thus our original hop that w will just indicat how to modify th prvious proof idas to th prsnt gnrality faild, and w fl obligd to prsnt dtaild proofs again. Th concrt xampls of bass blow (s 2.5) and thir fr itrativ algbras (s ) wr alrady considrd in [18]. 2 Bass and Bas Algbras Assumption 2.1 Throughout this sction w assum that a locally finitly prsntabl catgory A is givn. W dnot by FM(A) th catgory of all finitary monads on A (i.., monads whos undrlying functor prsrvs filtrd colimits) and monad morphisms. Dfinition 2.2 By a bas on A is undrstood a finitary functor from A to FM(A). Notation 2.3 (i) Givn a bas, w hav a finitary undrlying functor from A to th catgory Fin[A, A] of finitary ndofunctors of A. This is a currid form of a functor of two variabls, finitary in ach variabl, which w dnot by : A A A. (ii) Th unit of th monad X is dnotd by u X : Id X ; its componnts ar u X A : A X A. (iii) Th multiplication of th monad X is dnotd by m X ; its componnts ar m X A : X (X A) X A. Rmark 2.4 Explicitly, to spcify a bas mans to spcify a finitary functor of two variabls : A A A togthr with morphisms u X A : A X A and mx A : X (X A) X A

6 152 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) for arbitrary objcts X, A of A such that th following diagrams commut: and X (X (X A)) u X X A X A X ux A X (X A) X A m X A X A m X X A X (X A) X m X A m X A X (X A) m X A X A xprssing th monad axioms for ach X, togthr with (2.1) (2.2) A f B u X A u Y B X A h f Y B (2.3) and X (X A) h (h f) Y (Y B) m X A m Y B X A h f Y B (2.4) which xprss th naturality of u X and m X and th fact that for vry morphism h : X Y w hav a monad morphism h ( ) :X ( ) Y ( ). Exampls 2.5 (i) W hav trivial bass givn by all constant functors from A to FM(A). That is, for vry finitary monad S on A w form th trivial bas X S A = SA whos unit and multiplication is that of S. (ii) Coproduct is a bas X A = X + A with th obvious unit and multiplication u X A = inr : A X+A and mx A =[inl, inl, inr ]:X+X+A X+A.

7 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) (iii) Lt X b a fr monoid on th objct X of A with unit η X :1 X and multiplication µ X : X X X. Thn w hav th bas X A = X A with bas unit u X A = η X A and bas multiplication m X A = µ X A. (iv) Lt B b any locally finitly prsntabl catgory. Thn so is th catgory A = Fin[B, B] of all finitary ndofunctors of B. Thr w hav a bas X A = F(X) A whr a fr monad on X is dnotd by (F(X),η X,µ X ) it xists sinc X is finitary, s [7]. Th bas unit is u X A = ηx A and th bas multiplication m X A = µx A. Exampl 2.6 Lt : A A A b a bas. W obtain othr bass by prcomposing with finitary ndofunctors H : A A: X A = HX A with unit and multiplication u X A = uhx A : A HX A ṁ X A = mhx A : HX (HX A) HX A. Of particular importanc is th bas obtaind from th bas +, i.., X A = HX + A. W will s blow that our prvious rsults of [5] on H-algbras ar spcial cass of th rsults concrning ths bass. Dfinition 2.7 Givn a bas, by a bas algbra is undrstood an objct A of A togthr with a monadic algbra on A of th monad A. That is, a bas algbra is givn by an objct A and a morphism α : A A A such that th following two diagrams A α A A A (A A) α m A A A A A α A ua A A A α A (2.5) commut.

8 154 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) Notation 2.8 W dnot by Alg(A, ) th catgory of all bas algbras and all homomorphisms from (A, α) to(b,β), i.., morphisms h : A A of A such that th squar A A h h B B α β A h B (2.6) commuts. Exampls 2.9 (i) Algbras of th bas X A = X + A ar givn by an objct A and an ndomorphism of A (i.., ths ar just unary algbras in A). Homomorphisms ar also th usual homomorphisms of unary algbras. (ii) Algbras of th bas X A = HX + A ar th usual H-algbras, i.., pairs consisting of an objct A and a morphism α : HA A. Also, homomorphisms ar th usual H-algbra homomorphisms. Thus, Alg(A, ) =Alg H is th catgory of H-algbras and homomorphisms. (iii) Algbras of th bas on St givn by X A = X A (s 2.5(iii)) ar prcisly th usual algbras on on binary opration. In fact, givn th lattr, say, : A A A dfin α : A A A by α(a 1 a 2...a n,a)=a 1 (a 2...(a n a)...). This satisfis (2.5). Convrsly, givn α : A A A satisfying (2.5), it is givn by th abov formula whr dnots th rstriction of α to all pairs in A A. Consquntly, th bass X A =(X X)+A and X A = X A on St dfin th sam catgoris of algbras.

9 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) (iv) Lt B b any locally finitly prsntabl catgory and put A = Fin[B, B] with th bas X A = F(X) A (s 2.5(iv)) An algbra is a pair (A, α) consisting of a finitary ndofunctor A : B B and a natural transformation α : A A A. Mor prcisly: ach such pair dfins a uniqu natural transformation α from A to A, A (th right Kan xtnsion of A along A). Sinc A, A is always a monad and F(A) isafrmonadona, α yilds a uniqu monad morphism α : F(A) A, A. Th uniqu natural transformation α : F(A) A A corrsponding to α dfins an algbra of our bas in fact, th condition (2.5) abov is quivalnt to α bing a monad morphism. Proposition 2.10 Th catgory Alg(A, ) is locally finitly prsntabl, and its forgtful functor into A has a lft adjoint, i.., fr bas algbras xist on vry objct of A. Proof. Th ndofunctor SA = A A of A is finitary, and thus th catgory Alg S is locally finitly prsntabl and its forgtful functor has a lft adjoint, s [6]. Th catgory Alg(A, ) is a full subcatgory of Alg S, and it is asy to vrify that it is closd undr limits and filtrd colimits in Alg S. It follows that it is a rflctiv subcatgory, s Thorm 2.48 in [6]. Sinc th forgtful functor of Alg(A, ) is a rstriction of that of Alg S, th proposition follows. 3 Itrativ Bas Algbras Assumption 3.1 Throughout this sction dnots a bas on a locally finitly prsntabl catgory A. Dfinition 3.2 (i) By a (finitary, flat) quation morphism in an objct A is mant a morphism : X X A, X finitly prsntabl. (ii) Suppos that A is th undrlying objct of a bas algbra α : A A A. Thn by a solution of is mant a morphism : X A such that th

10 156 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) squar commuts. X X A A A α A A (3.1) (iii) A bas algbra is calld itrativ providd that vry quation morphism has a uniqu solution. Exampl 3.3 Considr th bas X A = X +A in St. Thn a bas algbra, i.., a unary algbra α : A A, is itrativ if and only if α k has a uniqu fixd point for all k, s[5]. Exampl 3.4 Algbras of th bas X A =(X X)+A ona = St ar th usual algbras on on binary opration: s Exampl 2.9(ii) and put HX = X X. Thr is no asy critrion for an algbra to b itrativ. But thr ar nic xampls of itrativ algbras, s [5],.g., A = {1, 2, 3,...} { }with addition A =(0, ] A =(1, ] with addition with multiplication A fr itrativ algbra on a st Y (of gnrators) can b dscribd as th algbra RY of all rational binary trs on Y, s Sction 1. Exampl 3.5 Considr th bas X A = X A on A = St, s Exampl 2.5(iii). Although its algbras ar, again, th usual binary algbras, th concpt of itrativ algbras diffrs from th abov xampl. Rcall that an algbra (A, ) ladstoα : A A A with α(a 1 a 2...a n,a)=a 1 (a 2...(a n a)...). It is itrativ if and only if for vry quation morphism : X X A (X finit) thr xists a uniqu : X A such that for vry variabl x w hav that (i) (x) =(ε, a) implis (x) =a, and (ii) (x) =(x 1...x n,a) implis (x) = (x 1 ) ( (x 2 )...( (x n ) a)...).

11 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) Thus, for xampl, th mpty algbra is itrativ in th prsnt sns (but it is not itrativ for (X X)+A). A fr itrativ algbra, RY,onastY can b dscribd as a subalgbra of th abov algbra RY of all rational binary trs on Y. Lt us call a binary tr right-wllfoundd if from vry nod th right-most path is always finit (i.., it lads to a laf). It is obvious that th st RY of all rational rightwllfoundd trs is a subalgbra of RY. This subalgbra is itrativ (w.r.t. th prsnt bas X A), in fact, RY is a fr itrativ algbra on Y. Exampl 3.6 W know that th classical Σ-algbras for a signatur Σ = (Σ n ) n N ar just algbras of th corrsponding polynomial ndofunctor H Σ of St. ThabovExampl3.4 immdiatly gnralizs to th bas X Y = H Σ X + Y of Exampl 2.6: a fr itrativ algbra on a st Y is th algbra R Σ Y of all rational Σ-trs on Y, s Sction 1. Opn Problm 3.7 Dscrib, for A = Fin[St, St], fr itrativ algbras of th bass X A =(X X)+A and X A = F(X) A. Exampl 3.8 For th trivial bass S, s Exampl 2.5(i), all algbras ar itrativ. In fact, givn an Eilnbrg-Moor algbra α : SA A and an quation morphism : X SA, th uniqu solution is = α : X A. Notation 3.9 Lt : X X A b an quation morphism with paramtrs in A. Evry morphism h : A B in A yilds an quation morphism h X X A X h X A. (3.2) Lmma 3.10 Lt (A, α) and (B,β) b itrativ algbras. Thn a morphism h : A B in A is a homomorphism if and only if it prsrvs solutions, i.., for vry quation morphism : X X A th solution of h in B is h. Lmma 3.10 xplains that th choic of plain homomorphisms btwn itrativ algbras is adquat. Proposition 3.11 Evry objct of A gnrats a fr itrativ algbra.

12 158 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) Proof. It is asy to prov that itrativ algbras ar closd undr limits and filtrd colimits in Alg(A, ). By Thorm 2.48 in [6] thy form a rflctiv subcatgory of Alg(A, ). Now apply Proposition A Coalgbraic Construction W know from th prcding sction that, for vry bas, fr itrativ algbras xist. Th aim of th prsnt sction is to show that a fr itrativ algbra RY on an objct Y can b constructd as a filtrd colimit of all quation morphisms : X X Y (whr X rangs through a st A fp rprsnting all finitly prsntabl objcts of A up to isomorphism). Mor prcisly, considr th coalgbras of th ndofunctor Y togthr with th usual coalgbra homomorphisms. W dnot by EQ Y th catgory of all quation morphisms as th full subcatgory of Coalg ( Y ) on all objcts from A fp. Sinc Coalg ( Y ) is cocomplt, with colimits formd on th lvl of A, itisobviousthateq Y is closd in it undr finit colimits. In particular, EQ Y is a filtrd catgory. W also dnot by Eq Y : EQ A, ( : X X Y ) X th forgtful functor. This dfins a filtrd diagram in A. Notation 4.1 RY dnots a (filtrd) colimit of th diagram Eq Y with colimit cocon : X RY (for all : X X Y in EQ Y ). Rmark 4.2 Th aim of th prsnt sction is to prov that RY carris a structur of an itrativ algbra making it a fr itrativ algbra on Y. W procd in two stps: w first assum that Y is a finitly prsntabl objct. This nabls us, for xampl, to dfin th univrsal arrow immdiatly: obsrv that u Y Y : Y Y Y is an objct of EQ Y and dnot by η Y =(u Y Y ) : Y RY th corrsponding colimit morphism. An xtnsion of all th rsults to arbitrary objcts Y is thn asy. For xampl, η Y is dfind as follows: xprss Y as a colimit of a filtrd diagram of finitly prsntabl objcts Y t,(t T), thn it is asy to vrify that RY is a filtrd colimit of RY t,(t T), and w put η Y = colim η Y t. t T

13 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) Notation 4.3 For vry objct Y w dnot by i Y : RY RY Y th uniqu morphism for which th squars X X Y Y RY i Y RY Y (4.1) commut for vry in EQ Y. This is wll dfind sinc th morphisms ( Y ) ar asily sn to form a cocon of th diagram Eq Y. Lmma 4.4 For vry quation morphism in EQ Y thr xists a uniqu morphism : X RY such that th squar (4.1) commuts. Notation 4.5 W introduc notation hr for adding variabls to quations: givnanquationmorphism : X X Y in EQ Y and an objct Q, whn ar w abl to form canonically an quation morphism X + Q (X + Q) Y in EQ Y? On possibility is to assum that a morphism q : Q X X is givn. Thn w can dfin an quation morphism q =(inl Y ) [X Y,m X Y (X )] ( + q) i.., q is givn by th following diagram X X Y inl X + Q q inl Y (X + Q) Y inr inl Y Q q X X X X (X Y ) m X Y X Y

14 160 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) Thorm 4.6 For vry finitly prsntabl objct Y thr xists a uniqu structur of a bas algbra ρ Y : RY RY RY such that th squars Q q X X inr X + Q q (4.2) RY RY ρ Y RY (whr and q ar as in Notation 4.5) commut. Rmark 4.7 In th proof blow w will dnot by Eq Y Eq Y : EQ Y A th diagram givn by objcts X X, whr : X X Y rangs ovr EQ Y, and by morphisms h h, forh in EQ Y. Sinc is finitary in both variabls and th diagonal EQ Y EQ Y EQ Y is cofinal, th colimit of Eq Y Eq Y is RY RY with colimit injctions. Analogously, X RY = colim X Eq Y, tc. Proof. W stablish that thr is a uniqu morphism ρ Y for which (4.2) commuts, and postpon th vrification that this yilds a bas algbra to th full vrsion of our papr. (1) Considr an arbitrary morphism p : Q RY RY whr Q A fp. In vry locally finitly prsntabl catgory ach objct is a canonical filtrd colimit of morphisms from finitly prsntabl objcts, thus, RY RY is a colimit of th corrsponding diagram D with th colimit cocon formd by all p s. W show that ach p factors as p =( ) q for som and q, and that th morphisms q inr : Q RY (which, as w show, ar indpndnt of th choic of such a factorization) form a cocon of D. Th xistnc of th abov factorization follows from th abov filtrd colimit RY RY = colim(eq Y Eq Y ). Th morphism p : Q RY RY, having a finitly prsntabl domain, factors through on of th colimit injctions. W claim that th uppr passag q inr of th squar (4.2) is indpndnt of th choic of th factorization. Thus, lt f : Z Z Y b an quation

15 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) morphism and lt Q r p Z Z RY RY f f b anothr factorization of p through a colimit morphism of th diagram Eq Y Eq Y. Sinc that diagram is filtrd, w can assum, without loss of gnrality, that a morphism h from to f in EQ Y xists, and that th quation r = (h h) q holds. It follows that h + Q is a morphism from q to f r : X+Q +q (X Y )+(X X) q [X Y,m X Y (X )] X Y inl Y (X+Q) Y h+q Z+Q f+r (h Y )+(h h) h Y (Z Y )+(Z Z) Z Y [Z Y,m Y Z (Z f)] inl Y (h+q) Y (Z+Q) Y fr Consquntly, q = f r (h + Q). This provs th dsird indpndnc: q inr = f r (h + Q) inr = f r inr. (2) Th abov morphisms q inr : Q RY form a cocon of th diagram D. That is, givn an arrow p : Q RY RY, Q A fp,andgivna morphism t Q Q p p RY RY of th diagram D, w prov that for any factorization Z Z Q r p f f RY RY w hav q inr = f r inr t. (4.3)

16 162 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) In fact, th following factorization of p: Z Z r Q Q t p p RY RY f f yilds, by th indpndnc provd in part (1): q inr = f rt inr. Morovr, Z + t is an quation morphism from f rt into f r : Z + Q f+rt (Z Y )+(Z Z) [Z Y,mZ (Z f)] Y Z Y inl Y (Z + Q) Y Z+t (Z+t) Y Z + Q f+r (Z Y )+(Z Z) [Z Y,m Z Y (Z f)] Z Y inl Y (Z + Q ) Y Consquntly, f rt = f r (Z + t), which implis (4.3): q inr = f r (Z + t) inr = f r inr t. Th cocon q inr has prcisly on factorization through th colimit cocon this provs that (4.2) dfins a uniqu ρ Y. Lmma 4.8 For vry finitly prsntabl objct Y th morphism i Y Diagram (4.1)) is an isomorphism with th invrs (s i 1 Y = j Y RY Y RY η Y RY RY ρ Y RY (4.4) Thorm 4.9 For vry finitly prsntabl objct Y, th algbra (RY, ρ Y ) is a fr itrativ algbra on Y w.r.t. th univrsal arrow η Y : Y RY. Proof. (1) (RY, ρ Y ) is itrativ. In fact, vry quation morphism : X X RY, X finitly prsntabl, has a uniqu solution obtaind as follows. Sinc X RY = colim X Eq Y, s Rmark 4.7, factors through th colimit injction X f for som f :

17 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) V V Y in EQ Y. Thus, w hav a commutativ triangl X X RY 0 X V X f (4.5) W form an quation ẽ : X + V (X + V ) Y as follows: X 0 X V X f X (V Y ) inl (inr Y ) (X+V ) ((X+V ) Y ) inl X+V inr V f m X+V Y (X+V ) Y (4.6) V Y inr Y W will prov that th givn quation morphism has th solution From (4.4) and(4.1) whav X inl X + V RY. (4.7) j Y (f Y ) f = i 1 Y (f Y ) f = f. (4.8) Furthrmor, inr : V X + V is a morphism of quations from f to ẽ (s th lowr squar of (4.6)), thus, This provs that th diagram ẽ inr = f. (4.9) X V X f X (V Y ) inl (inr Y ) (X+V ) ((X+V ) Y ) m X+V Y (X+V ) Y inl (V Y ) (4.9) ( Y ) Y inl V (X+V ) (V Y ) (f Y ) m RY RY (RY Y ) Y RY Y (4.8) RY η Y RY (RY η Y ) (2.4) (X+V ) V RY (RY RY ) RY j (4.4) Y m RY RY RY RY f (2.5) ρ Y RY ρ Y RY RY ρ Y RY (2.4) η Y (4.10)

18 164 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) commuts. Dnot by q : X V (X + V ) Y th uppr horizontal morphism of (4.10). Th proof that (4.7) is a solution of follows from th fact that th outward squar of th diagram inl X 0 (4.5) X V X f X RY X+V RY i Y j Y (X+V ) Y Y RY Y (4.4) ρ Y ( ) RY η Y f η Y RY RY (4.6) (4.1) q RY commuts in fact, all innr parts xcpt ( ) commut and by (4.10) th triangl ( ) commuts whn xtndd by ρ Y, th right-hand vrtical arrow. It rmains to prov that th solution is uniqu. Givn anothr solution X X RY s s RY RY ρ Y RY RY (4.11) w prov that th squar X + V [s,f ] (X + V ) Y [s,f ] Y RY i Y RY Y (4.12) commuts. By Lmma 4.4, it follows that ẽ =[s, f ], thus = ẽ inl =[s, f ] inl = s.

19 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) Th right-hand componnt of (4.12) with domain V commuts: f V inr X + V f V Y (4.6) inr Y (X + V ) Y [s,f ] [s,f ] Y RY i Y RY Y f Y For th lft-hand componnt notic first that th quation holds. In fact, in th diagram m RY Y (RY i Y )=i Y ρ Y (4.13) RY RY RY i Y RY (RY Y ) RY (RY η Y ) RY (RY RY ) RY ρ Y RY RY ρ Y m RY Y (2.4) m RY RY (2.5) ρ Y RY i Y RY Y RY η Y RY RY ρ Y RY th horizontal morphisms ar both idntity morphisms (s (4.4)), which provs that th outward squar commuts. Consquntly, th lft-hand squar commuts, sinc ρ Y (RY η Y ) is an isomorphism. Th lft-hand componnt of (4.12) is, du to (4.6), th outward squar of th commutativ diagram inl s X RY ρ Y 0 (4.11) X V s RY X RY X f RY RY X f (4.1) RY i Y (4.13) s (f Y ) X (V Y ) RY (RY i Y ) Y inl (inr Y ) [s,f m RY Y (X+V ) ((X+V ) Y ) (2.4) m ] ([s,f ] Y ) X+V Y (X+V ) Y RY Y [s,f ] Y (2) RY is fr, i.., for vry itrativ algbra α : A A A and for vry morphism h 0 : Y A thr xist a uniqu homomorphism h : RY A with h η Y = h 0.

20 166 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) (2a) Existnc of h. For vry quation morphism g : X X Y in EQ Y w form h 0 g : X X A, s Notation 3.9, and obtain th uniqu solution (h 0 g) : X A. This is a cocon of th diagram Eq Y. In fact, givn a morphism g X X Y p p Y X g X Y in EQ Y,thn(h 0 g ) p is a solution of h 0 g: X g X Y p (4.14) p Y X (h 0 g ) A g X Y (3.1) X h 0 α X h 0 X A p A (h0 g ) A X A ((h 0 g) p) A A A (4.14) Thus, (h 0 g ) =(h 0 g) p as dsird. Consquntly, w can dfin h by th commutativity of th triangls h RY A g (h 0 g) X (4.15) for all g in EQ Y. W will show that this is th uniqu algbra homomorphism xtnding h 0. W first prov that h 0 = h η Y = h (u Y Y ). In fact, th commutativ diagram h 0 Y A A u Y Y (2.3) (2.5) Y Y u A α A h 0 h 0 Y h 0 Y A h 0 A A A

21 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) shows that h 0 is a solution of h u Y Y in A, thus, by (4.15) whav h 0 =(h u Y Y ) = h (u Y Y ). Nxt w will prov that h is a homomorphism. Du to Lmma 3.10, it is sufficint to prov that h prsrvs solutions, xplicitly, for vry quation morphism : X X RY w form = h X X RY X h X A (4.16) and w prov that = h. (4.17) Rcall from (4.6) th quation morphisms f : V V Y and ẽ : X + V (X + V ) Y, and put Lt us prov that f = h 0 f V f V Y V h0 V A. (4.18) [, f ]=(h 0 ẽ) : X + V A. (4.19) Thn (4.17) follows: (4.19) implis du to (4.7) and(4.15) applid to g = ẽ th quation h = h ẽ inl =(h 0 ẽ) inl =. Thus, th proof of (2a) will b complt by proving that th squar X + V (X + V ) Y [,f ] A α (4.20) (X+V ) h 0 (X + V ) A [,f ] A A A commuts. For th right-hand componnt with domain V this follows from th com-

22 168 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) mutativ diagram inr V A f f (3.1) V Y inr Y V h 0 α (X + V ) Y V A f A inr A (X + V ) A A A (X+V ) h 0 f [,f ] A For th right-hand componnt with domain X, first obsrv that th dfinition of solution yilds and = α ( A) (X h) = α ( h), (4.21) f = α (f A) (X h 0 ) f = α (f h 0 ) f. (4.22) Thrfor, w gt a commutativ diagram inl X 0 X V X f (4.22)&(4.15) A α (f h X (V Y ) 0 ) (4.6) A (A A) (4.21) X f X RY h A α A A (2.5) inl (inr Y ) [,f ] ([,f ] h 0) α (X+V ) ((X+V ) Y ) m m X+V (2.4) A A Y (X+V ) Y (X+V ) h 0 [,f ] h 0 (X+V ) A A A [,f ] A which shows that th lft-hand componnt of (4.20) commuts. (2b) Uniqunss. Lt k : RY A b a homomorphism (i.., k ρ Y = α (k k)) with h 0 = k η Y. Thn for vry quation morphism g : X X Y

23 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) of EQ Y th diagram h 0 g g X k RY A i Y ρ Y g (4.1) RY Y (4.4) RY RY RY η Y X Y α RY Y g Y k k X h 0 k h 0 X A A A (k g ) A commuts, proving k g =(h 0 g) = h g, s (4.15). Thus, k = h, sinc th morphisms g s form a colimit cocon. 5 Conclusions and Futur Work In our papr w providd a coalgbraic construction of fr itrativ algbras of a gnral nvironmnt calld bas. This is analogous to (but, unfortunatly, tchnically mor involvd than) th coalgbraic construction givn in [5] for H-algbras, whr H is an arbitrary finitary ndofunctor. Th nxt goal is to introduc itrativ bass in th spirit of itrativ thoris of C. Elgot, and prov that th monad of fr itrativ algbras yilds a fr itrativ bas. Th tchnical rsult will srv us to dscrib algbraic trs of B. Courcll [9] (i.., trs rsulting from a smantics of rcursiv program schms) in a mannr similar to th dscription of rational trs via R Σ, th rational monad of Σ-algbras. For that, w nd to introduc a suitabl bas on th catgory FM(St) of all finitary monads on St. Rfrncs [1] P. Aczl, J. Adámk, S. Milius and J. Vlbil, Infinit Trs and Compltly Itrativ Thoris: A Coalgbraic Viw, Thort. Comput. Scinc 300 (2003), [2] J. Adámk, On a Dscription of Trminal Coalgbras and Fr Itrativ Thoris, Elctron. Nots Thor. Comput. Sci (2003). [3] J. Adámk, S. Milius and J. Vlbil, Fr Itrativ Thoris A Coalgbraic Viw, Math. Structurs Comput. Sci. 13 (2003), [4] J. Adámk, S. Milius and J. Vlbil, On Rational Monads and Fr Itrativ Thoris, Elctron. Nots Thort. Comput. Sci. 69 (2003).

24 170 J. Adámk t al. / Elctronic Nots in Thortical Computr Scinc 122 (2005) [5] J. Adámk, S. Milius and J. Vlbil, From Itrativ Algbras to Itrativ Thoris (Extndd Abstract), accptd for publication in Elctron. Nots Thor. Comput. Sci., full vrsion availabl via [6] J. Adámk and J. Rosický, Locally prsntabl and accssibl catgoris, Cambridg Univrsity Prss, [7] M. Barr, Coqualizrs and fr tripls, Math. Z. 116 (1970), [8] S. Bloom and Z. Ésik, Itrativ Thoris: Th Equational Logic of Itrativ Procsss, EATCS Monograph Sris on Thortical Computr Scinc, Springr-Vrlag, [9] B. Courcll, Fundamntal Proprtis of Infinit Trs, Thort. Comput. Scinc 25 (1983), [10] C. C. Elgot, Monadic Computation and Itrativ Algbraic Thoris, in: Logic Colloquium 73 (ds: H. E. Ros and J. C. Shphrdson), North-Holland Publishrs, Amstrdam, [11] C. C. Elgot, S. L. Bloom and R. Tindll, On th Algbraic Structur of Rootd Trs, J. Comp. Syst. Sci. 16, (1978), [12] S. Ginali, Rgular Trs and th Fr Itrativ Thory, J. Comput. Systm Sci. 18 (1979), [13] P. Gabril and F. Ulmr, Lokal präsntirbar Katgorin, Lctur N. Math. 221, Springr- Vrlag, Brlin [14] S. MacLan, Catgoris for th Working Mathmatician, Springr-Vrlag, 2 nd dition, [15] S. Milius, On Itratabl Endofunctors, Elctron. Nots Thor. Comput. Sci. 69 (2003). [16] E. Nlson, Itrativ Algbras, Thort. Comput. Scinc 25 (1983), [17] J. Tiuryn, Uniqu Fixd Points vs. Last Fixd Points, Thort. Comput. Scinc 12 (1980), [18] T. Uustalu, Gnralizing substitution, Thor. Inform. Appl (2003),