Injective topological fibre spaces

Transcription

1 Topology and its pplications 125 (2002) Injctiv topological ibr spacs F. Cagliari a,,s.mantovani b a Dipartimnto di Matmatica, Univrsità di Bologna, Piazza di Porta S. Donato, 5, I Bologna, Italy b Dipartimnto di Matmatica, Univrsità di Milano, Via C. Saldini, 50, I Milano, Italy Rcivd 26 pril 2001; rcivd in rvisd orm 29 Octobr 2001 bstract W invstigat injctiv objcts with rspct to th class o mbddings in th catgoris Top/B (Top 0 /B) o(t 0 ) topological ibr spacs and thir rlations with xponntiabl morphisms. s a rsult, w obtain a charatrization o such injctiv objcts as rtracts o partial products o th thr-point spac S (S th Sirpinski spac or Top 0 ) Elsvir Scinc B.V. ll rights rsrvd. MSC: 55R05; 55R70; 54B30; 18G05 Kywords: Injctiv objct; Rtract; Exponntiabl morphism; Partial product Introduction Th importanc o th notion o injctiv objct in a catgory is univrsally rcognizd, spcially atr th dvlopmnt o commutativ and homological algbra. Injctiv objcts with rspct to a class H o morphisms hav bn invstigatd or a long tim in various catgoris. For xampl, in th catgory Pos o partial ordrd sts and monoton mappings, injctiv objcts with rspct to th class o all mbddings (= rgular monomorphisms) coincid with th complt lattics, whil, in th catgory SLat o (mt) smilattics and smilattics homomorphisms, injctiv objcts ar prcisly th locals (s [3]). In th catgory Top (Top 0 )o(t 0 ) topological spacs, injctiv objcts with rspct to th mbddings ar charactrizd as rtracts o products o th thr-point spac S ={0, 1, 2}({0} uniqu non-trivial opn) (S th Sirpinski spac or Top 0 ) (s [8,6]). * Corrsponding author. Invstigation supportd by Univrsity o Bologna. Funds or slctd rsarch topics. addrsss: cagliari@dm.unibo.it (F. Cagliari), Sandra.Mantovani@mat.unimi.it (S. Mantovani) /02/$ s ront mattr 2002 Elsvir Scinc B.V. ll rights rsrvd. PII: S (01)

2 526 F. Cagliari, S. Mantovani / Topology and its pplications 125 (2002) Rcntly, injctiv objcts in comma-catgoris hav bn invstigatd in dtail (s [9,10,1,2]), spcially in rlation with wak actorization systms, a concpt usd in homotopy thory, in particular or modl catgoris. Th charactrization o H-injctiv objcts in C/B, oranyb in C, may b vry usul, sinc thy otn orm th right part o a wak actorization systm that has morphisms o H as lt part. In [1], th authors charactriz H-injctiv objcts in various catgoris, in particular or H = Embddings in Pos/B and in SLat/B. In this papr w invstigat th sam cas in th comma-catgoris Top/B (Top 0 /B) o (T 0 ) topological ibr spacs ovr B. nalyzing in dtail th rlation btwn injctivity and xponntiability o morphisms, notd irst by Tholn in [9], w ind that injctiv ibr (T 0 ) spacs with rspct to th mbddings ar charactrizd as rtracts o partial products o S. Th analogy with th non-ibrd cas is obtaind using th notion o partial product, strictly rlatd with th concpt o xponntiation, as shown in [5]. 1. Injctivity and xponntiability W rcall th ollowing dinitions: Dinition 1.1. Givn a class H o morphisms in a catgory C, an objct I is H-injctiv i, or all h : X Y in H, th unction C(h, I) : C(Y, I) C(X, I) is surjctiv. In th comma-catgory C/B (whos objcts (, ) ar C-morphisms : B with ixd codomain B), this mans that an objct (, i) is H-injctiv i, or any commutativ diagram in C h X u i Y v B with h H, thr xists an arrow s : Y h X u s Y v B such that sh = u and is = v. i Notation. From now on, injctiv will dnot H-injctiv or H th class o mbddings in Top. Dinition 1.2. n objct X is xponntiabl in a catgory C with init products i th unctor ( ) X has a right adjoint ( ) X. morphism s : X B is xponntiabl in a catgory C with init limits i it is xponntiabl as an objct (X, s) o C/B, that is, i th unctor ( ) (X, s) has a right adjoint ( ) (X,s).

3 F. Cagliari, S. Mantovani / Topology and its pplications 125 (2002) Th ollowing rsults will b usul latr: Lmma 1.3. I X and ar xponntiabl in C, with rtract o X, thn Y is a rtract o Y X, or any Y C. Proo. Y ( ) is unctorial on xponntiabl objcts. Lmma 1.4. I (, i) is a rtract o (X, s) in Top/B and s is xponntiabl in Top,alsoi is xponntiabl in Top. Proo. I w dnot with PsTop th quasitopos o psudo-topological spacs, w can apply Lmma 1.3 to (X, s) and (, i), whrc = PsTop/B. Thn, or any map : Y B in Top, (Y, ) (,i) is a rtract o (Y, ) (X,s), that actually is an objct o Top/B, sincs is xponntiabl in Top (s Corollary 2.3(ii) in [4]). Thn (Y, ) (,i) has, as a domain, a subspac o a topological spac, so it is in Top/B. This mans (s Corollary 2.3(ii) in [4]) that also i is an xponntiablmap in Top. Lmma 1.5. In a pullback diagram in C C m Dm B i is H-injctiv, is H-injctiv. Corollary 1.6. I I is H-injctiv in C, th projction (I B,π B ) is H-injctiv in C/B. Corollary 1.7. I C(,S) dnots th discrt spac o continuous maps rom to S, th projction (S C(,S) B,π B ) is injctiv in Top/B, ors th Sirpinski spac or th thr-point spac S ={0, 1, 2} (with {0} th only non-trivial opn subst). Proo. Sinc C(,S) is discrt, S C(,S) coincids with a product o copis o S, soitis injctiv, bing th class o injctiv objcts in Top closd undr products. W can thn apply Corollary 1.6 to S C(,S). Notation. From now on, in th cas o maps btwn T 0 spacs, S will rprsnt th Sirpinski spac, othrwis th thr-point spac. Proposition 1.8. ny objct (X, ) in Top/B can b mbddd into an injctiv objct by th mbdding α,,whrα(x): C(X,S) S is givn by α(x)(ϕ) = ϕ(x) (s [8]): α, X S C(X,S) B π B B

4 528 F. Cagliari, S. Mantovani / Topology and its pplications 125 (2002) Corollary 1.9. In Top/B, any injctiv objct is a rtract o an xponntiabl objct. Proo. Givn (, i) injctiv in Top/B, by Proposition 1.8, w hav α,i id i S C(,S) B πb B with π B : S C(,S) B B xponntiabl in Top,sincS C(,S) is locally compact (s Corollary 2.10 in [7]). s a consqunc w ind th irst rlation btwn injctiv and xponntiabl objcts in Top/B: Corollary I (, i) is injctiv in Top/B,thni is an xponntiabl map in Top. Proo. By Corollary 1.9 and Lmma 1.4. nothr consqunc o Corollary 1.9 is a usul proprty o injctiv objcts: Proposition I (, i) is injctiv in Top/B,thni is an opn map. Proo. By Corollary 1.9, (, i) is a rtract in Top/B o th opn projction (S C(,S),π B ) and a rtract o an opn map is an opn map. 2. Charactrization o injctiv ibr spacs In ordr to obtain a charactrization o injctiv objcts in Top/B, irst w nd to rcall th ollowing dinition (s [5]): Dinition 2.1. Givn : B and Y in C with init limits, th partial product P(,Y) o Y on is dind (whn it xists) as a morphism p : P B quippd with an valuation : P B B, such that th squar in Y P B p P p B is a pullback and, givn a pullback diagram on and a map h : V Y Y h V g W g B

5 F. Cagliari, S. Mantovani / Topology and its pplications 125 (2002) thr is a uniqu h : W P with g = ph and h = h,whrh : V P B is givn by th univrsal proprty o th pullback Y h V h g P B p h W g P p B Th xistnc o partial products on is quivalnt to xponntiability o (Lmma 2.1 in [5]). partial product is in act a powr objct in C/B; mor prcisly, i π B : Y B B dnots th projction on B, P(,Y)= (Y B,π B ) (, ), or any xponntiabl (, ).So th construction o a partial product givs rais to a controvariant unctor P(,Y): C/B C/B. This unctor assigns to any map α : (X, ) (X, ) th ma(α,y)= α (with domain P(,Y) = (P,p ) and codomain P(,Y) = (P, p)), givn by th univrsal proprty o th partial product in corrspondnc to th pullback o along p. In particular w hav in Top: Proposition 2.2. Givn : B, th ollowing ar quivalnt: (1) Th partial product o Y on xists, or any Y in Top. (2) Th partial product o I on xists, or any I injctiv in Top. (3) Th partial product o I on xists, or any I injctiv and not indiscrt in Top. (4) Th partial product o S on xists, or S th Sirpinski spac. (5) is xponntiabl in Top. Proo. (1) (2) (3) ar trivial. (3) (4). Th Sirpinski spac S is a rtract o any injctiv spac I which is not indiscrt. In analogy with th proo o Lmma 1.4, givn th partial product p S : P PsTop B o S on in PsTop and th partial product p I : P B o I on in Top, it asy to s that P PsTop is a rtract o P and thn p S is actually th partial product o S on in Top. (4) (5). Thorm 2.3(c) in [7]. (5) (4).Lmma2.1in[5]. Th irst rlation with injctiv objcts is givn by th ollowing lmma: Lmma 2.3. Lt : B b xponntiabl in Top and I b injctiv in Top. Thn th partial product (P, p) o I on is injctiv in Top/B.

6 530 F. Cagliari, S. Mantovani / Topology and its pplications 125 (2002) Proo. It ollows rom Lmma 1.4 o [5] with F = ( ) B : Top/B Top and G = P(, ) : Top Top/B, sinc F prsrvs mbddings. Now w ar rady to giv th charactrization thorm: Thorm 2.4. (, ) is injctiv in Top/B i and only i it is a rtract in Top/B o a partial product o S. Proo. Lt (, ) b injctiv in Top/B. By Corollary 1.10, w hav that is xponntiabl. Thn thr xists in Top th partial product (P, p) on o S, with P ={(h, b) h: 1 (b) S, h continuous}: S P B p P p B By Lmma 2.3, (P, p) is injctiv in Top/B, thn p is xponntiabl, by Corollary So w can orm th partial product (P,p ) o S on p: ε S P B P B P π P P p π P p P p B By th univrsal proprty o partial product, corrsponding to th map, thrisa uniqu : P with p = and ε =, whr : P B P B P is givn by th univrsal proprty o th pullback. W want to show that is an mbdding. Th pullback o th projction π B : C(,S) B B along : ϕ S C(,S) π id C(,S) B πb B

7 F. Cagliari, S. Mantovani / Topology and its pplications 125 (2002) is ndowd with a map ϕ to S, namly th valuation on C(,S), that is continuous, sinc C(,S) is discrt (s [11]). Thn, by th univrsal proprty o th partial product, w gt a map ϕ : C(,S) B P, with ϕ (k, b) = (k,b), S ϕ ϕ C(,S) π P B p id ϕ C(,S) B π B P p B such that pϕ = π B. ϕ : (C(, S) B,π B ) (P, p) is a map in Top/B. pplying th unctor P(,S) to ϕ, w obtain ϕ : P(p,S) P(π B,S), with P(p,S) = (P,p ) and th partial product P(π B,S)= (S C(,S) B,p B ), as a routin calculation shows. Th situation is dscribd by th ollowing diagram: S ξ S C(,S) C(,S) ϕ C(,S) B ε P B P ϕ π P π B p S C(,S) B π P P P ϕ p p S C(,S) B p B B By th univrsal proprty o th partial product p B o S on π B, p B ( ϕ ) = p =. Sinc also th mbdding α, : S C(,S) B o Proposition 1.8 ralizs p B α, =,thn α, = ϕ and is provd to b an mbdding. This is nough or us in ordr to conclud that (, ) is a rtract o (P,p ) sinc, by injctivity o (, ), thr xists r : P, with r = id, as th ollowing diagram shows: r id P p B

8 532 F. Cagliari, S. Mantovani / Topology and its pplications 125 (2002) Vicvrsa, a partial product (P, p) o S on s : X B is injctiv by Lmma 2.3. rtract (, ) o such a (P, p) is thn injctiv in Top/B. Rrncs [1] J. dámk, H. Hrrlich, J. Rosicky, W. Tholn, Wak actorization systms and topological unctors, Prprint, [2] J. dámk, H. Hrrlich, J. Rosicky, W. Tholn, On a gnralizd small-objct argumnt or th injctiv subcatgory problm, Prprint, [3] G. Bruns, H. Laksr, Injctiv hulls o smilattics, Canad. Math. Bull. 13 (1970) [4] F. Cagliari, S. Mantovani, Prsrvation o topological proprtis undr xponntiation, Topology ppl. 47 (1992) [5] R. Dyckho, W. Tholn, Exponntiabl morphisms, partial products and pullback complmnts, J. Pur ppl. lgbra 49 (1987) [6] J. Isbll, Gnral unction spacs, products and continuous lattics, Math. Scand. 100 (1986) [7] S. Niild, Cartsiannss: Topological spacs and ain schms, J. Pur ppl. lgbra 23 (1982) [8] D.S. Scott, Continuous lattics, in: Lctur Nots in Math., Vol. 274, Springr, Brlin, 1972, pp [9] W. Tholn, Injctivs, xponntials, and modl catgoris, in: bstracts o th Int. Con. on Catgory Thory (Como, Italy, 2000), [10] W. Tholn, Essntial wak actorization systms, in: Proc. 60th rbitstagung llgmin lgbra (60) (Drsdn, Jun 2000). [11] S. Willard, Gnral Topology, ddison-wsly, Rading, M, 1970.