Model categories. Daniel Robert-Nicoud

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1 Model categores Danel Robert-Ncoud Localzaton Motvatons or model categores arse rom varous elds o study, e.g. rng theory and homotoy theory, as we wll see. More recsely, what we want to do s to reverse some arrows n a category. Examle 1 (rng theory). Let R be a commutatve rng wth unty, S R a subset o R (not necessarly a subrng). We want to buld a rng, usually denoted by R[S 1 ], where or all s S there s an nverse element s 1 to s. Ths rocess s well known n algebra under the name o localzaton. The ncluson : R R[S 1 ] has the ollowng unversal roerty: or every rng homomorhsm : R T such that or every s S we have that (s) T, there s a unque rng homomorhsm : R[S 1 ] T such that the dagram: R T R[S 1 ] commutes. We note that R can be descrbed as a category wth only one element, where the elements o R are gven by the morhsms and multlcaton o elements s the comoston o arrows (naturally there s the addtonal structure o an abelan grou on the morhsms, thus ths s actually a so called category enrched over Ab). Then t s evdent that the localzaton o R n S s the rocess o addng to the category nverses or the arrows corresondng to the elements o S. Examle 2 (homotoy theory). Let To be the category o toologcal saces and contnuous mas and let W be the subset o the arrows n To gven by all weak homotoy equvalences. In homotoy theory we are nterested n the study o the dstnct weak homotoy tyes, where two objects are sad to have the same weak homotoy tye there s a sequence o weak homotoes jonng them. The roblem s that weak homotoy equvalence s not an equvalence relaton. Let or examle K denote the Cantor set wth the subsace toology o R and K ds be the Cantor set wth the dscrete toology. Then the ma rom K ds to K gven by x x s a weak homotoy, but t s easy to see that there can be no weak homotoy rom K to K ds. Thus we want to make the 1

2 weak homotoes nto somorhsms to get a real equvalence relaton, obtanng To[W 1 ], the so called category o weak homotoy tyes. We generalze these concets as ollows: Denton 3 (Localzaton). Let C be a category, S a subset o the arrows o C. The localzaton o C n S s a unctor rom C to a category C [S 1 ] wth the same object set such that (c) = c or every object c C and that or every unctor F rom C to some category D wth the roerty that or all s S, F (s) s an somorhsm, there s a unque unctor F : C [S 1 ] D such that the ollowng dagram commutes: C C [S 1 ] F D Remark 4. It s mortant to note that the localzaton o a category mght not have sets or hom-sets, but only classes nstead. We wll not elaborate on the case. An alcaton o ths s obvously the localzaton n a commutatve rng. here we gve two other smle examles: Examle 5. C s the category wth two elements and only one arrow a between a them. We wrte C = 0 1. Then, wth the same notaton: F C [{a} 1 ] = 0 wth a 1 a = d 0 and a a 1 = d 1. a 1 Ths was an examle o a very smle localzaton. Wth more comlcated categores t becomes mmedately very dcult to descrbe the arrows or the localzaton. Examle 6. Let C be the ollowng category: a 1 C = 0 a b 1 c 2 3 We want to nd C [{b} 1 ]. Unortunately, the ollowng s not enough: C [{b} 1 ] = 0 a 1 b c 2 3 snce ths construct s no more a category, mssng the varous comostons. It s n act necessary to add three more arrows (the varous comostons o arrows wth b 1 ) n order to obtan the localzaton o C n {b}. What we want now s some secal knd o categores where, gven such a category C and a some knd o subset o ts arrows W, t s ossble to descrbe exactly what the sets o homomorhsms n the localzaton C [W 1 ] look lke. That s where model categores come nto lay. b 1 2

3 Prelmnary notons We recall some notons rom category theory that we wll need later on. Denton 7. Let A and B be objects o a category C. We say that A s an object retract o B there are two arrows, one rom A to B and one rom B to A, such that the ollowng dagram commutes: A B d A Let : X Y and : X Y be arrows n a category C. Then s sad to be an arrow retract (or smly a retract) o t s an object retract o n the category o arrows on C,.e. the category wth arrows and commutatve squares as objects and morhsms resectvely. Equvalently, s a retract o there are arrows such that the ollowng dagram commutes: X A X Y Y d X d X X Y Examle 8. We can easly see that the noton o an object retract s a natural generalzaton o the toologcal concet o a retract. Indeed, let X be some toologcal sace, A a retract o X and r : X A the retracton. Then r s a let nverse or the ncluson ma o A n X, A. Conversely, let X and A be two objects o the category To o toologcal saces and contnuous mas, and r : X A a retracton o some uncton : A X n the sense dened above. Then we have that s surely njectve (else r d A ) and thus bjectve on ts mage à = (A) X. Let be wth ts codoman restrcted to Ã. Obvously, r 1 à = r s also a bjecton. s a contnuous uncton, snce t s a well dened uncton and t equals r d A. Thus à = A. Ths way we get that à s a retract o X (n the toologcal sense) 3

4 by the ollowng commutatve dagram: à à 1 A X dã d A r A à where à s the ncluson o à n X. Notce that the retracton or à s gven by r. Denton 9. Let C be a category. An object t s termnal n C or every object c C there s exactly one arrow c t. An object s s ntal n C or every object c C there s exactly one arrow s c. Remark 10. Termnal and ntal objects are unque u to somorhsm (as you can readly check). Examle 11. Let To be the category o toologcal saces and contnuous unctons. Then the emty set s an ntal object o To and the one ont set s a termnal object. Model Categores Denton 12. Let α and β be two morhsms n a category C. We say that α has let ltng roerty wth resect to β, and that β has rght ltng roerty wth resect to α, or every commutng square there s a dashed arrow such that the ollowng dagram commutes: α β We denote ths by α β. Let S, T be two subsets o the arrows o C. We say that S has the let ltng roerty wth resect to T, and that T has the rght ltng roerty wth resect to S, or every α S, β T we have α β. In ths case we wrte S T. I S s any subset o the arrows o C, we dene the ollowng two other subsets o the arrows o C : S = {β s β, s S} S = {α α s, s S} 4

5 We gve two examles o ltng roertes or object dened trough some ltng roerty. Both come rom toology. Examle 13. Let B be a toologcal sace, E a coverng sace o B wth coverng ma : E B. Then the ma : I = [0, 1] sendng the one ont set to the element {0} I has let ltng roerty wth resect to. Indeed, take γ : I B any ath and α : E sendng to any ont n 1 (γ(0)). Then there s exactly one dashed arrow makng the ollowng dagram commute: I α E γ B Examle 14. A contnuous ma : X Y between two toologcal saces X and Y s called a Serre braton t has the rght ltng roerty wth resect to all ncluson mas n the set { : Z {0} Z I Z s a CW comlex}. Denton 15. A closed model category s a tule (M, W, C, F ), where M s a category and W, C and F are three classes o unctons, called weak equvalences, cobratons and bratons resectvely, such that the ollowng axoms hold: CM1 M s closed under lmts (comlete) and colmts (cocomlete). CM2 Let X, Y, Z be objects o M and, g, h be morhsms such that the ollowng dagram commutes: X g Y h Z I two o, g, h are weak equvalences, then so s the thrd. CM3 Let, g be morhsms n M. I s a retract o g and g s a weak equvalence, braton or cobraton, then so s. CM4 Assume the ollowng dagram commutes n M : U X V Y I s a cobraton and a braton, and one o the two s trval (.e. t s also a weak equvalence) then there s an arrow rom V to X such that 5

6 the ollowng dagram commutes: U X V Y Sad n another way, C (F W ) and (C W ) F. CM5 Let X, Y be objects n M and a morhsm rom X to Y. Then can be actored n the two ollowng ways: (a) =, where s a braton and s a trval cobraton (.e. a cobraton that s at the same tme a weak equvalence). (b) = q j, where q s a trval braton (at the same tme braton and weak equvalence) and j s a cobraton. Remark 16. The orgnal denton, gven by Qullen n [3.], s slghtly derent. The denton we use s more rened and owerul or our needs. To smly the readng o the dagrams, we wll rom now on use or bratons, or cobratons and ut a lttle on weak equvalences. Lemma 17. Let (M, W, C, F ) be a model category. Then (C W ) = F (.e. a ma s a braton and only t has the rght ltng roerty wth resect to all trval cobratons), C = F W, C W = F and C = (F W ). Proo. We wll rove only the rst statement. The other are roven n a smlar way. Assume s a braton and any trval cobraton. Then, by CM4, there a commutatve dagram as ollows: Thus (C W ). Conversely, assume (C W ). By CM5, there are a trval cobraton and a braton q such that = q, or, seen as a dagram: U W q V 6

7 Then, by rght ltng roerty o wth resect to trval cobratons, we have the ollowng dotted arrow: U= U W q V Thus, s a retract o q, as the ollowng dagram shows: Then by CM3, s a braton. U d U W lt q U V V V Remark 18. What just roven s what makes a model category closed. As already sad, the orgnal axoms o Qullen or a model category were derent, and ddn t mly ths. It was noted, though, that the vast majorty o examles were o closed model categores, and not smly o model categores, thus the axoms were changed to the ones we gave. From now on we wll dro the adjectve closed, when seakng o model categores, because nowadays t s only a hstorcal arteact. I M s a model category, the act that t s comlete and cocomlete (CM1) mles that t has some ntal object and some termnal object. Now let x and y be objects o M, and a hom M (x, y). Then we have: x a y Where the arrows rom and to are unque. ollowng actorzaton: a x y Qx b Then by CM5 we have the Ry 7

8 Ths mles that n the localzaton o M n W (the set o weak equvalences) we can nterchange a and b (snce weak equvalences become somorhsms). Objects lke Qx and Ry wll have a lot o mortance later on, and thus we gve them a name: Denton 19. Let M be a model category wth ntal object and termnal object. An object y o M s called brant the unque ma y s a braton. An object x o M s called cobrant the unque ma x s a cobraton. What we have done above can thus be nterreted as the act that we can always change x and y wth some cobrant and brant objects Qx and Ry. Remark 20. Qx and Ry are not x: they deend on a. Examle 21. In the case o the category To, we can take bratons to be Serre bratons, cobratons the contnuous unctons havng the ltng roerty o CM4 and weak equvalences to be weak homotoy equvalences. Then To together wth those three tyes o mas s a model category n whch every object s brant and all CW-comlexes are cobrant. Denton 22. Let M be a model category,, g : X Y two arrows n M, Y Y the roduct object o Y and Y, X X the coroduct object o X and X. Then: Take the ma : Y Y Y nduced by the dentty ma and actorze t as n CM5(a): Y Y Y Y I Then Y I s the so called ath object o X. We say that s rght homotoc to g, and wrte r g, there s a ma H : X Y I such that the ollowng dagram commutes: X g H Y Y Y I where the ma rom Y I to Y Y s the one o the revous dagram. Smlarly, let : X X X be the ma nduced by the dentty ma. We say that and g are let homotoc, and wrte l g, there s an 8

9 arrow K such that the ollowng dagram commutes: X X g X Y K X I where we have used CM5(b) to decomose. X I s called the cylnder object o X. Remark 23. I X s brant, then rght homotoy s an equvalence relaton on the hom-set hom(x, Y ). I Y s cobrant, then let homotoy s an equvalence relaton. I X s brant and Y s cobrant, then r g l g. n ths case, we wrte g. For a roo o those acts, reer to [4.]. Remark 24. It s qute easy to see both the notons o let/rght homotoy equvalence generalze the noton o homotoy o mas n To. Frst o all we notce that there s n act an object I n To (gven by the closed unt nterval) such that X I s the cylnder object o X and such that Y I s the ath object o Y. The denton or let homotoy equvalence s then exactly the denton o homotoy. Rght homotoy equvalence can be seen to exress the same concet by notng that there s a (qute obvous) somorhsm hom(x I, Y ) = hom(x, Y I ). What we do now s to relace the model category M by ts localzaton n the weak equvalences M [W 1 ]. We wll denote ths new category by HoM and call t the homotoy category o M. Here all weak equvalences are somorhsms, and or ths reason, wth a lttle work, we can dene Q and R as unctors rom M to HoM, the so called cobrant and brant relacements (see or examle [4.]). We can then have a comlete reresentaton o hom HoM (x, y) or any two objects x and y. It s gven by hom HoM (x, y) = hom M (Qx, Ry)/. We wll not gve roo o ths act, but t s somethng very useul when tryng to descrbe the homotoy category HoM. Yoga o derved unctors When seakng o categores, unctors are o undamental mortance. Thus gven two model categores (M, W, C, F ), (M, W, C, F ) and a unctor F rom M to M, we d lke to derve a unctor rom HoM to HoM such that the localzaton s reserved n some sense. A rst, very ntutve aroach would be to ask or a unctor G makng the ollowng dagram commute: M F M HoM G HoM 9

10 where the vertcal arrows are the localzatons. Examle 25. Take the unctor π n : To G, n 1. The weak equvalences n To are the weak homotoy equvalences, n G they are smly the somorhsms. Then G s the localzaton o tsel and thus we can use the unversal roerty o the localzaton to the ollowng dashed unctor: π n To G HoTo HoG It s qute obvous that a unctor satsyng ths roerty can exst only F (W ) W. It turns out that ths requrement s too strct. We wll loosen our requrements to accet a broader class o unctors. Let (M, W, C, F ) be a model category, C any category and F a unctor rom M to C. Let G be a unctor rom HoM to C. We requre that the ollowng dagram commutes u to a natural transormaton: M F or C M F C HoM G HoM where denotes the localzaton. That means that we requre that ether F G or G F. We also requre that G has some knd o unqueness roerty, so that t can be dened well, when t exsts. To dene ths we need the ollowng dentons: Denton 26. Let (M, W, C, F ) be a model category and C any category. Then we denote by Fun(M, C ) the set o unctors rom M to C and by Fun W (M, C ) the set o unctors rom M to C such that weak equvalences are sent to somorhsms. Denton 27. Let C be a category, D a subset o the objects o C and x some object o C. Then we dene the ollowng two categores: D/x s the category wth as objects the arrows y x wth y D and as arrows the commutatve trangles: where y, y D. y x y G 10

11 x\d s the category wth as objects the arrows x y wth y D and as arrows the commutatve trangles: where y, y D. x y We notce that n act Fun W (M, C ) s a subset o the objects o the category o unctors, thus the ollowng denton, whch gves us the sought unqueness, makes sense: Denton 28. Let (M, W, C, F ) be a model category, C any category, F a unctor rom M to C. Then: I the category Fun W (M, C )/F has a termnal object, we call t the let derved unctor and denote t by LF. In act, ths object s a natural transormaton. Wth an abuse o notaton, we also denote ts codoman LF. I the category F \Fun W (M, C ) has an ntal object, we call t the rght derved unctor and denote t by RF. Agan, wth an abuse o notaton we also denote ts doman by RF. Remark 29. I F sends weak equvalences to somorhsms, we get that F = LF = RF. For the case o F = LF, take the category Fun W (M, C )/F. We notce that F F s n act n ths category, and thus t s ts termnal object. The case or RF s very smlar. In the general case, we can see those unctors as the ones that are the closest to F n ther resectve categores. We can then use LF, RF to get the unctors we wanted. Assume or examle that LF exsts. Then, snce LF Fun W (M, C )/F, t s actually a natural transormaton LF F rom some unctor LF Fun W (M, C ) to F. Ths means, by denton 26, that the unctor LF sends weak equvalences to somorhsms. Thus we can aly the unversal roerty o the localzaton to get a sutable unctor: M LF C HoM We automatcally have the natural transormaton G = LF F. Furthermore, snce the natural transormaton LF s unque u to somorhsm (beng a termnal object), we have that the unctor LF s also unque u to somorhsm (nvertble natural transormaton), by the denton o Fun W (M, C )/F, and the same s then vald or G, as we wanted. G y 11

12 Man examle: cochan comlexes We resent now an mortant examle o model category: the (co)chan comlexes o let (or rght) modules over a rng R. Denton 30. The category coch(r) o cochan comlexes o let modules over a rng R s the ollowng category: Objects are sequences o let R-modules and mas (M, d) = (M n, d n ) n Z wth d n : M n M n+1 such that the comoston o two successve such arrows s the trval ma (.e. d n (M n ) ker(d n+1 )). We oten reresent such an object by:... d 2 M 1 d 1 M 0 d 0 M 1 d 1 M 2 d 2... Morhsms between two chans (M, d), (N, e) are sequences o unctons = ( n ) n Z, n : M n N n such that every square o the ollowng dagram commutes: d 2 d... 1 d M 1 0 d M 0 1 d M 1 2 M e 2 N 1 e 1 N 0 e 0 N 1 e 1 N 2 e 2... In other words, n+1 d n = e n n. Denton 31. We dene two classes o objects n coch(r). Let M be a let R-module, then: S n (M) s the chan wth M at ndex n and the zero module at every other ndex. D n (M) s the chan wth M at ndexes n 1 and n (wth dentty ma between them) and the zero module at every other ndex. We usually use the notaton S n = S n (R) and D n = D n (R). Denton 32. The n th homology or cochan comlexes s the unctor H n : coch(r) R mod dened by H n (M, d) = ker(d n )/d n 1 (M n 1 ). I : (M, d) (N, e), then H n () : H n (M, d) H n (N, e) s dened by ([x]) = [ n (x)]. We now have all we need to construct a model structure on coch(r). Frst o all we dene the class o weak equvalences by W = { H n () s a bjecton n Z} Then we take the ollowng two sets o arrows: I = {S n 1 D n } J = {0 D n } 12

13 From these we dene our set o bratons as F = J, and our set o cobratons as C = (I ). We show two o the axoms needed or ths to actually be a model structure on coch(r): CM2 Assume that two out o three o g, and h are weak equvalences and that the ollowng dagram commutes: X g Y h Z We show that the thrd arrow s also a weak equvalence. I and g are weak equvalences, then by denton we have that or all n, H n () and H n (g) are bjectve. Snce H n s a unctor, we have that H n (h) = H n () H n (g), and t s thus bjectve, rovng that h s also a weak equvalence. Smlar reasonng roves the other two cases. CM3 Let, g be two arrows, and let be a retract o g. We have d a g c b d d Assume g s a weak equvalence. Then, snce d = b a, we have H n (d) = H n (b) H n (a) and thus (snce H n (d) s obvously bjectve) that H n (a) s njectve and H n (b) s surjectve. Smlarly, H n (c) s njectve, H n (d) s surjectve. Note then that g a = c (snce the rst square commutes. Snce H n (g) s bjectve by assumton and, as we have shown, H n (a) s njectve, we have that H n (g) H n (a) s also njectve, and thus that H n () also s. Smlarly, usng the second square, we get that H n () s surjectve, and thus that s a weak equvalence. Assume that g s a braton. Then t s easy to see that s also a braton (.e. J ). Indeed, let j J. Then the ollowng dagram shows that has rght ltng roerty wth resect to j: d j g d 13

14 where the dashed arrow s gven by rght ltng roerty o g wth resect to the set J. The case where g s a cobraton s smlar. In the book o Hovey ([4.]), the very smlar o the category Ch(R) o chan comlexes over a rng R s treated n more detal. Remark 33. The category coch(r) s n act a case o what s called a cobrantly generated model category. The general constructon o ths knd o category s done by choosng a set W o weak equvalences and two sutable generatng sets o arrows I and J, wth J I, and then denng bratons and cobratons as ollows: F = J (F W ) = I (C W ) = (J ) C = (I ) For J and I satsyng some condtons, ths gves n act a model category. Reerences 1. S. Mac Lane, Categores or the Workng Mathematcan (second ed.), Graduate Texts n Mathematcs 5, Srnger. 2. P. Goerss, J. Jardne, Smlcal homotoy theory, Progress n mathematcs, Brkhauser. 3. D. Qullen, Homotocal Algebra, Lecture Notes n Mathematcs, No. 43, Srnger. 4. M. Hovey, Model Categores, Mathematcal Surveys and Monograhs, No. 63, AMS. 5. J. R. Munkres, Toology (second ed.), Prentce Hall. 14