HOMOTOPY THEORY OF COALGEBRAS OVER OPERADS 6 0 JUSTINR.SMITH 0 2 Abstra t. Thispaper onstru tsmodelstru tures onthe ategoriesof oal- t c gebras and pointed irredu ible oalgebras over an operad whose omponents O are proje tive, (cid:28)nitely generated in ea h dimension, and satisfy a ondition that allows one to take tensor produ ts with a unit interval. The underly- 3 ing hain- omplex is assumed to be unbounded and the results for bounded oalgebras over anoperadare derivedfromtheunbounded ase. ] T C . h t a Contents m 1. Introdu tion 1 [ 2. Notation and onventions 3 4 3. The general ase 9 v 4. The bounded ase 20 7 5. Examples 21 1 Appendix A. Nearly free modules 21 3 5 Appendix B. Category-theoreti onstru tions 24 0 B.1. Cofree- oalgebras 24 3 B.2. Core of a module 27 0 B.3. Categori al produ ts 29 / h B.4. Limits and olimits 31 t Referen es 39 a m : v i X 1. Introdu tion r a Althoughtheliterature ontainsseveralpapersonhomotopytheoriesforalgebras over operads (cid:22) see [14℄, [17℄, and [18℄ (cid:22) it is more sparse when one pursues similar results for oalgebras. In [20℄, Quillen developed a model stru ture on the ategory of 2- onne ted o ommutative oalgebras over the rational numbers. V. Hini h extended this in [13℄ to oalgebras whose underlying hain- omplexes were unbounded (i.e., extended into negative dimensions). Expanding on Hini h's methods, K. Lefèvre derived a model stru ture on the ategory of oasso iative oalgebras (cid:22) see [15℄. In general, these authors use indire t methods, relating of oalgebra ategories to other ategories with known model stru tures. Date:8thFebruary 2008. 1991 Mathemati s Subje tClassi(cid:28) ation. Primary18G55;Se ondary 55U40. Key words and phrases. operads, ofree oalgebras. 1 HOMOTOPY THEORY OF COALGEBRAS JUSTIN R. SMITH Ourpaper(cid:28)ndsmodelstru turesfor oalgebrasoveranyoperadful(cid:28)llingabasi requirement ( ondition 3.5). Sin e operads uniformly en ode many diverse oal- gebra stru tures ( oasso iative-, Lie-, Gerstenhaber- oalgebras, et .), our results have wide appli ability. Several unique problems arise that require spe ial te hniques. For instan e, onstru ting inje tive resolutions of oalgebrasnaturally leads into in(cid:28)nitely many negative dimensions. The resulting model stru ture (cid:22) and even that on the un- derlying hain- omplexes (cid:22) fails to be o(cid:28)brantly generated (see [5℄). Wedevelopthegeneraltheoryforunbounded oalgebras,andderivethebounded results by applying a trun ation fun tor. InŸ2,wede(cid:28)neoperadsand oalgebrasoveroperads. Wealsogiveabasi ondi- tion(see3.5)ontheoperadunder onsiderationthatweassumetoholdthroughout thepaper. This onditionissimilarto thatofadmissibility ofBergerandMoerdijk in [2℄. Co(cid:28)brant operads always satisfy this ondition and every operad is weakly equivalent to one that satis(cid:28)es this ondition. In Ÿ 3, we brie(cid:29)y re all the notion of model stru ture on a ategory and de(cid:28)ne model stru tures on two ategories of oalgebras over operads. When the operad is proje tive and (cid:28)nitely-generated in all dimensions, we verify that nearly free oalgebrassatisfy Quillen's axioms of a model stru ture (see [19℄ or [11℄). A key step involves proving the existen e of o(cid:28)brant and (cid:28)brant repla ements for obje ts. In our model stru ture, all oalgebras are o(cid:28)brant (solving this half of the problem) and the hard part of is to (cid:28)nd (cid:28)brant repla ements. Wedevelopresolutionsof oalgebrasby ofree oalgebrasthatsolvestheproblem (cid:22) see lemma 3.16 and orollary 3.17. This onstru tion naturally leads into in- (cid:28)nitelymany negativedimensions and wasthe motivationforassumingunderlying hain- omplexes are unbounded. Fibrant oalgebrasare hara terizedas retra ts of layered oalgebras (see de(cid:28)n- ition 3.18 and orollary 3.19) (cid:22) an analogue to total spa es of Postnikov towers. In the o ommutative ase over the rational numbers, the model stru ture that we get is not equivalent to that of Hini h in [13℄. He gives an example (9.1.2) of a oalgebrathatisa y li butnot ontra tible. Inourtheoryitwould be ontra tible, sin e it is over the rational numbers and bounded. InŸ4,wedis ussthe(minor) hangestothemethodsinŸ3tohandle oalgebras thatareboundedfrombelow. Thisinvolvesrepla ingthe ofree oalgebrasbytheir trun ated versions. In Ÿ 5, we onsider two examples over the rational numbers. In the rational, 2- onne ted, o ommutative, oasso iative ase, we re over the model stru ture Quillen de(cid:28)ned in [20℄ (cid:22) see example 5.2. Z InappendixA,westudynearly free -modules. Thesearemoduleswhose ount- Z able submodules are all -free. They take the pla e of free modules in our work, sin e the ofree oalgebra on a free modules is not free (but is nearly free). In appendix B, we develop essential ategory-theoreti onstru tions, in luding equalizers (Ÿ B.2), produ ts and (cid:28)bered produ ts (Ÿ B.3), and olimits and limits (Ÿ B.4). The onstru tion of limits in Ÿ B.4 was this proje t's most hallenging aspe t and onsumed the bulk of the time spent on it. This se tion's key results are orollary B.21, whi h allows omputation of inverse limits of oalgebras and theorem B.23, whi h shows that these inverse limits share a basi property with those of hain- omplexes. 2 HOMOTOPY THEORY OF COALGEBRAS JUSTIN R. SMITH I am indebted to Professor Bernard Keller for severaluseful dis ussions. 2. Notation and onventions R Z Throughout this paper, will denote a (cid:28)eld or . R M De(cid:28)nition 2.1. An -module will be alled nearly free if every ountable R submodule is -free. R=Z Remark. This ondition is automati ally satis(cid:28)ed unless . Z Zℵ0 Clearly,any -freemoduleisalsonearlyfree. The Baer-Spe kergroup, , isa Z well-known example of a nearly free -module that is not free (cid:22) see [10℄, [1℄, and ℵ 1 [23℄. Compare this with the notion of -free groups (cid:22) see [4℄. By abuse of notation, we will often all hain- omplexes nearly free if their underlying modules are (ignoring grading). Z Nearly free -modules enjoy useful properties that free modules do not. For instan e, in many interesting ases, the ofree oalgebra of a nearly free hain- omplex is nearly free. De(cid:28)nition 2.2. WeRwill denote the losed sRymmetri monoidal aCtehg(oRr)y of un- bounded, nearly free - hain- omplexes with -tensor produ ts by . We make extensive use of the Koszul Convention (see [12℄) regarding signs in homologi al al ulations: f:C →D g:C →D a⊗b∈C ⊗C a 1 1 2 2 1 2 De(cid:28)nition2.3. If , aremaps,and (where (f⊗g)(a⊗b) (−1)deg(g)·deg(a)f(a)⊗ isahomogeneouselement),then isde(cid:28)nedtobe g(b) . f g i i Remark 2.4.(fIf ⊗,g )◦ar(efm⊗apgs,)i=t i(s−n'1t)hdeagr(df2)t·odevg(egr1i)f(yfth◦aft t⊗hegK◦osgzu)l onvention 1 1 2 2 1 2 1 2 implies that . I De(cid:28)nition 2.5. The symbol willdenotetheunitinterval, a hain- omplexgiven by I = R·p ⊕R·p 0 0 1 I = R·q 1 I = 0 k 6=0,1 k if A∈Ch(R) Given , we an de(cid:28)ne A⊗I and Cone(A)=A⊗I/A⊗p 1 The set of morphisms of hain- omplexes is itself a hain omplex: A,B ∈Ch(R) De(cid:28)nition 2.6. Given hain- omplexes de(cid:28)ne Hom (A,B) R R to be the hain- omplex of graded -morphisms where the degree of an element x∈Hom (A,B) R is its degree as a map and with di(cid:27)erential ∂f =f ◦∂ −(−1)degf∂ ◦f A B R Hom (A,B) = Hom (A ,B ) As a -module R k j R j j+k . Q 3 HOMOTOPY THEORY OF COALGEBRAS JUSTIN R. SMITH Remark. Given A,B ∈Ch(R)Sn, we an de(cid:28)ne HomRSn(A,B) in a orresponding way. De(cid:28)nition 2.7. De(cid:28)ne: Set Set2 (1) f tobethe ategoryof(cid:28)nitesetsandbije tions. Let f bethe ategory of (cid:28)nite sets whose elements are also (cid:28)nite sets. Morphisms are bije tions ofsetsthatrespe tthe(cid:16)(cid:28)nestru ture(cid:17) ofelementsthatarealsosets. There is a forgetful fun tor f:Set2 →Set f f Set2 that simply forgets that the elements of an obje t of f are, themselves, (cid:28)nite sets. There is also a (cid:16)(cid:29)attening(cid:17) fun tor g:Set2 →Set f f that sends a set (of sets) to the union of the elements (regarded as sets). X Σ =End (X) (2) FSeotr a−(cid:28)mnoitde set , X Setf . Func(Setop,Ch(R)) (3) f tobethe ategoryof ontravariantfun tors f , with morphisms that are natural transformations. C, D ∈Set −mod Hom(C,D) f (4) Given , de(cid:28)ne to be the setof natural trans- Hom (C,D) X ∈ Set formations of fun tors. Also de(cid:28)ne X , where f, to be C D X the natural transformations of and restri ted to sets isomorphi to (i.e., of the same ardinality). Both of these fun tors are hain- omplexes. Σ−mod {M(n)} m ≥ 1 M(n) ∈ (5) Ch(R) to bMe(tnh)e ategory of sequen es S , where n and is equipped with a right -a tion. [n] n Σ = S [n] n Remark. If is the set of the (cid:28)rst positive integers, then , the sym- M Set X f metri group. If is a -module then, for ea h (cid:28)nite set, , there is a right Σ M(X) X -a tion on . S =S ={1} 0 1 We follow the onvention that , the trivial group. Σ−mod Noate=th{a{tx},{y,z,ti}s,w{hh}a}t ∈is oSfette2n alledf(tah)e∼= a[t3e]gory of olle tions. g(aI)f={x,y,z,t,h} f then , a set of three elements, and . Set −mod Σ−mod f It is well-known that the ategories and are isomorphi (cid:22) see se tion 1.7 in part I of [18℄. The restri tion isomorphism r:Set −mod→Σ−mod f [n] n ≥ 1 F ∈ simply involves evaluating fun tors on the (cid:28)nite sets for all . If Set −mod r(F) = {F([n])} F F([n]) f , then . The fun torial nature of implies that S Hom (C,D) is equipped with a natural n-a tion. The fun tors n orrespond to Hom (C([n]),D([n])) Set RSn and the fa t that morphisms in f preserve ardinality imply that Hom(C,D)= Hom (C,D) n n≥0 Y Set f Although -modules are equivalent to modules with a symmetri group a - Set −mod f tion, it is often easier to formulate operadi onstru tions in terms of . Equivarian e relations are automati ally satis(cid:28)ed. X n X De(cid:28)nition 2.8. If is a (cid:28)nite set of ardinality the set of orderings of is Ord(X)={f|f:X −∼=→[n]} 4 HOMOTOPY THEORY OF COALGEBRAS JUSTIN R. SMITH Set X f Nowwedne(cid:28)nea analogueto themultiCple∈teCnsho(rRp)rodu t. Given asext∈Xof x ardinality , andanassignmentofanobje t forea helement , g ∈Ord(X) we an de(cid:28)ne, for ea h a produ t Cx =Cg−1(1)⊗···⊗Cg−1(n) g O The symmetry of tensor produ ts determines a morphism σ¯: C → C x x g σ◦g O O σ ∈S ±1 n for ea h whi h essentially permutes fa tors and multiplies by , following the Koszul Convention in de(cid:28)nition 2.3. De(cid:28)nition 2.9. The unordered tensor produ t is de(cid:28)ned by C =coequalizer σ¯: C → C x x x   OX σ∈Sn  g∈OMrd(X)Og g∈OMrd(X)Og  C ∈Ch(R) X ∈Set CX If and fthen will denote the unordered tensor produ t C X O C X C⊗ Set f of opiesof indexedbyelementsof ,and willdenotethe -modulewhose X ∈Set CX f value on is . X·C n C n We use to denote a dire t sum of opies of , where isthe ardinality X of a (cid:28)niteXse∈t Se.t2 f When , C X O f(X) X is regardedas being taken over (cid:22) i.e., we (cid:16)forget(cid:17) that the elements of are sets themselves. Ch(R) Remark. The unordered tensor produ t is isomorphi (as an obje t of to C x X x the tensor produ t of the , as runs over the elements of . The oequalizer onstXru =tio[nn]determCin[ne]s=hoCwnthe it behavCesXw⊗ithCrYes=peC tXt⊔oYset-mXor,pYhis∈mSs.et f If , then . Note that , for . We C∅ = =R 0 also follow the onvention that 1 , on entrated in dimension . X ∈Set x∈X {f :V →U } Ch(R) f y y y De(cid:28)nition 2.10. If , and are morphisms of y ∈X indexed by elements then de(cid:28)ne (U,V)= Z −1−⊗−·−··⊗−−fx−⊗−·−··⊗−→1 U y y X,x y∈X y∈X O O O to be the unordered tensor produ t, where U y 6=x y Zy = if (Vy u=x if X Remark. Given any ordering of the elements of the set , there exists a anoni al isomorphism (U,V)=U ⊗···⊗V ⊗···⊗U OX,x x position | {z } 5 HOMOTOPY THEORY OF COALGEBRAS JUSTIN R. SMITH X,Y ∈Set x∈X f De(cid:28)nition 2.11. Let and let . De(cid:28)ne X ⊔ Y =(X \{x})∪Y x X ⊔ ∅=X\{x} x Remark. Note that . X, Y, Z ∈Set xx , X ∈X y ∈Y f 1 2 Proposition. If , and and , then X ⊔ (Y ⊔ Z) = (X ⊔ Y)⊔ Z x y x y (X ⊔ Y)⊔ Z = (X ⊔ Z)⊔ Y x1 x2 x2 x1 Ch(R) Set C f De(cid:28)nition 2.12. An operad in is a -module, equipped with opera- tions ◦ :C(X)⊗C(Y)→C(X ⊔ Y) x x x∈X X, Y ∈Set f for all and all and satisfying the two axioms (1) Asso iativity: ◦ (1⊗◦ )=◦ (◦ ⊗1): x y y x C(X)⊗C(Y)⊗C(Z)→C(X ⊔ (Y ⊔ Z)) x y ◦ (◦ ⊗1)=◦ (◦ ⊗1)(1⊗τ): x2 x1 x1 x2 C(X)⊗C(Y)⊗C(Z)→C((X ⊔ Y)⊔ Z) x1 x2 X, Y, Z ∈ Set x, x , x ∈ X y ∈ Y τ:C(Y)⊗ f 1 2 for all and all and , where C(Z)→C(Z)⊗C(Y) is the transposition isomorphism. η : → C({x}) {x} ∈ x (2) Unit: There exist morphisms 1 for all singleton sets Set f that make the diagrams C(X)⊗ ∼= //C(X) ⊗C(X) ∼= //C(X) 1⊗ηx (cid:15)(cid:15) q1qqqq◦qxqqqq88 η1x⊗1(cid:15)(cid:15) ssssss◦sxsss99 C(X)⊗C(x) C(X) X ∈ Set f ommute, for all . The operad will be alled nonunital if the axioms above only hold for nonempty sets. Remark. See theorem1.60and 1.61and se tion1.7.1of [18℄ forthe proofthat this de(cid:28)nes operads orre tly. For more traditional de(cid:28)nitions, see [22℄, [14℄. This is basi ally the de(cid:28)nition of a pseudo-operad in [18℄ where we have added the unit nth axiom. To translate this de(cid:28)nition into the more traditional ones, set the C([n]) omponent of the operad to . Set −mod f The use of auses the equivarian e onditions in [14℄ to be automat- i ally satis(cid:28)ed. The operads we onsider here orrespond to symmetri operads in [22℄. The term (cid:16)unital operad(cid:17) is used in di(cid:27)erent ways by di(cid:27)erent authors. We use 0 itinthesenseofKrizandMayin[14℄,meaningtheoperadhasa - omponentthat C(∅)= a ts like an arity-loweringaugmentation under ompositions. This is 1. A simple example of an operad is: X C(X)=ZΣ X Example 2.13. Forea h(cid:28)niteset, , , with ompositionde(cid:28)nedby S n 0 th in lusion of sets. This operad is denoted . In other notation, its omponent ZS n is the symmetri group-ring . 6 HOMOTOPY THEORY OF COALGEBRAS JUSTIN R. SMITH For the purposes of this paper, the anoni al example of an operad is C ∈ Ch(R) De(cid:28)nition 2.14. Given any , the asso iated oendomorphism operad, CoEnd(C) is de(cid:28)ned by CoEnd(C)(X)=Hom (C,CX) R X ∈Set CX = C for f, and X is the unordered tensor produ t de(cid:28)ned in de(cid:28)ni- {◦ } x tion 2.9. The ompositionNs are de(cid:28)ned by ◦ :Hom (C,CX)⊗Hom (C,CY)→ x R R Hom (C,CX\{x}⊗C ⊗Hom (C,CY))−H−o−m−R−(−1−,1−⊗−e→) R x R Hom (C,CX\{x}⊗CY))=Hom (C,CX⊔xY) R R C C x∈X e:C ⊗Hom (C,CY)→CY x x R where isthe opyof orrespondingto and C ∈Ch(R) is the evaluation morphism. This is a non-unital operad, but if has an ε:C → augmentation map 1 then we an set CoEnd(C)(∅)= 1 and ◦ :Hom (C,CX)⊗Hom (C,C∅)=Hom (C,CX)⊗ x R R R 1 −H−o−m−R−(−1,−1X−−\{−x−}⊗−ε−x→) Hom (C,CX\{x}) R 1 :CX\{x} → CX\{x} ε :C → X\{x} x x where is the identity map and 1 is the aug- C x∈X mentationC, a∈pCplhie(dRt)o the opy of ind{eDxed,.b.y.,D } . 1 k Given withsub omplexes ,therelative oendomorphism CoEnd(C;{D }) CoEnd(C) i operadf ∈Hom (C,CX)is de(cid:28)ned tfo(bDe)th⊆eDsuXb-⊆opCerXad of j onsisting of maps R su h that j j for all . We use the oendomorphism operad to de(cid:28)ne the main obje t of this paper: V C ∈ Ch(R) De(cid:28)nition 2.15. A oalgebra over an operad is a hain- omplex α:V → CoEnd(C) with an operad morphism , alled its stru ture map. We will sometimes want to de(cid:28)ne oalgebrasusing the adjoint stru ture map α¯:C →Hom(V,C⊗) Ch(R)) (in or even the set of hain-maps α¯ :C →Hom (V(X),CX) X X X ∈Set f for all . We an also de(cid:28)ne the analogueof an ideal: C U De(cid:28)nition 2.16. Let be a oalgebra over the operad with adjoint stru ture map α:C →Hom(U,C⊗) D ⊆ ⌈C⌉ D and let be a sub- hain omplex that is a dire t summand. Then will C be alled a oideal of if the omposite α|D:D →Hom(U,C⊗)−H−o−m−(−1U−,−p−⊗→) Hom(U,(C/D)⊗) p:C →C/D Ch(R) vanishes, where is the proje tion to the quotient (in ). 7 HOMOTOPY THEORY OF COALGEBRAS JUSTIN R. SMITH Remark. Notethatitiseasierforasub- hain- omplextobea oidealofa oalgebra than to be an ideal of an algebra. For instan e, all sub- oalgebras of a oalgebra are also oideals. Consequently it is easy to form quotients of oalgebrasand hard to form sub- oalgebras. This is dual to what o urs for algebras. V We will sometimeswantto fo uson aparti ular lassof - oalgebras: the poin- ted, irredu ible oalgebras. We de(cid:28)ne this on ept in a way that extends the on- ventional de(cid:28)nition in [24℄: V De(cid:28)nition 2.17. Givena oalgebraoveraunitaloperad withadjointstru ture- map a :C →Hom (V(X),CX) X X c∈C a (c)=f (cX) n>0 cX ∈CX X X anelement is alledgroup-like if forall . Here n R n X is the -fold -tensor produ t, where is the ardinality of , f =Hom (ǫ ,1):Hom ( ,CX)=CX →Hom (V(X),CX) X R X R 1 X ǫ :V(X) → V(∅) = = R n X and 1 is the augmentation (whi h is -fold omposition V(∅) with ). C V A oalgebra over an operad is alled pointed if it has a unique group-like 1 element (denoted ), and pointed irredu ible if the interse tion of any two sub- oalgebras ontains this unique group-likeelement. V C Remark. Notethatagroup-likeelementgeneratesasub - oalgebraof andmust 0 lie in dimension . Althoughthisde(cid:28)nitionseems ontrived,itarisesin(cid:16)nature(cid:17): The hain- omplex of a pointed, simply- onne tedredu ed simpli ial set is naturallya pointed irredu- S = {C(K(S ,1))} n ible oalgebra over the Barratt-E les operad, (see [21℄). In this ase, the operad a tion en odes the hain-level e(cid:27)e t of Steenrod operations. D V Proposition 2.18. Let be a pointed, irredu ible oalgebra over an operad . Then the augmentation map ε:D →R is naturally split and any morphism of pointed, irredu ible oalgebras f:D →D 1 2 is of the form 1⊕f¯:D =R⊕kerε →D =R⊕kerε 1 D1 2 D2 ε :D →R i=1,2 i i where , are the augmentations. R·1⊆D i Proof. Thede(cid:28)nition(2.17)ofthesub- oalgebra isstatedinaninvariant way, so that any oalgebra morphism must preserve it. Any morphism must also 0th preserve augmefntations be kaeursεe the akuegrmεentation is the -order stru ture-map(cid:3). Consequently, must map D1to D2. The on lusion follows. V S 0 De(cid:28)nition 2.19. We denote the ategory of nearly free oalgebrasover by . V V C If isunital,every - oalgebra, , omesequippedwitha anoni alaugmentation ε:C →R R V so the terminal obje t is . If is not unital, the terminal obje t in this ategory 0 is , the null oalgebra. V I 0 The ategory of nearly free pointed irredu ible oalgebras over is denoted V (cid:22) this is only de(cid:28)ned if is unital. Its terminal obje t is the oalgebra whose R 0 underlying hain omplex is on entrated in dimension . 8 HOMOTOPY THEORY OF COALGEBRAS JUSTIN R. SMITH We also need: A ∈ C = I S ⌈A⌉ 0 0 De(cid:28)nitionC2h.2(0R.)If or , then denotes the underlying hain- omplex in of kerA→t t C ⌈∗⌉ where denotes the terminCal obCjeh (tRin) (cid:22) see de(cid:28)nition 2.19. We will all the forgetful fun tor from to . We will use the on ept of ofree oalgebra ogenerated by a hain omplex: C ∈Ch(R) V V G De(cid:28)nition 2.21. Let and let be an operad. Then a - oalgebra C will be alled the ofree oalgebra ogenerated by if ε:G→C (1) there exists a morphism of DG-modules V D f:D → C (2) given any - oalgebra and anyVmorphism offˆ:DDG→-mGodules , there existsaunique morphismof - oalgebras, , that makesthe diagram D fˆ // G @ @ @ @ @@ ε f @ (cid:15)(cid:15) C ommute. This universal property of ofree oalgebras implies that they are unique up to isomorphism if they exist. The paper [22℄ gives a onstru tive proof of their exist- en e in great generality (under the unne essary assumption that hain- omplexes R LVC are -free). In parti ular, this paper de(cid:28)nes ofree oalgebras and pointed PVC C irredu ible ofree oalgebras ogeneratedby a hain- omplex . R WeRwill denote the loseCdhsy(Rm)metri monoidal ategory of - hain- omplexes with -tensor produ ts by . These hain- omplexes are allowed to extend R into arbitrarily many negative dimensions and have underlying graded -modules that are • R arbitrary if is a (cid:28)eld (but they will be free) • R=Z nearly free, in the sense of de(cid:28)nition 2.1, if . 3. The general ase G We re allthe on eptof a model stru ture on a ategory . Thisinvolvesde(cid:28)n- ing spe ialized lasses of morphisms alled o(cid:28)brations, (cid:28)brations, and weak equi- valen es (see [19℄ and [11℄). The ategory and these lasses of morphisms must satisfy the onditions: G CM 1: is losed under all (cid:28)nite limits and olimits G CM 2: Suppose the following diagram ommutes in : X@ g //??Y @ ~ @ ~ @ ~ @ ~ @ ~ h @ ~~ f Z f,g,h If any two of are weak equivalen es, so is the third. f g g CM 3: If isaretra tof and isaweakequivalen e,(cid:28)bration,or o(cid:28)bra- f tion, then so is . 9 HOMOTOPY THEORY OF COALGEBRAS JUSTIN R. SMITH CM 4: Suppose that we are given a ommutative solid arrowdiagram U >>//A i f (cid:15)(cid:15) (cid:15)(cid:15) W //B i p where is a o(cid:28)bration and is a (cid:28)bration. Then the dotted arrow exists i p making the diagram ommute if either or is a weak equivalen e. f:X →Y G CM 5: Any morphism in may be fa tored: f =p◦i p i (1) , where is a (cid:28)bration and is a trivial o(cid:28)bration f =q◦j q j (2) , where is a trivial (cid:28)bration and is a o(cid:28)bration We also assume that these fa torizations are fun torial (cid:22) see [9℄. X •→X De(cid:28)nition 3.1. An obje t, , forwhi hthe map isa o(cid:28)bration,is alled Y Y →• o(cid:28)brant. An obje t, , for whi h the map is a (cid:28)bration, is alled (cid:28)brant. Ch(R) Example 3.2. The ategory, , of unbounded hain omplexes overthe ring R has a model stru ture in whi h: (1) Weak equivalen es are hain-homotopy equivalen es (2) Fibrations are surje tions of hain- omplexes that are split (as maps of R graded -modules). (3) Co(cid:28)brations are inje tions of hain- omplexes that are split (as maps of R graded -modules). Remark. All hain omplexes are (cid:28)brant and o(cid:28)brant in this model. Thisistheabsolutemodelstru turede(cid:28)nedbyChristensenandHoveyin[8℄,and Cole in [6℄. In this model stru ture, all unbounded hain- omplexes are o(cid:28)brant and a quasi-isomorphismmay fail to be a weak equivalen e. R R = Z Remark 3.3. We must allow non- -free hain omplexes (when ) be ause PV(∗) LV(∗) the underlying hain omplexes of the ofree oalgebras and are not R R R=Z knowntobe -free. They ertainlyareif isa(cid:28)eld,butif theirunderlying Zℵ0 Z abelian groups are subgroups of the Baer-Spe ker group, , whi h is -torsion free but well-known not to be a free abelian group (see [23℄, [3℄ or the survey [7℄). Proposition3.4. The forgetfulfun tor(de(cid:28)nedinde(cid:28)nition2.20)and ofree oal- gebra fun tors de(cid:28)ne adjoint pairs PV(∗):Ch(R) ⇆ I0:⌈∗⌉ LV(∗):Ch(R) ⇆ S0:⌈∗⌉ Remark. The adjointness of the fun tors follows from the universal property of ofree oalgebras(cid:22) see [22℄. V Condition 3.5. Throughout the rest of this paper, we assume that is an operad equipped with a morphism of operads δ:V→V⊗CoEnd(I;{R·p ,R·p }) 0 1 (see de(cid:28)nition 2.14) that makes the diagram VRRRRRRδRRRRRR//RVRRRR⊗RRRRRCRRoRRERRRRnRRdRRR(RIRR;{(cid:15)(cid:15)R·p0,R·p1}) V 10