YASUAKI OGAWA
Abstract. A model structure was originally introduced by Quillen as an abstraction of the category of topological spaces. However, it also plays an important role in the representation of algebras in connection with Grothendieck-Verdier’s derived category.
Afterwards, Hovey proved a one-to-one correspondence theorem between model struc- tures and cotorsion pairs in abelian categories. ˇSˇtov´ıˇcek provided an exact version of Hovey’s theorem which includes model structures whose homotopy categories are derived categories. In this talk, we focus an extriangulated version of Hovey’s theorem which was proved by Nakaoka-Palu. Finally, we will explain the theory behind Nakaoka-Palu’s approach, which serves a localization theory of extriangulated categories.
Contents
1. Introduction 1
2. Preliminary 3
2.1. Model categories 3
2.2. Extriangulated category 6
2.3. Cotorsion pairs 8
3. Model structures and Cotorsion pairs 9
3.1. Hovey’s correspondence theorem 9
3.2. Nakaoka-Palu’s correspondence theorem 11
4. Derived categories and model structures 13
4.1. Injective model structures 14
4.2. Lifting cotorsion pairs to categories of complexes 16
4.3. A tilting Quillen equivalence 18
5. Localization of extriangulated categories 18
5.1. Multiplicative systems compatible with the extriangulation 18
5.2. Localization via thick subcategories 19
5.3. The homotopy categories via extriangulated localizations 20
Acknowledgements 21
References 21
1. Introduction
The Gabriel-Zisman localization serves a foundation where one would like to consider certain morphisms to be isomorphisms [GZ67]. It is defined by formally inverting such morphisms, e.g., weak equivalences andquasi-isomorphisms. But in this case, morphisms in the obtained quotient category are rather difficult to handle. Especially, given an
Key words and phrases. cotorsion pair, model structure, exact category, triangulated category, extrian- gulated category.
2020Mathematics Subject Classification. Primary 18E35; Secondary 18G80, 18E10.
1
additive categoryC, the quotient categoryCrW´1swith respect to a classW of morphisms is not necessarily additive. In this note, we focus on the two different types of localization which allow us to get a better description of morphisms.
(1) Model structure
——Model structures were introduced by Quillen as an abstraction of the category of topological spaces [Qui67, Qui69]. It forms the foundation of homotopy theory.
Thehomotopy category HopCq of a model categoryCis the quotient category with respect to the class of weak equivalences.
(2) Localization ofExtriangulated category
——Grothendieck-Verdier introduced the derived category as the foundation of the homological algebra, which is defined to be a Verdier quotient of a triangu- lated category [Ver67]. Nakaoka-Palu’s extriangulated category is a unification of triangulated categories and exact categories [NP19].
We shall present some relations between (1) and (2). We are particularly interested in the admissible model strucutre which belongs to both (1) and (2). Basic examples of admissible model structures can be found on the module category of a Frobenius algebra and the category of (chain) complexes of modules.
(1) The module category ModA over a Frobenius algebra A admits an admissible model structure. Then, the homotopy category is equivalent to the (projectively) stable categoryModA.
(2) The category CpAq of complexes over an algebra A admits an admissible model structure. Then, the homotopy category is equivalent to the derived category DpAq.
These examples show that the model structure will play important roles in the representa- tion theory of algebras. A central concept to define and understand the admissible model structure is the cotortion pair, which were firstly introduced in [Sal79] to devide a given abelian category in two smaller pieces. A cotorsion pair on an abelian categoryCis defined to be a pair pU,Vq of subcategories with some conditions involving Ext1CpU,Vq “0. Af- terwards, a triangulated analog was considered in [IY08] and the unification of them was introduced by Nakaoka-Palu in terms of extriangulated categories. It was firstly proved by Hovey that, in the level of abelian categories, admissible model structures bijectively cor- respond to special cotorsion pairs. The first aim of this note is to prove an extriangulated version of Hovey’s correspondence theorem stated as below.
Theorem 1.1. [NP19, Section 5] Let C be an extriangulated category satisfying (WIC).
There is a one-to-one correspondence between admissible model structures on C and twin cotorision pairs ppS,Tq,pU,Vqq satisfying ConepV,Sq “CoConepV,Sq.
In this case, the canonical functorC ÑHopCqinduces an equivalence T XU rSXVs
ÝÑ„ HopCq.
Furthermore, HopCq is naturally equipped with a triangulated structure.
The second aim is to explain the triangulated structure on the homotopy category HopCq in a more conceptual framework. The extriangulated localization was formulated in [NOS21], which is a simultaneous generalization of the Verdier localization and the Serre localization. With this view in mind, the associated functor Q:C ÑHopCq can be regarded as an extriangulated localization ofCwith respect to the biresolving subcategory ConepV,Sq “CoConepV,Sq.
This note is organized as follows: Section 2 will be devoted to prepare basic results on model categories, extriangulated categories, and cotorsion pairs. Section 3 proves
Theorem 1.1. In Section 4, as an example of admissible model structures, we investigate the construction of the derived categories in terms of cotorsion pairs. Finally, in Section 5, we formulate the extriangulated localization which contains the Verdier localization and Serre localization. This new framework provides a good understanding of the homotopy category with respect to an admissible model structure.
Convention 1.2. Throughout, all categories and functors are assumed to be additive. For a category C, we denote by HomCpX, Yq the morphism space from X to Y. This is also denoted by CpX, Yq for short. All subcategories are always full. For subcategories U and V of C, HomCpU,Vq “ 0 means HomCpU, Vq “ 0 for any U P U and V P V. If C is an extriangulated category, the notion EpU,Vq is used in a similar meaning. We denote by UK the full subcategory of objectsX withEpU, Xq “0. The notionKU is used in the dual meaning. MorpCqis the category (or the class) of morphisms inC. All algebra is assumed to be finite dimensional over a fixed field k.
2. Preliminary
This section is devoted to recall basic facts on model categories, extriangulated cate- gories and cotorsion pairs.
2.1. Model categories. Amodel category was introduced by Quillen [Qui67], and it has been modified by some authors from various viewpoints, e.g. [Qui69, DS95, BR07]. A model category is usually assumed to have all finite limits and colimits, and more often to have all limits and colimits. However, to include model structures on triangulated and exact categories, we need to drop such assumptions.
Definition 2.1. Let C be an additive category. A model structure on C is a triple pCof,W,Fibqof classes of morphisms inC satisfying the following axioms. The morphisms in Fib (resp. Cof,W) is called the fibrations (resp. cofibrations, weak equivalences). We denote by wFib“FibXW the class of trivial fibrations and by wCof “CofXW the class of trivial cofibrations.
(M1) [Two out of three axiom] For a composed morphism gf in C, two of f, g and gf are weak equivalences, then so is the third.
(M2) [Retract axiom] For a morphism f : X Ñ Y in C together with the following commutative diagram, f PW implies gPW.
X1
id &&
//
g
X
f ////X1
g
Y1
id
99 //Y ////Y1
(M3) [Lifting axiom] For a commutative diagram inC of the following shape X
i
f //X1
p
Y g //Y1
withiPCofandpPFib, ifiorpbelongs toW, there exists a morphismh:Y ÑX1 such thath˝i“f and p˝h“g.
(M4) [Factorization axiom] MorpCq “ wFib˝Cof “ Fib˝wCof holds. Precisely, any morphism f PMorpCq admits factorizationsf “ f2f1 “f21f11 with f1 P Cof, f2 P wFib, f11 PwCof and f21 PFib
We call an additive category equipped with a model structure amodel category. Further- more, we sometimes refer the following stronger version of (M4).
(M4’) [Functorial factorization axiom] Afunctorial factorization is a pair pα, βq of func- tors MorpCq ÑMorpCq such thatf “βpfq ˝αpfq for allf PMorpCq. There is two functorial factorizationspα, βqandpγ, δq such thatαpfq PCof, βpfq PwFib, γpfq P wCof and δpfq PFibfor anyf PMorpCq.
For a commutative diagram
X
i
f //X1
p
Y g //Y1
a morphism h:Y ÑX1 which satisfies h˝i“f and p˝h“gis called alifting. If such a lifting exists, iis said to have the left lifting property against p and p is said to have the right lifting property againsti.
Lemma 2.2. (1) A morphism is a fibration iff it has the right lifting property against all trivial cofibrations.
(2) A morphism is a cofibration iff it has the left lifting property against all trivial fibrations.
(3) A morphism is a trivial fibration iff it has the right lifting property against all cofibrations.
(4) A morphism is a trivial cofibration iff it has the left lifting property against all fibrations.
(5) W “wFib˝wCof.
Although (co)limits do not necessarily exist, the existence of initial and terminal objects allow us to define cofibrant and fibrant objects.
Definition 2.3. Let pCof,W,Fibq be a model structure. We associate the full subcate- gories:
‚ Ctcof :“ tS PC|0ÑS is a trivial cofibrationu.
‚ Cfib:“ tT PC|T Ñ0 is a fibrationu.
‚ Ccof :“ tU PC|0ÑU is a cofibrationu.
‚ Ctfib:“ tV PC|V Ñ0 is a trivial fibrationu.
The objects in Ctcof (resp. Cfib,Ctcof,Cfib) are called trivially cofibrant (resp. fibrant, cofi- brant, trivially fibrant) objects. In addition, we put Ccf :“CcofXCfib. Note that, by the retract axiom (M2), it follows that the subcategories Ctcof,Cfib,Ccof and Ctfib are closed under direct summands.
The following lemma shows that a model structure provides nice approximations.
Definition 2.4. LetC be a category and D a full subcategory of C.
(1) For an object X P C, a morphism f : DX Ñ X from DX P D is called a right D-approximation of X if it induces a surjective morphism pD, fq. Dually, a left D-approximation is defined.
(2) If any objectX PC admits a rightD-approximation,Dis called acontravariantly finite subcategory of C. A covariantly finite subcategory is defined dually.
Lemma 2.5. For any object X:
(1) there exists a right Ctcof-approximationSX ÝπÑ1 X which is a fibration;
(2) there exists a left Cfib-approximation XÝÑι TX which is a trivial cofibration;
(3) there exists a right Ccof-approximation UX π
ÝÑX which is a trivial fibration;
(4) there exists a left Ctfib-approximation XÝÑι1 VX which is a cofibration.
A morphism ι (resp. ι1) is called a (resp. trivially) fibrant replacement of X and a morphism π (resp.π1) is called a (resp. trivially) cofibrant replacementof X.
Proof. We only prove (1). The others are similar. For an objectX, we consider a zero map 0 Ñ X and resolve it as 0 ÝÑi SX ÝÑp X by the factorization system MorpCq “Fib˝wCof.
Sincei: 0ÑSX is a trivial cofibration, by definition, we have thatSX is trivially cofibrant.
It remains to show that p:S Ñ X is a right SX-approximation. To this end, consider a morphism to S1 ÝÑf X withS1 PS and a commutative square
0 //
SX
p
S1 f //X
Since 0ÑS1is a trivially cofibration andpis a fibration, we have a liftingh:S1 ÑSX. □ Remark 2.6. The factorization axiom (M4) is not required to be functorial, so an assign- ment X ÞÑSX in Lemma 2.5 does not give rise to a functor. Similar assertions hold for other assignments in the lemma.
Definition 2.7. LetC be a model category. Thehomotopy category HopCq is defined to be the Gabriel-Zisman localization of C with respect to the classW of weak equivalences.
To provide an alternative description of the homotopy category HopCq, we recall the homotopy relations.
Definition 2.8. Letf, g:XÑY are maps in a model category C.
(1) A cylinder object for X is a factorization of the summation map p1X 1Xq : Xś
XÑ X into a cofibration Xś
X ÝÝÝÝÑpi0 i1q X1 followed by a weak equivalence X1 ÑX.
(2) We sayf and g are left homotopic if there is some cylinder object X1 and a map H :X1 Ñ Y such that Hi0 “f and Hi1 “ g. This defines a relation called left hmotopy, denoted „l, on HomCpX, Yq.
The path object and theright homotopy „r are defined dually.
In fact, if X is cofibrant and Y is fibrant, then „l“„r. In this case, we simply call it homotopy and denote by „. Then, by putting πHomCpX, Yq “ HomCpX, Yq{ „, we obtain the natural functor π:Ccf ÑCcf{ „ which is an identity on objects.
Theorem 2.9. The composition Ccf ãÑ C ÝÑQ HopCq induces an equivalence pCcf{ „q Ý„Ñ HopCq.
Example 2.10. LetAbe a finite dimensional self-injectivek-algebra1. Then the category modA of finite dimensional (right) A-modules admits a model strucutre pCof,W,Fibq:
‚ Cof“the set of all monomorphisms;
1An algebraAis said to beself-injective ifprojA“injA. A Frobenius algebra is self-injective.
‚ Fib“the set of all epimorphisms;
‚ W “ tf |f factors through an object P PprojAu.
In this case, the full subcategoryCcf of cofibrant and fibrant objects is the whole category modA. The homotopy relation on Ccf defines the (two-sided) ideal rprojAs consisting of all morphisms having a factorization through an object in projA. Hence, the homotopy category is equivalent to the stable category modA“modA{rprojAs.
We end this subsection by recalling the definition of the Quillen functors.
Proposition 2.11. Let C andDbe model categories and consider an adjoint pairpF, Gq: C ÑD, i.e.,F :CÑDis a left adjoint toG:DÑC. Then, the following are equivalent.
(1) F preserves fibrations and trivial fibrations.
(2) G preserves cofibrations and trivial cofibrations.
Under the above equivalent conditions, we call F a left Quillen functor and G a right Quillen functor. Moreover, the pairpF, Gq is called a Quillen adjunction.
2.2. Extriangulated category. Anextriangulated categorieswas introduced by Nakaoka- Palu [NP19] as a simultaneous generalization of triangulated categories and exact cate- gories. It is defined to be a triplepC,E,sq of
‚ an additive categoryC;
‚ an additive bifunctorE:CopˆCÑAb, whereAbis the category of abelian groups;
‚ a correspondence s which associates each equivalence class of a sequence of the form XÑY ÑZ inC to an element in EpZ, Xq for anyZ, X PC,
which satisfies some ‘additivity’ and ‘compatibility’. It is simply denoted by C if there is no confusion. We refer to [NP19, Section 2] for its detailed definition. We only recall a part of the ‘compatibility’ analogous to the octahedral axiom.
(ET4) Letδ PEpD, Aq andδ1 PEpF, Bq beE-extensions realized by spδq “ rAÝÑf B f
1
ÝÑDs, spδ1q “ rB ÝÑg C g
1
ÝÑFs.
Then there exist an object EPC, a commutative diagram A f //B f
1 //
g
D
d
A h //C
g1
h1 //E
e
F F
inC, and an E-extensionδ2 PEpE, Aq realized byAÝÑh CÝhÑ1 E, which satisfy the following compatibilities:
(1) DÝÑd E ÝÑe F realizesf˚1δ1; (2) d˚δ2 “δ;
(3) f˚δ2“e˚δ1.
where we putf˚1δ1 :“EpF, f1qpδ1q, d˚δ2 :“Epd, Aqpδ2q and so on.
Remark 2.12. A triangulated category and an exact category are special cases of an extriangulated category.
(1) By putting E :“ Cp´,´r1sq, a triangulated category pC,r1s,∆q can be regarded as an extriangulated category [NP19, Prop. 3.22]. In this case, we say that an extriangulated category corresponds to a triangulated category.
(2) By puttingE:“Ext1Cp´,´q, an exact categoryCcan be regarded as an extriangu- lated category [NP19, Example 2.13]. In this case, we say that an extriangulated category corresponds to an exact category.
We shall use the following terminology in many places.
Definition 2.13. Let pC,E,sq be an extriangulated category.
(1) We call an element δ P EpZ, Xq an E-extension, for any X, Z P C; A sequence XÝÑf Y ÝÑg Zcorresponding to anE-extensionδ PEpZ, Xqis called ans-conflation.
In addition,f andgare called an s-inflation and ans-deflation, respectively. The pair xX ÝÑf Y ÝÑg Z, δyis called an s-triangle.
(2) An objectP PC is said to beprojective if for any deflationg:Y ÑZ, the induced morphism CpP, gq : CpP, Yq Ñ CpP, Zq is surjective. We denote by projpCq the subcategory of projectives inC. Aninjective object andinjpCq are defined dually.
(3) We say that C has enough projectives if for any Z P C, there exists a conflation XÑP ÑZ with P projective. Having enough injectives are defined dually.
(4) Let C have enough projectives and injectives. If the class of projectives coincides with the class of injectives,C is said to be Frobenius.
Remark 2.14. (1) If an extriangulated categoryC corresponds to a triangulated cat- egory, thenChas enough projectives and injectives. The projectives and inejctives are nothing but the zero objects.
(2) If an extriangulated categoryC corresponds to an exact category, projectives and injectives agree with usual definitions.
Keeping in mind the triangulated case, we introduce the notions cone and cocone.
Proposition 2.15. Let C be an extriangulated category. For an inflation f P CpX, Yq, take a conflation X ÝÑf Y ÝÑg Z, and denote this Z by Conepfq. We call Conepfq a cone of f. Then, Conepfq is uniquely determined up to isomorphism. The dual notion cocone CoConepgq exists.
Furthermore, for any subcategoriesU andVofC, we define a full subcategoryConepV,Uq to be the one consisting of objectsXappearing in a conflationV ÑU ÑXwithU PU and V PV. A subcategoryCoConepV,Uq is defined dually. Next, we recall the notion of weak pullback in extriangulated categories. Consider a conflation X ÝÑf Y ÝÑg Z corresponding toE-extensionδ PEpZ, Xqand a morphismz:Z1 ÑZ. Putδ1 :“Epz, Xqpδqand consider a corresponding conflation X Ñ E Ñ Z1. Then, there exists a commutative diagram of the following shape
X //E
(Pb) g1 //
z1
Z1
z
X f //Y g //Z
The commutative square (Pb) is called aweak pullback ofgalongzwhich is a generalization of the pullback in exact categories and the homotopy pullback in triangulated categories.
An induced sequenceE p´gz11q
ÝÝÝÑZ1‘Y ÝÝÝÑpz gq Z is a coflation. The dual notionweak pushout exists and it will be denoted by (Po).
We often refer the following condition, analogous to the weak idempotent completeness ([Buh, Prop. 7.6])
Condition 2.16 (WIC). For an extriangulated category C, we consider the following conditions.
(1) Letg˝f be a composed morphism in C. If g˝f is an inflation, then so isf. (2) Letg˝f be a composed morphism in C. If g˝f is an deflation, then so is g.
Remark 2.17. (1) If C corresponds to a triangulated category, (WIC) is automati- cally satisfied.
(2) IfC corresponds to an abelian category, (WIC) is automatically satisfied.
(3) If C corresponds to an exact category, (WIC) is equivalent to the usual condition so thatC is weakly idempotent complete.
We end this subsection by mentioning that the class of extriangulated categories is closed under certain operations.
Proposition 2.18. [NP19, Rem. 2.18, Prop. 3.30] Let C be an extriangulated category.
(1) Any extension-closed subcategory admits an extriangulated structure induced from that of C.
(2) Let I be a full additive subcategory, closed under isomorphisms which satisfies I ĎprojpCq XinjpCq, then the ideal quotientC{rIs has an extriangulated structure, induced from that of C.
2.3. Cotorsion pairs. The aim of this subsection is briefly recalling the definition of a cotorsion pair and explaining how it relates to the theory of derived category in the representation of algebras.
Definition 2.19. LetCbe an extriangulated category. A (complete)cotorsion pair pU,Vq is a pair of full subcategories on C which is closed under direct summands and satisfies that EpU,Vq “0 and, for anyXPC, there exist conflations
XÑVX ÑUX, VX ÑUX ÑX
with UX, UX P S, VX, VX P V. The above defining sequences are called approximation sequences.
Example 2.20. (1) Let C be an exact category. Then, pprojC,Cq forms a cotorsion pair if and only ifC has enough projectives.
(2) Denote by modA the category of finite dimensional modules over a Gorenstein algebra A, then pCMA,Pă8q is a cotorsion pairs of modA. Here, CMA is the subcategory of Cohen-Macaulay modules, namely,
CMA:“ tXPmodA|ExtiApX, Aq “0, ią0u,
and Pă8 is the subcategory of modules of finite projective dimension.
In the representation theory of algebras, cotorsion pairs play a central role in connection with derived categories, since, if a special cotorsion pair exists inmodA, we obtain a new algebra B derived equivalent to A. In the rest of this subsection, the symbol A and B always denote finite dimensional k-algebra over a fixed field k. Let us begin with some preparations.
For a subcategory C of modA we denote by Cqthe subcategory of modA consisting of objects X for which there is an exact sequence 0 Ñ X Ñ C0 Ñ C1 Ñ ¨ ¨ ¨ Ñ Cn Ñ 0, withCiPC. Recall that T PmodAisbasic ifT admits an indecomposable decomposition T “śn
i“1Ti withTi fiTj fori‰j.
Definition 2.21. A finite dimensional A-moduleT PmodA is called a tilting A-module, if it satisfies the following conditions.
(1) pdT ă 8, wherepdT is the projective dimension ofT. (2) ExtiCpT, Tq “0 forią0.
(3) There is an exact sequence 0ÑAÑT0 Ñ ¨ ¨ ¨ ÑTnÑ0 with Ti PaddT.
Theorem 2.22. [Hap88, Ch. III, Thm. 2.10] Let T PmodA be a tilting A-module and put B :“EndApTq. Then, the functor HomApT,´q:ModAÑModB induces a triangule equivalence DpAqÝÑ„ DpBq.
Thus, tilting modules are basic and powerful tools to construct derived equivalent alge- bras. The following theorem shows that tilting modules bijectively correspond to a class of cotorsion pairs.
Definition 2.23. A subcategoryU of an exact categoryCis called aresolving subcategory if, for any conflation 0ÑXÑY ÑZ Ñ0,Y, ZPU impliesX PU.
Lemma 2.24. For a cotorsion pair pU,Vq, the following are equivalent.
(i) U is resolving.
(ii) Ext2CpU, Vq “0 for any U PU and V PV. (iii) ExtiCpU, Vq “0 for any U PU, V PV and ią0.
Under the above equivalent conditions, we call pU,Vq a resolving cotorsion pair2.
Theorem 2.25. [AR91, Thm. 5.5] There is one-to-one correspondence between basic tilting modules T and resolving cotorsion pairs pU,Vq with Vq “ modA, given by T ÞÑ p}addT,p}addTqKq and pU,Vq ÞÑ direct sum of the indecomposable modules in U XV.
A trivial example of tilting A-module is A itself. The corresponding cotorsion pair is pprojA,modAq on modA.
Krause-Solberg showed that a cotorsion pair pU,Vq on modA can give rise to that on ModA.
Theorem 2.26. [KS03, Thm. 2.4] Let pU,Vq be a resolving cotorsion pair on modA.
Then plim ÝÑU,lim
ÝÑVq is a resolving cotorsion pair on ModA. Here, lim
ÝÑU denotes the full subcategory of all A-modules which are filtered colimits of modules in U.
3. Model structures and Cotorsion pairs
3.1. Hovey’s correspondence theorem. Let us start with brief observations on an abelian categoryC equipped with a model structurepCof,W,Fibq. Following [BR07, VII], we firstly sharpen Lemma 2.5 and show that the cofibrant/fibrant replacements are closely related to approximation sequences obtained from cotorsion pairs.
Lemma 3.1. [BR07, VII.2.1] For any object X PC: (1) there exists an exact sequence0ÑTX
ÝÑf SX π1
ÝÑX whereπ1 is a trivially cofibrant replacement and TX PCfib;
(2) there exists an exact sequenceX ÝÑι TX ÑSX Ñ0whereιis a fibrant replacement and SX PCtcof;
(3) there exists an exact sequence0ÑVX ÑUX π
ÝÑX where π is a cofibrant replace- ment and VX PCtfib;
2This is also called ahereditarycotorsion pair in the literature.
(4) there exists an exact sequence X ÝÑι1 VX Ñ UX Ñ0 where ι1 is a trivially fibrant replacement and UX PCcof.
Proof. We shall only show (1). Due to Lemma 2.5, it remains to show TX PCfib. To this end, we consider a trivial cofibration g:AÑB together with the following commutative squares.
A
g //TX f //
SX
π1
B //0 //X
Since π1 is a fibration, we have a lifting h : B Ñ SX for the whole square consisting of A, B, SX and X. The lifting h factors throughTX, say h “f ˝f1. Since f is monic, the morphism f1 should be a lifting for the left square. Hence TX Ñ0 is a fibration. □ Definition 3.2. Let C be an abelian (more generally, extriangulated) category with a model structure. For a given morphism f, we consider the following four conditions:
(1) f is an inflation with cone in Ctcof ifff PwCof.
(2) f is deflation with cocone inCfib ifff PFib;
(3) f is an inflation with cone in Ccof iff f PCof;
(4) f is a deflation with cocone in Ctfib ifff PwFib;
The model structure ofCis said to beright admissible (resp. left admissible ) if the above (3)(4) (resp. (1)(2)) are satisfied. A left and right admissible model structure is said to be admissible.
The following shows that admissible model structure yields a pair of cotorsion pairs.
Definition 3.3. LetpS,TqandpU,Vqbe cotorsion pairs onC. We call the pairppS,Tq,pU,Vqq of them a twin cotorsion pair if it satisfiesS ĎU (or equivalentlyT ĚV).
Proposition 3.4. [BR07, VII.3.4] Let C be an abelian category with a model structure pCof,W,Fibq.
(1) If the model structure is left admissible, then pCtcof,Cfibq forms a cotorsion pair.
(2) If the model structure is right admissible, then pCcof,Ctfibq forms a cotorsion pair.
In particular, if the model structure is admissible, we have a twin cotorsion pairppCtcof,Cfibq,pCcof,Ctfibqq.
Proof. We shall prove only part (2) since the proof of (1) is dual. Due to the assumption and Lemma 3.1, we have only to check Ext1CpU, Vq “0 forU PCcof andV PCtfib. Consider a conflation V ÑX ÝÑp U and a commutative square
0 //
X
p
U U
Since 0 ÑU is a cofibration and p is a trivial fibration, we have a lifting, which forces p
splitting. □
It is natural to ask when a given twin cotorsion pairs induces a model structure3. The answer was essentially given by Hovey.
3Another natural question is when a single cotorsion pair corresponds to a model structure. An answer is given by Beligiannis-Reiten [BR07, VII. 4.2].
Definition 3.5. A twin cotorsion pairppS,Tq,pU,VqqonCis calledHovey twin cotorsion pair if it satisfies ConepV,Sq “CoConepV,Sq.
Theorem 3.6. [Hov02, Thm. 2.2] Let C be an abelian category. There exists one-to-one correspondence between admissible model structures on C and Hovey twin cotorsion pairs on C.
tpCof,W,Fibq: admissible model strucutres on Cu
Φ
tppS,Tq,pU,Vqq: Hovey twin cotorsion pairs on Cu
Ψ
OO
Proof. It is a special case of Theorem 1.1. □
Remark 3.7. Theorem 3.6 has been modified and generalized by some authors:
- Gillespie and ˇSˇtov´ıˇcek proved an exact version of the theorem [Gil11, Sto13];
- Yang proved a triangulated version of the theorem [Yan15].
- Nakaoka-Palu’ extriangulated version contains the above cases. [NP19].
In addition, we remark that the theorem is a bit modified formulation of the original one.
This type of formulations can be found in [NP19].
Example 3.8. The model structure in Example 2.10 is admissible and corresponds to a Hovey twin cotorsion pair ppprojA,modAq,pmodA,projAqq.
3.2. Nakaoka-Palu’s correspondence theorem. This subsection is devoted to prove Theorem 1.1, the extriangulated version of Hovey’s correspondence theorem.
3.2.1. From admissible model strucutre to Hovey twin cotorsion pair. Let C be an extri- angulated category with an admissible model structure pCof,W,Fibq and put
ppS,Tq,pU,Vqq:“ ppCtcof,Cfibq,pCcof,Ctfibqq.
Proposition 3.9. [NP19, Prop. 5.6] ppS,Tq,pU,Vqqis a twin cotorsion pair.
Proof. A similar discussion in the proof of Proposition 3.4 still works well. So we skip the
details. □
It directly follows from the next proposition thatppS,Tq,pU,Vqqis a Hovey twin cotor- sion pair.
Proposition 3.10. [NP19, Prop. 5.7] The following are equivalent for any object N PC. (i) N PConepV,Sq.
(ii) p0ÑNq PW.
(iii) pN Ñ0q PW. (iv) N PCoConepV,Sq.
Proof. (i) ñ (ii): Consider a conflationV ÑS Ñ N withV PV, S PS. Then, we have a factorization of 0 Ñ N as a trivial cofibration followed by a trivial fibration S Ñ N. Hence p0ÑNq PW.
(ii) ñ (i): Since p0 Ñ Nq P W and W “ wFib˝wCof, we have a factorization of p0ÑNq which shows the condition (i).
(iii) ô (iv): It is a dual of (i)ô (ii).
(ii) ô(iii): It follows from the 2-out-of-3 axiom on W. □
3.2.2. From Hovey twin cotorsion pair to admissible model strucutre. In the rest of this section, we fix an extriangulated category C satisfying (WIC) together with a Hovey twin cotorsion pair ppS,Tq,pU,Vqq.
To construct the corresponding model structure, we define the following classes of mor- phisms.
- wCof“ tf PMorC|f is an inflation with Conepfq PSu.
- Fib“ tf PMorC|f is a deflation with CoConepfq PTu.
- Cof“ tf PMorC|f is an inflation with Conepfq PUu.
- wFib“ tf PMorC|f is a deflation with CoConepfq PVu.
- W “wFib˝wCof.
Lemma 3.11. wCof,Fib,Cof and wFib are closed under composition.
Proof. Since the corresponding subcategories S,T,U and V are extension-closed, the as-
sertions directly follow from (ET4). □
Proposition 3.12. The lifting axiom (M3) is satisfied, that is, we have the following.
(1) wCof satisfies the left lifting property againstFib.
(2) wFib satisfies the right lifting property againstCof.
Proof. We only prove (1), because (2) is dual. We consider the following square with the columns forming conflations.
T
t
A a //
f
C
g
B
s
b //D S
Under the assumption S PS and T PT, we shall construct a lifting h :B Ñ C for the above square. Since EpS, Tq “0, by a basic property of extriangulated categories, there exists a mapc:B ÑC withb“g˝c. However, unlike the exact case, a“c˝f does not necessarily hold. So we put d:“a´c˝f. Sinceg˝d“0, we have a mapc1 :AÑT with d“t˝c1. Again, due toEpS, Tq “0, we get c2 :B Ñ T with c2˝f “c1. Then the map
h:“c`t˝c2:BÑC is a desired lifting. □
The next proposition shows that the factorization axiom (M4) is satisfied.
Proposition 3.13. MorpCq “wFib˝Cof“Fib˝wCof.
Proof. We only show MorpCq “wFib˝Cof. Letf PCpA, Bq a map and resolve A to get an approximation sequence AÝÑιA VAÑ UA withUA PU and VA PV. A weak pushout of ιA along f yields a conflation A f
1
ÝÑ B‘VAÑ C where we put f1 :“`f
ιA
˘. ResolveC by pU,Vq and obtain the following commutative diagram made of conflations.
VC
VC
A i //M
(Pb)
//
p
UC
A f
1//B‘VA //C
Since iPCof andpPwFib, a factorizationf “ p1 0q ˝p˝i, the morphismp˝ifollowed by the projection p1 0q PB‘VAÑB, is a desired one.
The remaining axioms (M1) and (M2) follow from Corollary 5.22 and Propsition 5.24 in [NP19], repectively. A triangulated structure is given in [NP19, Thm. 6.20]. Although the detailed proof is not included here, we shall present more conceptual proofs in Section
5 via the localization of extriangulated categories. □
By the argument so far, admissible model structures and Hovey twin cotorsion pairs correspond bijectively. Recall that, the homotopy category HopCq admits another de- scription Ccf{ „ in terms of homotopy relations which should be better understood. We summarize the essentials which shows homotopy relations can be interpreted in view of ideal quotients.
Proposition 3.14. [Gil11, Prop. 4.4]LetCbe an extriangulated category satisfying (WIC) with an admissible model structure.
(1) Two morphismsf, g:X ÑY in C are right homotopic if and only ifg´f factors through an object of Ctcof.
(2) Two morphisms f, g :X ÑY in C are left homotopic if and only if g´f factors through an object of Ctfib.
In particular, the composition Ccf ãÑ C ÝÑQ HopCq induces an equivalence rC Ccf
tcofXCtfibs
Ý„Ñ HopCq.
Proof. We will prove (2). The part (1) is dual. We first construct a cylinder object forX by using the corresponding cotorsion pair pU,Vq :“ pCcof,Ctfibq. Resolving X by pU,Vq, we get a conflation X ÝÑj VX Ñ UX with UX P U and VX P V. Now we consider the following factorization of the summation p1X 1Xq:Xś
X ÑX:
Xź
X ÝÑi Xź
VX pÝÑX withi“
´1X 1X
0 j
¯
andp“ p1X 0q. Since Kerp–VX is trivially fibrant, we havepPwFib.
To claim that iis a cofibration, we consider the equation ˆj 0
0 j
˙
“
ˆj ´1VX 0 1VX
˙ ˆ1X 1X
0 j
˙ . Since ´
j 0
0 j
¯
is an inflation, the property (WIC) guarantees i P Cof. Thus, Xś VX is a cylinder object for X. By definition, f „l g if and only if there is a map pα βq : Xś
VX Ñ Y such that the equation pα βqi “ pf gq holds. Since i “
´1X 1X
0 j
¯ , the equation is equivalent tof `βj “g. Thus, g´f factors throughVX PCtfib.
The remaining assertion directly follows from Theorem 2.9. □ 4. Derived categories and model structures
In this section, we investigate a model structure on a category CpAq of complexes of A-modules the homotopy category of which coincides with the derived category DpAq.
A classical example of such model structures is known as an injective model structure which was first constructed by Joyal and Beke [Bek00]. Based on an elegant approach in [SS11, Sto13], we shall show that such model structures can be obtained from resolving cotorsion pairs on ModA. Throughout the section, fix a finite dimensional k-algebra A and the following symbols are used in many places:
- CpAq - the category of complexes ofA-modules;
- CacpAq - the full subcategory of acyclic complexes ofCpAq;
- KpAq - the homotopy category of complexes ofA-modules;
- KacpAq- the full subcategory of acyclic complexes of KpAq;
- DpAq - the derived category ofModA.
Moreover, for an extension-closed subcategory U of ModA, letCpUq denote the category of complexes in U and CacpUq the full subcategory of acyclic complexes X P CpUq such that ZnpXq PU for all nPZ.
4.1. Injective model structures. The derived category is, by definition, the Gabriel- Zisman localization ofCpAqwith respect to the quasi-isomorphisms. Let us recall another construction of the derived category which fits into a well understood framework. The homotopy category KpAq is a triangulated category together with a thick subcategory KacpAq. The derived category is obtained as the Verdier localization ofKpAq with respect to KacpAq. In addition, we have KacpAq ÑKpAqÝQÑDpAq with additional property that, for anyX PKpAq, there exists a triangle
VX ÑUX ÑXÑVXr1s
with UX PKacpAq and VX PKacpAqK. It turns out that pKacpAq,KacpAqKq forms a cotor- sion pair on KpAq. Furthermore, it realizes the derived category as the full subcategory KacpAqK ofKpAq.
Lemma 4.1. The compositionKacpAqK ãÑKpAqÝQÑDpAq is a triangle equivalence.
Now, we are in position to state that one can realize the derived category as the homo- topy category with respect to an admissible model structure.
Proposition 4.2. There exists a Hovey twin cotorsion pair
`pCacpAq,CacpAqKq,pCpAq,CacpInjAqq˘
the homotopy category of which is the derived categoryDpAq. The corresponding admissible model structure is called the injective model structure. Furthermore, the cotorsion pairs are resolving.
Before proving that, we recall the following well-known fact.
Lemma 4.3. The following conditions are equivalent for XPCpAq.
(i) Ext1CpAqpE, Xq “0 for any acyclic complexE PCacpAq.
(ii) HomKpAqpE, Xq “0 for any acyclic complexE PCacpAq.
Under the above equivalent conditions, the object X in CpAq are called an injectively fibrant objects4 and those in KpAq are called K-injectives.
Proof of Proposition 4.2. It is obvious thatpCpAq,CacpInjAqqforms a cotorsion pair, since any acyclic complex in CacpInjAqis splitting.
Due the cotorsion pair pKacpAq,KacpAqKq, it is easy to check that pCacpAq,CacpAqKq is also a cotorsion pair. In fact, for any XPCpAq, there exists a triangleTX ÝÑi SX ÑX Ñ TXr1s with SX P KacpAq and TX PKacpAqK. Any triangle in KpAq comes from an exact
4Each termXiof an injectively fibrant objectX belongs toInjA.
sequence in CpAq. More precisely, by taking the pushout ofTX i
Ý
ÑSX along the injective hull TX ÑIpTXq, we get an exact sequence
0ÑTX ÑSX ‘IpTXqÝÑp X1 Ñ0
which corresponds to the triangle. We may assume that there exists a contractible complex I PCacpAq with an isomorphism X1–X‘I inCpAq. Taking the pullback ofp along the section XÑX1 yields the following commutative diagram made of conflations.
0
0
I
I
0 //TX //SX1
//
(Pb)
X //
0 0 //TX //SX ‘IpTXq p //
X1 //
0
0 0
The middle row is a desired sequence. Another approximation sequence can be obtained similarly.
Since CacpInjAq is the class of injectives inCpAq, it is easy to check the equiality CacpAq “ConepCacpInjAq,CacpAqq “CoConepCacpInjAq,CacpAqq.
Hence, the given pair forms a Hovey twin cotorsion pair.
It remains to show that W is the class of quasi-isomorphisms. Let f : X Ñ Y be a quasi-isomorphism and consider a factorization f : X ÝÑf1 Z ÝÑf2 Y with f1 surjective and f2 injective. Note that K :“ Kerf1,Cokf2 belongs to CacpAq and, in particular, f2 PwCof. Since CpAqhas enough injectives formingCacpInjAq, we get an exact sequence 0 Ñ K ÑI Ñ K1 Ñ 0 with I PCacpInjAq and K1 PCacpAq and construct the following commutative diagram with all rows and columns exact:
0
0
0 //K
(Po)
//
X f1 //
f11
Z //0 0 //I //
X1
f12
//
Z //0 K1
K1
0 0
Since f11 PwCof, f12PwFib, we thus concludef PW. Any weak equivalence is obviously a quasi-isomorphism. Therefore, the homotopy category is the derived category. □
The injective model structure is explicitly given as follows:
- W “the class of quasi-isomorphisms;
- Cof“the class of monomorphisms;
- Fib“the class of epimorphisms whose kernels are injectively fibrant.
Moreover, thanks to the description of the homotopy category in Theorem 1.1, we have an equivalence
HopCq “DpAq » CacpAqKXKCacpInjAq
rCacpAq XCacpInjAqs “KacpAqK.
Dually, the projective model structure exists. The corresponding Hovey twin cotosion pair is
`pCacpprojAq,CpAqq,pKCacpAq,CacpAqq˘ , and the homotopy category is also the derived category.
4.2. Lifting cotorsion pairs to categories of complexes. In this subsection, we shall show that a cotorsion pair pU,Vq on ModA gives rise to a model structure on CpAq the homotopy category is the derived category. Such model structures contain the injective and projective model structures. Recall that CacpAq is a thick subcategory of CpAq with an inherited exact structure.
Proposition 4.4. [Sto13, Prop. 7.7]A resolving cotorsion pair pU,Vq on ModA induces a resolving cotorsion pair pCacpUq,CacpVqqon the exact category CacpAq.
Proof. We denote for brievityCacpUq and CacpVq by Ur and Vr, respectively, and put N :“
CacpAq. We first show Ext1NpUr,Vq “r 0. Every U P Ur can be written as an extension of stalk complexes5 ZipUq which is depicted as follows.
0
0
0
¨ ¨ ¨ //Z0pUq
0 //Z1pUq
0 //Z2pUq
//¨ ¨ ¨
¨ ¨ ¨ //U0
B0 //U1
B1 //U2
//¨ ¨ ¨
¨ ¨ ¨ //Z1pUq
0 //Z2pUq
0 //Z3pUq
//¨ ¨ ¨
0 0 0
Therefore we have only to check Ext1CpAqpU1, Vq “0 for anyV PVr and any stalk complex U1 PCpUq concentrated in degreen. Any extension δ in Ext1CpAqpU1, Vq “0 is necessarily degreewise splitting:
¨ ¨ ¨ //Vn´1
Bn´1 //Vn p10q
Bn //Vn`1
//¨ ¨ ¨
¨ ¨ ¨ //Vn´1
α//Vn‘U1
p0 1q
β //Vn`1
//¨ ¨ ¨
¨ ¨ ¨ //0 0 //U1 0 //0 //¨ ¨ ¨ where α “`Bn´1
0
˘ and β “ pBn fq for a morphismf :U1 ÑVn`1. Since Bn`1˝f “0,f factors through Zn`1pVq. Due to Ext1ApU,Vq “0, f is factored as f :U1 f
1
ÝÑ ZnpVq B
n
ÝÑ
5A stalk complexX is a complex satisfyingXi“0 for anyi‰nfor somenPZ.
Vn`1. Thus we have a splitting exact sequence 0 Ñ U1 p´f11q
ÝÝÝÑ Vn‘U1 pf
1 1q
ÝÝÝÑ Vn Ñ 0 which shows that δ splits.
Clearly,UrandVrare closed under direct summands sinceU andV are. For anyX PN, we resolve the cocycleZnpXq PModAby the cotorsion pairpU,Vqto have approximation sequences 0 Ñ ZnpXq Ñ Vn Ñ Un Ñ 0 and 0 Ñ Vn1 Ñ Un1 Ñ ZnpXq Ñ 0. As an application of the Horseshoe lemma, we can construct desired approximation sequences forX.
It is obvious thatUris closed under epikernels. Hence we have proved thatpUr,Vrq forms
a resolving cotorsion pair CacpAq. □
The following model structures were first found in [SS11, Thm. 4.2] and streamlined in [Sto13, Thm. 7.16].
Theorem 4.5. Let pU,Vq be a resolving cotorsion pair on ModA. Then, it induces a Hovey twin cotorsion pair
ppCacpUq,CacpUqKq,pKCacpVq,CacpVqqq
the homotopy category of which is the derived category DpAq. Furthermore, the cotorsion pairs are resolving.
Proof. We shall show that pCacpUq,CacpUqKq is a cotorsion pair. Any X P CpAq admits an approximation sequence 0Ñ X1 ÑE ÝÑf1 X Ñ0 with E PCacpUq and X1 PCacpAqK. Resolve E by the cotorsion pair pCacpUq,CacpVqq to get an approximation sequence 0 Ñ Vr ÑUr ÝÑf2 E Ñ0 withUr PCacpUqandVr PCacpVq. The composed mapf :“f2˝f1 gives rise to the following commutative diagram of exact sequences.
0
0
Vr
Vr
0 //X2 //
Ur f //
f2
X //0 0 //X1 //E f1 //X //0
Since X2 PCacpAqK, the middle exact row provides a desired approximation sequence. A similar method yields another approximation sequence. Thus the pair forms a resolving cotorsion pair.
Dually we can show that the pair in the righthand side also forms a resolving cotorsion pair.
Proposition 4.4 shows CacpAq “ ConepCacpVq,CacpUqq “ CoConepCacpVq,CacpUqq. We have thus concluded that the given pair is a Hovey twin cotorsion pair.
A similar method given in the latter part of the proof of Proposition 4.2 shows that W
is the class of quasi-isomorphism. □
In the case of pU,Vq “ pModA,InjAq, the obtained model structures in Theorem 4.5 is nothing other than the injective model structures. Dually the projective model structures correspond to the cotorsion pair pProjA,ModAq.
Remark 4.6. (1) The admissible model structure given in Theorem 4.5, in practice, satis
