Equivalences between localisations of categories provided by replacements

Equivalences between localisations of categories provided by replacements

Sebastian Thomas October 10, 2018

Abstract

We give a characterisation of functors whose induced functor on the level of localisations is an equivalence and where the isomorphism inverse is induced by some kind of replacements such as projective resolutions or cofibrant replacements.

1 Introduction

To a categoryCtogether with a set of distinguished morphisms, called denominators inC, one might attach its (Gabriel/Zisman) localisation GZ(C), that is, the universal category where the denominators in C become in- vertible. Given categoriesC andDtogether with sets of denominators and a functorF:C → Dthat maps de- nominators inCto denominators inD, the universal property ofGZ(C)yields a functorGZ(F) :GZ(C)→GZ(D) such that the following quadrangle commutes, where loc denotes the canonical functor from a category with denominators to its localisation.

C D

GZ(C) GZ(D)

F

loc loc

GZ(F)

As every functor is an equivalence if and only if it is dense, full and faithful, this in particular holds for the induced functorGZ(F). In this article, we focus on functorsF with a particular property that ensures density ofGZ(F), and establish a characterisation forGZ(F)to be an equivalence.

An arbitrary morphism in the Gabriel/Zisman localisation may consist of arbitrarily but finitely many numer- ators and denominators: Every morphism of the form Y → Y⁰ in GZ(D) is represented by a diagram of the form

Y Y˜₁ ^≈ Y₁ . . . ^≈ Y_n−1 Y⁰

inD, where the “backward” arrows are denominators inD. In particular, density ofGZ(F)means that for every objectY in Dthere exists a diagram of the form

F X Y˜₁ ^≈ Y₁ . . . ^≈ Y_n−1 Y

in D. Typically, in this case the “forward” arrows are also denominators in D, but in general even that is not guaranteed.

To obtain a suitable criterion, we suppose that density of GZ(F) is provided by single denominators in D, so-calledS-replacements: For every object Y in Dthere is supposed to be an objectX in C and a denomina- torq:F X →Y inD.

F X ^≈q Y

This property will be called S-density in the following. IfF is S-dense andGZ(F)is an equivalence, we callF anS-equivalence. With this restriction, we obtain the following result.

Mathematics Subject Classification 2010: 18G10, 18E35, 18G55, 55U35.

Theorem (characterisation of S-equivalences, see (5.25)). We suppose that the denominators inD are closed under composition and identities. ThenF is an S-equivalence if and only if it is S-dense, S-full and S-faithful.

Here,S-fullness resp.S-faithfulnessofFis defined as fullness resp. faithfulness ofGZ(F)on images ofS-2-arrows inD: For all objects X andX⁰ in Cand every diagram of the form

F X ^g Y˜^{0 b≈} F X⁰

inDthere is a unique morphismϕ:X →X⁰ inGZ(C)such that loc(g) loc(b)⁻¹=GZ(F)ϕ

in GZ(D). So the theorem states that being an S-equivalence can be decided by investigating morphisms inGZ(D)that consist of precisely one numerator and precisely one denominator.

This characterisation of S-equivalences is based on the S-approximation theorem (5.24), where an isomorphism inverseQˆS:GZ(D)→GZ(C)toGZ(F) :GZ(C)→GZ(D)is explicitly constructed using a choice of an S-replace- ment for every object inD.

A classical instance of this result is the fact that the inclusion of the category of bounded above complexes with entries in projective modules into the category of bounded above complexes with entries in all modules induces an equivalence between the according derived categories, where an isomorphism inverse on the derived categories is provided by pointwise projective replacements (aka projective resolutions of complexes). More generally, the inclusion of the full subcategory of cofibrant objects in a model category in the sense of Quillen [8, ch. I, sec. 1, def. 1] or, even more generally, in a right derivable category in the sense ofCisinski[3, 2.22, dual of 1.1]

is always an S-equivalence, where an isomorphism inverse to the induced functor on the homotopy categories is provided by cofibrant replacements.

Sufficient criteria for S-equivalences have been established by Rădulescu-Banu [9, th. 5.5.1] and by Kahn and Sujatha [7, dual of th. 2.1, dual of cor. 4.4]. Many techniques used in this article are similar to the techniques used in these two sources. In particular, to verify thatGZ(F)is an equivalence of categories in their frameworks,Rădulescu-Banuas well asKahnandSujathaalso constructed an explicit isomorphism inverse, respectively. The advantage of these two sufficient approaches is their easier verifiability: Although S-fullness and S-faithfulness are particular cases of fullness and faithfulness of the induced functor on the localisation level, these properties still involve arbitrary morphisms in the localisation of the start category with denominators.

As it can be hard to check S-faithfulness, it would be desirable to have a “decomposition” of this axiom into a conjunction of simpler conditions.

In his framework of left exact functors between left derivable categories, Cisinski has given in [3, th. 3.19]

a characterisation of morphisms whose right derived functor is an equivalence. Since density is (in general) obtained by zigzags oftwodenominators in his theory, this approach is independent of the one presented in this article.

Outline Some preliminaries on localisations of categories are recalled in section 2. In section 3, our main tools for the construction of an isomorphism inverse to GZ(F), the S-replacements, are introduced. S-equivalences and their characterising conditions are defined in section 4. The final and main part of the article is section 5, where an isomorphism inverse toGZ(F)is constructed.

Conventions and notations

We use the following conventions and notations.

• To avoid set-theoretical difficulties, we (implicitly) work with Grothendieck universes [1, exp. I, sec. 0]. In particular, every category has aset of objects and aset of morphisms.

• The composite of morphismsf:X→Y andg:Y →Z is usually denoted byf g:X→Z. The composite of functorsF:C → DandG:D → E is usually denoted byG◦F:C → E.

• Given objects X andY in a categoryC, we denote the set of morphisms from X toY by_C(X, Y).

• Given a functor F:C → D, we denote its map on the objects by ObF: ObC →ObD, its map on the morphisms byMorF: MorC →MorD, and its maps on the hom-sets byFX,X⁰:_C(X, X⁰)→_D(F X, F X⁰) forX, X⁰ ∈ObC .

inclusion offull sub- category of cofibrant

objectsinQuillen model category[8, ch. I, sec. 1, def. 1]

inclusion offull subcategory of cofibrant objects inright derivable category[3, 2.22]

left approxima- tion(approach byRădulescu- Banu) [9, 5.4.1]

approach byKahnand Sujatha[7, dual of 2.1]

S-equivalence(4.3), multiplicativity(Cat)

S-density(4.1), S-fullness(4.4), S-faithfulness(4.7), multiplicativity(Cat)

equivalence be- tween localisations

[6, dual of 10.8], [7, dual of 4.3 a)]

[9, 6.1.5]

[9, 5.5.1] [7, dual of 2.1]

(5.25)

(4.3)

Figure 1: S-equivalences: a concept for equivalences between localisations.

• IfX is isomorphic toY, we writeX ∼=Y.

• Given a setX, we denote the identity map ofX byidX:X→X. Likewise, given a categoryC, we denote the identity functor ofC byid_C:C → C.

• We suppose given categories C and D. A functor F:C → D is said to be an equivalence (of categories) if there exists a functor G:D → C such thatG◦F ∼= idC and F◦G∼= idD. Such a functorG is then called an isomorphism inverse of F. The categoriesC and Dare said to be equivalent, writtenC ' D, if an equivalence of categoriesF: C → Dexists.

• We use the notationN={1,2,3, . . .}.

• Givena, b∈Zwitha≤b+ 1, we write[a, b] :={z∈Z|a≤z≤b}for the set of integers lying betweena andb.

• When defining a category via its hom-sets, these are considered to be formally disjoint. In other words, a morphism between two given objects may be formally seen as a triple consisting of an underlying morphism and its source and target object.

A comment on the terminology The notions S-replacements, S-density, S-fullness, . . . are adapted to the notion of an S-2-arrow, whereas the terminology of an S-2-arrow is inspired from [10, def. 4.2]: an S-2-arrow may be interpreted as a 3-arrow whose “T-part” is trivial. The dual concepts may be named T-replacements, T-density,T-fullness, . . . , respectively.

2 Preliminaries

In this section, we collect some preliminaries, particularly on localisations, connectedness and contractibility of categories. Its main purpose is to fix notation and terminology.

Categories with denominators

A category with denominators (¹) consists of a category C together with a subset D of MorC. By abuse of notation, we refer to the said category with denominators as well as to its underlying category by C. The elements ofD are calleddenominators (²) in C.

Given a category with denominators C with set of denominators D, we write DenC := D. In diagrams, a denominatord:X →Y inC will usually be depicted as

X ^≈d Y .

Given categories with denominators C and D, a morphism of categories with denominators from C to D is a functor F: C → D that preserves denominators, that is, such that F d is a denominator in D for every denominatordinC.

Given morphisms of categories with denominatorsF, G: C → D, a2-morphism of categories with denominators fromF toGis a transformationα:F →G.

Given categories with denominators C and D, a functor F: C → D is said to reflect denominators if every morphismdinC such thatF dis a denominator inDis a denominator inC.

Multiplicativity and isosaturatedness

A category with denominatorsCis said to be multiplicativeif the following holds.

(Cat) Multiplicativity. For all denominatorsd:X →X˜ ande: ˜X →X¯ inC, the compositede:X →X¯ is also a denominator inC. For every object X inC, the identity1X:X →X is a denominator inC.

A category with denominatorsCis said to be isosaturated if the following holds.

(Iso) Isosaturatedness. Every isomorphism inCis a denominator inC.

While multiplicativity of categories with denominators occurs quite often troughout this article, the notion of isosaturatedness is solely used in proposition (5.4).

Localisations

We suppose given a category with denominators C. A localisation of C consists of a category L and a func- torL:C → LwithLdinvertible inLfor every denominatordinC, and such that for every categoryDand every functorF:C → DwithF dinvertible inDfor every denominatordinC, there exists a unique functorFˆ:L → D withF = ˆF◦L.

C D

L

F

L Fˆ

By abuse of notation, we refer to the said localisation as well as to its underlying category byL. The functorL is called thelocalisation functor ofL.

Given a localisationL ofCwith localisation functor L:C → L, we writeloc = loc^L :=L.

A localisation L also fulfils the following universal property with respect to transformations, see e.g. [11, prop. (1.15)]: For every categoryD, all functorsG, G⁰: L → Dand every transformationα:G◦loc→G⁰◦loc there exists a unique transformationαˆ:G→G⁰ withα= ˆα∗loc.

C D

L

G◦loc

G⁰◦loc

loc G

G⁰ ˆ α α

1KahnandMaltsiniotisuse the terminologylocalisateur (localisator) [6, sec. 3.1].

2KahnandMaltsiniotisuse the terminologyéquivalences faibles(weak equivalences) [6, sec. 3.1].

The Gabriel/Zisman localisation

We suppose given a category with denominators C. In [4, sec. 1.1], Gabriel and Zisman constructed a localisation ofC. We call this particular localisation theGabriel/Zisman localisationofCand denote it byGZ(C).

As the notion of a localisation is defined by a universal property, a localisation ofCis unique up to isomorphism.

We will use the following two facts about the Gabriel/Zisman localisation of a category with denominatorsC:

First, the localisation functorloc :C →GZ(C)is given on the objects by Ob loc = id_ObC = id_ObGZ(C).

Second, for every morphismϕ:X →X⁰inGZ(C)there existn∈N, morphismsfi:Xi−1→X˜iinCfori∈[1, n]

and denominatorsai:Xi→X˜i inC fori∈[1, n−1]withX =X0,X⁰= ˜Xn and such that ϕ= loc(f1) loc(a1)⁻¹loc(f2). . .loc(a_n−1)⁻¹loc(fn)

inGZ(C).

X ^f1 X˜1 ^a1≈ X1 ^f2 . . . ^an−1≈ X_n−1 ^fn X⁰

The Gabriel/Zisman localisation turns into a 2-functor from the 2-category of categories with denominators in a Grothendieck universe to the 2-category of categories in this Grothendieck universe as follows. Given a morphism of categories with denominators F:C → D, then GZ(F) :GZ(C) → GZ(D) is the unique functor with loc^GZ(D)◦F =GZ(F)◦loc^GZ(C). Given morphisms of categories with denominatorsF, F⁰:C → D and a 2-morphism of categories with denominators α: F →F⁰, the transformation GZ(α) :GZ(F)→ GZ(F⁰)is the unique transformation withloc^GZ(D)∗α=GZ(α)∗loc^GZ(C).

C D

GZ(C) GZ(D)

F

F⁰

loc loc

GZ(F)

GZ(F⁰) GZ(α) α

In this article, we study conditions on a morphism of categories with denominators F: C → D implying thatGZ(F) :GZ(C)→GZ(D)is an equivalence of categories.

S-2-arrows

We suppose given a category with denominatorsC. An S-2-arrow inC is a diagram X ^f Y˜ ^a≈ Y

inC whereais supposed to be a denominator, denoted by(f, a) :X →Y˜ ←Y.

S-2-arrows are usually used in the description of locations of well-behaved categories with denominators, see e.g. [4, ch. I, sec. 2.2, sec. 2.3], [2, th. 1], [5, sec. III.2, lem. 8], [11, th. (2.35), th. (2.37), th. (3.128), rem. (3.129)], where every morphism in the localisation is represented by an S-2-arrow. We will not do so in this article; instead, we will use the notion of an S-2-arrow in the formulation of the characterising conditions for S-equivalences in section 4.

On the construction of isomorphism inverses on localisation level

We suppose given a morphism of categories with denominators F: C → D. In corollary (2.2), we characterise those functors G: D →GZ(C)that induce an isomorphism inverse to GZ(F) :GZ(C)→GZ(D). This criterion is most likely folklore.

Remark (2.1)(b)(ii) will be used in the proof of corollary (5.14)(b), (c).

(2.1) Remark. We suppose given a morphism of categories with denominators F: C → D and a func- torG⁰:GZ(D)→GZ(C).

(a) (i) Given an isotransformation α⁰: G⁰◦ GZ(F) → id_GZ(C), then α⁰ ∗loc^GZ(C) is an isotransformation from (G⁰◦loc^GZ(D))◦F toloc^GZ(C).

(ii) Given an isotransformationα: (G⁰◦loc^GZ(D))◦F →loc^GZ(C), then there exists a unique transforma- tionα:ˆ G⁰◦GZ(F)→id_GZ(C)withα= ˆα∗loc^GZ(C), and this transformationαˆis an isotransformation.

(b) (i) Given an isotransformation α⁰: GZ(F)◦G⁰ → id_GZ(D), then α⁰∗loc^GZ(D) is an isotransformation from GZ(F)◦(G⁰◦loc^GZ(D))to loc^GZ(D).

(ii) Given an isotransformationα:GZ(F)◦(G⁰◦loc^GZ(D))→loc^GZ(D), then there exists a unique trans- formationα:ˆ GZ(F)◦G⁰→id_GZ(D) withα= ˆα∗loc^GZ(D), and this transformationαˆ is an isotrans- formation.

Proof.

(a) (i) Asα⁰:G⁰◦GZ(F)→id_GZ(C)is an isotransformation, the transformationα⁰∗loc^GZ(C)is an isotrans- formation fromG⁰◦GZ(F)◦loc^GZ(C)=G⁰◦loc^GZ(D)◦F toid_GZ(C)◦loc^GZ(C)= loc^GZ(C).

(ii) AsG⁰◦loc^GZ(D)◦F =G⁰◦GZ(F)◦loc^GZ(C), there exists a unique transformation α:ˆ G⁰◦GZ(F)→ id_GZ(C) with α = ˆα∗loc^GZ(C), see e.g. [11, prop. (1.16)]. Moreover, αˆ is an isotransformation, see e.g. [11, cor. (1.18)].

(b) (ii) This follows e.g. from [11, prop. (1.16), cor. (1.18)].

(2.2) Corollary. We suppose given a morphism of categories with denominators F: C → D and a func- torG:D →GZ(C)that maps denominators inDto isomorphisms inGZ(C)and we letG:ˆ GZ(D)→GZ(C)be the unique functor withG= ˆG◦loc^GZ(D). Moreover, we suppose given an isotransformationα: G◦F →loc^GZ(C)and an isotransformationβ:GZ(F)◦G→loc^GZ(D)and we letα: ˆˆ G◦GZ(F)→id_GZ(C)be the unique transformation withα= ˆα∗loc^GZ(C) and we letβˆ:GZ(F)◦Gˆ →id_GZ(D) be the unique transformation with β= ˆβ∗loc^GZ(D). Thenαˆandβˆare isotransformations. In particular,GZ(F)andGˆare mutually isomorphism inverse equivalences of categories.

Proof. The transformation αˆ is an isotransformation by remark (2.1)(a)(ii) and the transformation βˆ is an isotransformation by remark (2.1)(b)(ii).

A construction principle for functors via choices

We recall from [11, app. A, sec. 1] a systematic method to construct a functor whose map on the objects depends on a choice.

We suppose given a category C and a family S = (SX)_X∈ObC over ObC. The structure category of C with respect to S is the category CS given as follows. The set of objects of CS is given by ObCS = {(X, S)| X ∈ObC,S∈SX}. Given objects(X, S), (Y, T)in CS, we have the hom-set _CS((X, S),(Y, T)) = {(f, S, T)|f ∈_C(X, Y)}. The composite of morphisms(f, S, T) : (X, S)→(Y, T)and(g, T, U) : (Y, T)→(Z, U) in CS is given by(f, S, T)(g, T, U) = (f g, S, U), and the identity morphism on an object(X, S)in CS is given by1(X,S)= (1X, S, S).

Theforgetful functor ofCS is the functorU :CS→ C,(X, S)7→X,(f, S, T)7→f.

Given objects(X, S)and(Y, T)inCS, a morphism(f, S, T) : (X, S)→(Y, T)inCSwill usually be denoted just byf: (X, S)→(Y, T). Moreover, we usually write_CS((X, S),(Y, T)) =_C(X, Y)instead ofCS((X, S),(Y, T)) = {(f, S, T)|f ∈C(X, Y)}.

Given a functorF¯:C_S→ D, we usually writeF¯_SX := ¯F(X, S)for(X, S)∈ObC_S andF¯_S,Tf :=F(f, S, T)for a morphismf: (X, S)→(Y, T)inC_S.

A choice of structures for C with respect to S is a family S = (S_X)_X∈ObC over ObC such that S_X ∈ S_X for X ∈ ObC. Given a choice of structures S = (SX)_X∈ObC for C with respect to S, the structure choice functor with respect toS is the functorIS:C → CS given on the objects byISX = (X, SX)forX∈ObC and on the morphisms by ISf =f: (X, SX)→(Y, SY)for every morphism f:X →Y in C. It fulfils U◦IS = id_C and IS ◦U∼= idCS, where an isotransformationε: IS ◦U →id_CS is given by ε_(X,T)= 1X: (X, SX)→(X, T)

for(X, T)∈ObC_S. In particular, the forgetful functorU :C_S→ Cand the structure choice functorIS:C → C_S are mutually isomorphism inverse equivalences of categories.

To construct a functor F: C → D whose definition on the objects uses a choice of structures S for C with respect toS, we may first construct a choice-free variant F¯:CS → D and then defineF := ¯F◦IS. With the notations introduced above, we then have F X= ¯FS_XX for every object X in C andF f = ¯FS_X,S_X0f for every morphismf: X→X⁰ in C.

Given a functor F¯: CS → D and choices of structures S and S˜ for C with respect to S, thenFS := ¯F ◦IS˜

andFS˜:= ¯F◦IS˜are isomorphic, an isotransformationα_S,S˜:FS →FS˜is given by(α_S,S˜)X = ¯F_S

X,S˜_X1X:FSX → FS˜X forX∈ObC, and the inverse ofα_S,S˜ is given byα⁻¹

S,S˜=αS,S˜ .

We will make use of this principle in the construction of S-replacement functors in section 5.

3 S-replacements

We suppose given a morphism of categories with denominatorsF:C → D. If GZ(F) :GZ(C)→ GZ(D) is an equivalence of categories, then it is in particular dense, that is, for every objectY inDthere is an objectXinC such thatY ∼=GZ(F)X =F X in GZ(D). Since the localisation functor loc :D →GZ(D)maps denominators inDto isomorphisms inGZ(D), the easiest non-trivial situation where we have such an isomorphism inGZ(D) is the one where we already have a denominatorF X →Y (or, dually, a denominatorY →F X) inD.

Below we will often suppose thatFadmits for every objectY inDan objectX and a denominatorq:F X →Y inD. In fact, to show thatF induces an equivalence on localisation level (under certain additional conditions), we will use such pairs(X, q)to construct an isomorphism inverse.

In the following, we will introduce terminology for these pairs and introduce a categorical setup for objects endowed with these pairs.

Throughout this section, we suppose given a morphism of categories with denominatorsF:C → D. (³)

Concept

We begin with the definition of the basic concept of this article.

(3.1) Definition (S-replacement). We suppose given an object Y in D. An S-replacement ofY alongF (⁴) (or, if no confusion arises, just an S-replacement of Y) is a pair (X, q) consisting of an object X in C and a denominatorq:F X →Y inD.

(3.2) Remark. In addition to the morphism of categories with denominatorsF:C → D, we suppose given a morphism of categories with denominatorsG: D → E.

(a) Given an objectZ in E and an S-replacement(X, r)of Z alongG◦F, then(F X, r)is an S-replacement ofZ alongG.

GF X

Z

≈ r

(b) We suppose thatE is multiplicative. Given an objectZ in E, an S-replacement(Y, r)of Z along Gand an S-replacement(X, q)ofY alongF, then(X,(Gq)r)is an S-replacement ofZ alongG◦F.

GF X

GY

Z

≈ Gq

≈ r

3Some parts of this section may also make sense ifCis (only) supposed to be a category,Dis supposed to be a category with denominators andF is (only) supposed to be a functor.

4KahnandMaltsiniotisuse the terminologyF-résolution à gauche (leftF-resolution) [6, sec. 5.11, dual of déf. 5.4].

(3.3) Definition (having enough S-replacements). The category with denominators Dis said tohave enough S-replacements along F (or, if no confusion arises, just to have enough S-replacements) if for every object Y inDthere exists an S-replacement ofY alongF.

(3.4) Remark. In addition to the morphism of categories with denominatorsF:C → D, we suppose given a morphism of categories with denominatorsG: D → E.

(a) IfE has enough S-replacements alongG◦F, then it has enough S-replacements alongG.

(b) We suppose that E is multiplicative. If E has enough S-replacements along G and D has enough S-re- placements alongF, thenE has enough S-replacements alongG◦F.

Proof.

(a) We suppose thatE has enough S-replacements alongG◦F. Then for every objectZ inE, there exists an S-replacement(X, r)ofZ alongG◦F, which yields the S-replacement(F X, r)ofZ alongG. ThusE has enough S-replacements alongG.

(b) We suppose that E has enough S-replacements alongGand thatDhas enough S-replacements alongF. Moreover, we suppose given an objectZ in E. Since E has enough S-replacements alongG, there exists an S-replacement (Y, r) of Z along G, and since D has enough S-replacements along F, there exists an S-replacement(X, q)ofY alongF. But then(X,(Gq)r)is an S-replacement ofZ alongG◦F. ThusE has enough S-replacements alongG◦F.

(3.5) Definition (having all trivial S-replacements). The category with denominators D is said to have all trivial S-replacements along F (or, if no confusion arises, just to have all trivial S-replacements) if for every objectX inC the identity1F X:F X→F X is a denominator inD.

F X

F X

≈ 1_{F X}

(3.6) Remark. IfC orDis multiplicative, thenDhas all trivial S-replacements alongF.

The category of objects with S-replacement

Next, we consider structures consisting of an object inDequipped with an S-replacement.

(3.7) Definition(object with S-replacement). We let R= (R_Y)_Y∈ObD be given by R_Y ={(X, q)|(X, q)is an S-replacement ofY alongF}

for Y ∈ ObD. The category of objects with S-replacement in D along F is the category with denomina- torsD_Rpl

S(F) whose underlying category is given by the structure categoryD_Rand whose set of denominators is given by

DenD_Rpl

S(F)={e∈MorD_Rpl

S(F)|Ueis a denominator inD}.

An object inD_Rpl

S(F) is called anobject with S-replacement inDalongF. A morphism inD_Rpl

S(F)is called a morphism of objects with S-replacement in DalongF. A denominator in D_RplS(F) is called a denominator of objects with S-replacement inDalongF.

(3.8) Remark. We have

ObDRpl_S(F)={(Y, X, q)|Y ∈ObD, (X, q)is an S-replacement ofY alongF}.

For objects(Y, X, q)and(Y⁰, X⁰, q⁰)inDRpl_S(F), we have the hom-set

D_RplS(F)((Y, X, q),(Y⁰, X⁰, q⁰)) =_D(Y, Y⁰).

For morphisms g: (Y, X, q) → (Y⁰, X⁰, q⁰) and g⁰: (Y⁰, X⁰, q⁰) → (Y⁰⁰, X⁰⁰, q⁰⁰) in D_Rpl

S(F), the compo- site gg⁰: (Y, X, q) → (Y⁰⁰, X⁰⁰, q⁰⁰) in D_Rpl

S(F) has the underlying morphism gg⁰:Y → Y⁰⁰ in D. For an ob- ject(Y, X, q)in D_Rpl

S(F), the identity morphism 1_(Y,X,q): (Y, X, q)→(Y, X, q)in D_Rpl

S(F)has the underlying morphism1Y:Y →Y inD.

The forgetful functorU :D_Rpl

S(F)→ Dis given on the objects by U_(X,q)Y =Y

for(Y, X, q)∈ObDRpl_S(F), and on the morphisms by U_(X,q),(X0,q⁰)g=g

for every morphismg: (Y, X, q)→(Y⁰, X⁰, q⁰)in DRpl_S(F). (3.9) Remark. The forgetful functorU :D_Rpl

S(F)→ Dpreserves and reflects denominators.

Choices of S-replacements

Our construction of an isomorphism inverse on the localisation level in section 5 will use a choice of an S-re- placement for every object of D. This leads us to the following notion, whose properties are just particular cases of the more general facts on choices of structures, see section 2 or [11, app. A, sec. 1].

(3.10) Definition (choice of S-replacements). We letR= (R_Y)_Y∈ObD be given by R_Y ={(X, q)|X ∈ObC,q:F X →Y is a denominator inD}

forY ∈ObD. Achoice of S-replacements forDalongF is a choice of structures with respect to R.

(3.11) Remark. A choice of S-replacements for DalongF is a family ((XY, qY))Y∈ObD such that(XY, qY) is an S-replacement ofY alongF forY ∈ObD.

(3.12) Remark. There exists a choice of S-replacements forDalongF if and only ifDhas enough S-replace- ments alongF.

Every choice of structures leads to a structure choice functor, see section 2 or [11, def. (A.8)]. In the case of a choice of S-replacements, the structure choice functor is given as follows.

(3.13) Remark. We suppose given a choice of S-replacements R = ((XY, qY))Y∈ObD for D along F. The structure choice functorI_R:D → DRpl_S(F)is given on the objects by

I_RY = (Y, X_Y, q_Y)

forY ∈ObD, and on the morphisms by IRg=g: (Y, XY, qY)→(Y⁰, XY⁰, qY⁰)

for every morphismg:Y →Y⁰ in D.

(3.14) Corollary. We suppose given a choice of S-replacementsR = ((XY, qY))Y∈ObD for D alongF. The structure choice functorIR:D → D_RplS(F)is a morphism of categories with denominators.

Structure choice functors are isomorphism inverse to the forgetful functor from the structure category to the category of its underlying objects. We recall this fact in the case of a structure choice functor with respect to a choice of S-replacements and see that we obtain a pair of mutually isomorphism inverse equivalences on the localisation level:

(3.15) Remark. We suppose given a choice of S-replacements R= ((X_Y, q_Y))_Y∈ObD forDalongF. (a) We have

U◦IR= id_D.

(b) We have

IR◦U∼= idDRplS(F).

An isotransformationα¯: IR◦U→id_DRpl

S(F) is given by

¯

α_(Y,X0,q⁰)= 1Y: (Y, XY, qY)→(Y, X⁰, q⁰) for(Y, X⁰, q⁰)∈ObD_Rpl

S(F). In particular, U : D_Rpl

S(F) → D and IR: D → D_Rpl

S(F) are mutually isomorphism inverse equivalences of categories.

Proof. This follows from [11, prop. (A.9)].

(3.16) Corollary. We suppose given a choice of S-replacements R= ((XY, qY))Y∈ObD forDalongF. (a) We have

GZ(U)◦GZ(I_R) = id_GZ(D).

(b) We have

GZ(IR)◦GZ(U)∼= idGZ(D_Rpl

S(F)).

An isotransformationα¯:GZ(IR)◦GZ(U)→id_GZ(DRpl

S(F)) is given by

¯

α_(Y,X⁰_,q⁰₎= 1Y: (Y, XY, qY)→(Y, X⁰, q⁰) for(Y, X⁰, q⁰)∈ObGZ(DRpl_S(F)).

In particular, GZ(U) :GZ(DRpl_S(F))→GZ(D)and GZ(I_R) :GZ(D)→GZ(DRpl_S(F))are mutually isomorphism inverse equivalences of categories.

Proof.

(a) By remark (3.15)(a), we have

GZ(U)◦GZ(IR) =GZ(U◦IR) =GZ(id_D) = id_GZ(D).

(b) By remark (3.15)(b), we have an isotransformationα¯⁰: IR◦U→id_DRpl

S(F) given by

¯

α⁰_(Y,X0,q⁰)= 1Y: (Y, XY, qY)→(Y, X⁰, q⁰) for (Y, X⁰, q⁰)∈ObD_Rpl

S(F). But thenα¯ :=GZ( ¯α⁰)is an isotransformation fromGZ(IR◦U) =GZ(IR)◦ GZ(U)to GZ(id_DRpl

S(F)) = id_GZ(DRpl

S(F))by2-functoriality, given by

¯

α_(Y,X0,q⁰)= loc^GZ(D)( ¯α⁰_(Y,X0,q⁰)) = loc^GZ(D)(1_Y) = 1_Y: (Y, X_Y, q_Y)→(Y, X⁰, q⁰) for(Y, X⁰, q⁰)∈ObGZ(D_Rpl

S(F)) = ObD_Rpl

S(F).

4 S-equivalences and the characterising conditions

Next, we introduce S-equivalences, that is, those morphisms of categories with denominators inducing equiva- lences on the localisation level that we want to characterise in this article, as well as the characterising conditions.

Throughout this section, we suppose given a morphism of categories with denominatorsF:C → D.

S-density

We begin with the restriction we put onF:C → Dthat ensures thatGZ(F) :GZ(C)→GZ(D)is dense.

(4.1) Definition(S-dense). We say thatF isS-dense ifDhas enough S-replacements alongF.

SoF is S-dense if and only if for every objectY in Dthere exists an S-replacement ofY alongF. (4.2) Remark. IfF is S-dense, thenGZ(F) :GZ(C)→GZ(D)is dense.

Proof. We suppose thatF is S-dense and we suppose given an object Y in D. Then there exists an S-replace- ment (X, q) of Y along F. As q: F X → Y is a denominator in D, it follows that loc(q) :F X → Y is an isomorphism inGZ(D), and so we have

Y ∼=F X =GZ(F)X

inGZ(D). ThusGZ(F)is dense.

We will give a characterisation of S-density (under additional assumptions on the degree of saturatedness ofD) via the forgetful functorU :D_RplS(F)→ Din proposition (5.4).

S-equivalences

The primary objective of this article is to characterise whenF is an S-equivalence in the following sense.

(4.3) Definition (S-equivalence). The morphism of categories with denominatorsF is called anS-equivalence if it is S-dense andGZ(F) :GZ(C)→GZ(D)is an equivalence.

The characterisation of S-equivalences will be given in corollary (5.25), which is based on the S-approximation theorem (5.24), where an isomorphism inverse toGZ(F) :GZ(C)→GZ(D)is constructed.

S-fullness and S-faithfulness

While S-density is already part of the definition of an S-equivalence, we will now introduce the remaining characterising conditions – S-fullness and S-faithfulness.

(4.4) Definition (S-fullness). We say that F is S-full if for all objects X and X⁰ in C and every S-2-ar- row(g, b) :F X→Y˜⁰ ←F X⁰ in Dthere exists a morphismϕ:X →X⁰ in GZ(C)such that

loc(g) loc(b)⁻¹=GZ(F)ϕ

inGZ(D).

So roughly said, S-fullness ofF means “fullness ofGZ(F)on S-2-arrows”.

(4.5) Remark. IfGZ(F) :GZ(C)→GZ(D)is full, then F is S-full.

(4.6) Proposition. We suppose that Dis multiplicative and thatF is S-dense. Then F is S-full if and only ifGZ(F) :GZ(C)→GZ(D)is full.

Proof. If GZ(F) is full, then in particular F is S-full. Conversely, we suppose that F is S-full. To show that GZ(F) is full, we suppose given objects X and X⁰ in C and a morphism ψ: F X → F X⁰ in GZ(D).

Moreover, we choose n∈N, morphisms gi: Y_i−1 →Y˜i in D fori ∈[1, n] and denominators bi:Yi →Y˜i in D fori∈[1, n−1]such thatF X =Y0,F X⁰ = ˜Yn and

ψ= loc(g1) loc(b1)⁻¹loc(g2). . .loc(b_n−1)⁻¹loc(gn).

Since D is multiplicative, the identity 1F X⁰: F X⁰ → F X⁰ is a denominator in D. We set Yn := F X⁰ and bn := 1F X⁰. Moreover, since F is S-dense, for i∈ [1, n−1]there exists an S-replacement (Xi, qi) of Yi.

We set (X0, q0) := (X,1F X) and (Xn, qn) := (X⁰,1F X⁰). Then qibi is a denominator in D for i ∈ [1, n] by multiplicativity.

F X Y˜1 F X1 . . . F Xn−1 F X⁰ F X⁰

F X Y˜1 Y1 . . . Y_n−1 F X⁰ F X⁰

g1

≈ 1F X

q1g2

≈ q1

q1b≈ 1 qn−1g_n

≈ qn−1

qn−1b≈ n−1

1_{F X}≈ 0

≈ 1_{F X}0

g₁ g₂

b1≈ g_n

b_n−1≈

1_{F X}≈ 0

Now the S-fullness ofF implies that fori∈[1, n] there exists a morphismϕi:X_i−1→Xi in GZ(C)such that loc(q_i−1gi) loc(qibi)⁻¹=GZ(F)ϕi. We obtain

ψ= loc(g1) loc(b1)⁻¹loc(g2). . .loc(b_n−1)⁻¹loc(gn)

= loc(q₀g₁) loc(q₁b₁)⁻¹loc(q₁g₂). . .loc(q_n−1b_n−1)⁻¹loc(q_n−1g_n) loc(q_nb_n)⁻¹

= (GZ(F)ϕ1)(GZ(F)ϕ2). . .(GZ(F)ϕn) =GZ(F)(ϕ1ϕ2. . . ϕn).

ThusGZ(F)is full.

(4.7) Definition (S-faithfulness). We say that F is S-faithful if for all objects X and X⁰ in C, every S-2-ar- row(g, b) :F X→Y˜⁰ ←F X⁰ in Dand all morphismsϕ₁, ϕ₂:X →X⁰ inGZ(C)such that

loc(g) loc(b)⁻¹=GZ(F)ϕ₁=GZ(F)ϕ₂ inGZ(D), we have

ϕ1=ϕ2

inGZ(C).

So roughly said, S-faithfulness ofF means “faithfulness ofGZ(F)on S-2-arrows”.

(4.8) Remark. IfGZ(F) :GZ(C)→GZ(D)is faithful, thenF:C → Dis S-faithful.

Under the (mild) additional assumption thatDis multiplicative we will show thatF is an S-equivalence if and only if it is S-dense, S-full and S-faithful, see corollary (5.25).

5 S-replacement functors and the S-approximation theorem

We suppose given a morphism of categories with denominators F: C → D. The aim of this section is the construction of an isomorphism inverse to GZ(F) :GZ(C) → GZ(D), provided that D is multiplicative and F fulfils the conditions of S-density, S-fullness and S-faithfulness defined in the previous section.

We give a sketch of this construction: First, we show thatF lifts to the category of objects with S-replacement in D along F, see remark (5.1), that is, we show that there exists a morphism of categories with denomi- nators F¯:C → D_Rpl

S(F) such that the following triangle on the left commutes. By the functoriality of the Gabriel/Zisman localisation, this commutative triangle on the left induces the following commutative triangle on the right.

D_Rpl

S(F)

C ^F D

F¯ U

GZ(D_Rpl

S(F))

GZ(C) ^GZ(F) GZ(D)

GZ( ¯F) GZ(U)

By remark (3.15) we already know that the forgetful functor U :D_Rpl

S(F)→ D is an equivalence of categories if F:C → D is S-dense, where an isomorphism inverse I_R: D → D_Rpl

S(F) is constructed by a choice of an S-replacement for each object in D, see definition (3.10) and remark (3.15). This pair of mutually inverse equivalences induces a pair of mutually inverse equivalences on the localisation level, see corollary (3.16).

So in order to show that the functorGZ(F) :GZ(C)→GZ(D)is an equivalence of categories, it suffices to show thatGZ( ¯F) :GZ(D_Rpl

S(F))→GZ(D)is an equivalence of categories. To this end, we construct the so-called total

S-replacement functor Q¯_SF:D_Rpl

S(F) → GZ(C), see proposition (5.5), which induces an isomorphism inverse ˆ¯

Q_SF:GZ(D_Rpl

S(F))→GZ(C)toGZ( ¯F) :GZ(C)→GZ(D_Rpl

S(F)), see corollary (5.9) and corollary (5.14).

C D_Rpl

S(F) D

GZ(C) GZ(D_Rpl

S(F)) GZ(D)

F¯

loc

U '

loc Q¯_SF

loc IR

GZ( ¯F) '

GZ(U) ' ˆ¯

QSF GZ(I_R)

The proof of the S-approximation theorem (5.24) is concluded by showing that an isomorphism inverse to GZ(F) :GZ(C) → GZ(D) can be induced by a so-called S-replacement functor Q_SF: D → GZ(C) that is given as compositeQSF= ¯QSF◦IR, see definition (5.16).

As the structure choice functor IR: D → D_Rpl

S(F) depends on a choice of S-replacements, this also holds for the S-replacement functor QSF = ¯QSF ◦IR. Thus the total S-replacement functor Q¯SF may be seen as a uniform variant of the various possible isomorphism inverse inducing S-replacement functors, which do necessitate choices.

Throughout this section, we suppose given a morphism of categories with denominatorsF:C → D.

The canonical lift

Under the assumption thatDhas all trivial S-replacements alongF, we may liftFto the corresponding category of objects with S-replacement:

(5.1) Remark. We suppose thatDhas all trivial S-replacements alongF. (a) We have a functorF¯:C → D_Rpl

S(F), given on the objects by F X¯ = (F X, X,1F X)

forX ∈ObC, and on the morphisms by

F f¯ =F f: (F X, X,1_{F X})→(F X⁰, X⁰,1_{F X}⁰)

for every morphism f:X →X⁰ inC.

F X

F X

≈ 1_{F X}

F X F X⁰

F X F X⁰

F f

≈1_{F X} ^≈ 1_{F X}0

F f

(b) We have

F = U◦F .¯ DRpl_S(F)

C ^F D

F¯ U

(5.2) Definition (canonical lift). We suppose that D has all trivial S-replacements along F. The func- torF¯: C → DRpl_S(F)in remark (5.1) is called the canonical lift ofF alongU :DRpl_S(F)→ D.

In fact, if F is S-dense, then one can construct several (non-canonical) lifts along the forgetful func- tor U : D_Rpl

S(F) → D: Every choice of S-replacements R = ((XY, qY))Y∈ObD for D along F leads to a lift FR := IR ◦ F: C → D_Rpl

S(F) as U ◦ FR = U◦ IR ◦ F = F by remark (3.15). However, to prove an assertion analogous to proposition (5.13)(a) below, it seems that one still (at least implicitly) needs the canonical lift F¯: C → DRpl_S(F) for the construction of an isotransformation Q¯SF ◦FR → loc^GZ(C), where Q¯SF:DRpl_S(F)→GZ(C)denotes the total S-replacement functor introduced in definition (5.6) below.

(5.3) Remark. We suppose thatDhas all trivial S-replacements alongF. For every object(Y, X, q)inD_Rpl

S(F), the pair (X,(q,(X,1F X),(X, q))) is an S-replacement of(Y, X, q)along the canonical liftF¯:C → D_Rpl

S(F). (F X, X,1F X) ^≈q (Y, X, q)

In particular, the category with denominatorsDRpl_S(F) has enough S-replacements alongF¯. (5.4) Proposition. We suppose that Dis multiplicative. The following conditions are equivalent.

(a) The morphism of categories with denominatorsF is S-dense.

(b) The forgetful functorU :DRpl_S(F)→ Dis S-dense.

(c) The forgetful functorU :D_RplS(F)→ Dis surjective on the objects.

IfDis isosaturated, then these conditions are also equivalent to the following conditions.

(d) The forgetful functorU :D_RplS(F)→ Dis dense.

(e) The forgetful functorU :D_Rpl

S(F)→ Dis an equivalence of categories.

Proof. First, we show that condition (a), condition (b) and condition (c) are equivalent.

By remark (5.1)(b), we haveF = U◦F¯, whereF¯:C → D_Rpl

S(F) denotes the canonical lift ofF along U. The canonical liftF¯ is S-dense by remark (5.3). So asDis multiplicative, remark (3.4) implies thatF is S-dense if and only ifUis S-dense, that is, condition (a) and condition (b) are equivalent.

Moreover, for every objectY in Dthere exists an S-replacement(X, q)ofY alongF if and only if there exists an object with S-replacement (Y, X, q) in D alongF. Thus F is S-dense if and only if U is surjective on the objects, that is, condition (a) and condition (c) are equivalent.

Thus condition (a), condition (b) and condition (c) are equivalent.

Second, we suppose that D is isosaturated and show that this premise implies that all five conditions are equivalent.

We suppose that condition (a) holds, that is, we suppose that F is S-dense. Then there exists a choice of S-replacementsRforDalongF, andUandIRare mutually isomorphism inverse equivalences by remark (3.15).

Thus condition (e) holds.

If condition (e) holds, that is, if U is an equivalence, then in particular U is dense by the dense-full-faithful criterion, that is, condition (d) holds.

Finally, we suppose that condition (d) holds, that is, we suppose that U :DRpl_S(F) → D is dense. Moreover, we suppose given an object Y in D. As U is dense, there exists an object (Y⁰, X, q) in D_Rpl

S(F) and an isomorphism g: U(Y⁰, X, q)→ Y in D. The isosaturatedness of D implies that g: Y⁰ →Y is a denominator in D, and so ((Y⁰, X, q), g) is an S-replacement of Y along U. Thus U : D_Rpl

S(F) → D is S-dense, that is, condition (b) holds.

Thus condition (a), condition (b), condition (c), condition (d) and condition (e) are equivalent.

The total S-replacement functor

Next, we construct a functor that leads to an isomorphism inverse of the canonical lift.

(5.5) Proposition. We suppose that F is S-full and S-faithful. Then we have a functor Q¯SF:D_RplS(F)→GZ(C),

given on the objects by ( ¯QSF)_(X,q)Y =X

for (Y, X, q) ∈ ObD_Rpl

S(F), and on the morphisms as follows. Given a morphism g: (Y, X, q) → (Y⁰, X⁰, q⁰) inD_Rpl

S(F), then( ¯QSF)_(X,q),(X0,q⁰)g: X→X⁰ is the unique morphism inGZ(C)with loc(q) loc(g) = (GZ(F)( ¯Q_SF)_(X,q),(X0,q⁰)g) loc(q⁰)

inGZ(D).

F X F X⁰

Y Y⁰

GZ(F)( ¯Q_SF)_(X,q),(X0,q0)g

loc(q) =∼

loc(q⁰)

∼=

loc(g)

Proof. We define a map Q¯0: ObD_Rpl

S(F)→ObGZ(C),(Y, X, q)7→X.

Moreover, as F: C → D is S-full and S-faithful, for all (Y, X, q),(Y⁰, X⁰, q⁰) ∈ ObD_Rpl

S(F) we obtain a well- defined map

Q¯(Y,X,q),(Y⁰,X⁰,q⁰):D_Rpl

S(F)((Y, X, q),(Y⁰, X⁰, q⁰))→_GZ(C)(X, X⁰), whereQ¯_(Y,X,q),(Y⁰_,X⁰_,q⁰₎g∈_GZ(C)(X, X⁰)forg∈D_Rpl

S(F)((Y, X, q),(Y⁰, X⁰, q⁰))is the unique element with loc(qg) loc(q⁰)⁻¹=GZ(F) ¯Q_(Y,X,q),(Y0,X⁰,q⁰)g,

that is, with

loc(q) loc(g) = (GZ(F) ¯Q_(Y,X,q),(Y0,X⁰,q⁰)g) loc(q⁰), inGZ(D).

Given morphismsg: (Y, X, q)→(Y⁰, X⁰, q⁰)andg⁰: (Y⁰, X⁰, q⁰)→(Y⁰⁰, X⁰⁰, q⁰⁰)inDRpl_S(F), we have loc(q) loc(gg⁰) = loc(q) loc(g) loc(g⁰) = (GZ(F) ¯Q(Y,X,q),(Y⁰,X⁰,q⁰)g) loc(q⁰) loc(g⁰)

= (GZ(F) ¯Q_(Y,X,q),(Y0,X⁰,q⁰)g) (GZ(F) ¯Q_(Y0,X⁰,q⁰),(Y⁰⁰,X⁰⁰,q⁰⁰)g⁰) loc(q⁰⁰)

=GZ(F)(( ¯Q_(Y,X,q),(Y0,X⁰,q⁰)g) ( ¯Q_(Y0,X⁰,q⁰),(Y⁰⁰,X⁰⁰,q⁰⁰)g⁰)) loc(q⁰⁰)

inGZ(D)and thereforeQ¯_(Y,X,q),(Y⁰⁰_,X⁰⁰_,q⁰⁰₎(gg⁰) = ( ¯Q_(Y,X,q),(Y⁰_,X⁰_,q⁰₎g)( ¯Q_(Y⁰_,X⁰_,q⁰_),(Y⁰⁰_,X⁰⁰_,q⁰⁰₎g⁰)inGZ(C). More- over, for(Y, X, q)∈ObD_RplS(F) we have

loc(q) loc(1_Y) = 1_{F X}loc(q) = (GZ(F)1_X) loc(q)

inGZ(D)and thereforeQ¯(Y,X,q),(Y,X,q)(1Y) = 1X= 1Q¯₀(Y,X,q)in GZ(C).

Thus we have a functor Q¯SF: DRpl_S(F) → GZ(C) given by Ob ¯QSF = ¯Q0 and by ( ¯QSF)(X,q),(X⁰,q⁰)g = Q¯_(Y,X,q),(Y0,X⁰,q⁰)gfor every morphism g: (Y, X, q)→(Y⁰, X⁰, q⁰)inDRpl_S(F).

(5.6) Definition (total S-replacement functor). We suppose that F is S-full and S-faithful. The func- torQ¯SF: D_RplS(F)→GZ(C)from proposition (5.5) is called thetotal S-replacement functor alongF.

(5.7) Remark. We suppose thatDis multiplicative and thatF is S-full and S-faithful. Moreover, we suppose given a morphismg: (Y, X, q)→(Y⁰, X⁰, q⁰)in DRpl_S(F), denominatorse:Y →Y˜ ande⁰:Y⁰→Y˜⁰ inDand a morphism˜g: ˜Y →Y˜⁰ inDsuch thatge⁰=e˜g inD.

F X F X⁰

Y Y⁰

Y˜ Y˜⁰

≈q ^≈ q⁰

g

≈e ^≈ e⁰

˜ g

Then we have

( ¯QSF)_(X,qe),(X⁰_,q⁰_e⁰₎˜g= ( ¯QSF)_(X,q),(X⁰_,q⁰₎g inGZ(C).

Proof. The pair (X, qe) is an S-replacement of Y˜ and the pair (X⁰, q⁰e⁰) is an S-replacement of Y˜⁰ by the multiplicativity ofD. Thus

loc(qe) loc(˜g) = loc(q) loc(e) loc(˜g) = loc(q) loc(g) loc(e⁰) = (GZ(F)( ¯Q_SF)_(X,q),(X0,q⁰)g) loc(q⁰) loc(e⁰)

= (GZ(F)( ¯QSF)_(X,q),(X⁰_,q⁰₎g) loc(q⁰e⁰) implies that

( ¯Q_SF)_(X,qe),(X0,q⁰e⁰)˜g= ( ¯Q_SF)_(X,q),(X0,q⁰)g inGZ(C).

F X F X⁰

Y Y⁰

Y˜ Y˜⁰

GZ(F)( ¯Q_SF)_(X,q),(X0,q0)g

loc(q) =∼

loc(q⁰)

∼=

loc(g)

loc(e) =∼

loc(e⁰)

∼=

loc(˜g)

(5.8) Corollary. We suppose thatDis multiplicative and thatF is S-full and S-faithful. Moreover, we suppose given a denominatore: (Y, X, q)→(Y⁰, X⁰, q⁰)in D_Rpl

S(F). Then we have ( ¯QSF)_(X,q),(X⁰_,q⁰₎e= ( ¯QSF)_(X,qe),(X⁰_,q⁰₎1Y⁰

inGZ(C).

Proof. This follows from remark (5.7).

F X F X⁰

Y Y⁰

Y⁰ Y⁰

≈q ^≈ q⁰

≈e

≈e ^≈ 1_Y0

≈

1_Y0

(5.9) Corollary. We suppose thatDis multiplicative and thatF is S-full and S-faithful. The total S-replace- ment functorQ¯_SF:D_Rpl

S(F)→GZ(C)maps denominators inD_Rpl

S(F)to isomorphisms inGZ(C).

Proof. We suppose given a denominatore: (Y, X, q)→(Y⁰, X⁰, q⁰)inDRpl_S(F). Then we have ( ¯Q_SF)_(X,q),(X0,q⁰)e= ( ¯Q_SF)_(X,qe),(X0,q⁰)1_Y⁰

in GZ(C) by corollary (5.8). In particular, ( ¯QSF)_(X,q),(X⁰_,q⁰₎e = ( ¯QSF)_(X,qe),(X⁰_,q⁰₎1Y⁰ is an isomorphism inGZ(C)since1Y⁰: (Y⁰, X, qe)→(Y⁰, X⁰, q⁰)is an isomorphism inD_Rpl

S(F).

(5.10) Notation. We suppose thatDis multiplicative and thatF is S-full and S-faithful. We denote by ˆ¯

QSF:GZ(D_Rpl

S(F))→GZ(C)

the unique functor withQ¯SF = ˆQ¯SF◦loc^GZ(DRplS(F)).