Next, we recall the S-2-arrow calculus of an S-fractionable category. Cf. theorem (2.93).
(2.31) Definition (S-2-arrow conditions). We suppose given a multiplicative category with denominators C, a categoryLand a functorL:C → L such thatLdis invertible inLfor every denominatordinC.
(a) We say that(L, L)fulfils theS-2-arrow representative condition if the following holds.
(2acrepS ) S-2-arrow representative condition. We have
MorL={(Lf)(La)−1|(f, a)is an S-2-arrow inC}.
(b) We say that(L, L)fulfils theS-2-arrow equality condition if the following holds.
(2aceqS) S-2-arrow equality condition. Given S-2-arrows(f, a),(f0, a0)in Cwith (Lf)(La)−1= (Lf0)(La0)−1
in L, there exist S-2-arrows( ˜f0,˜a0),(c, d)in C such that the following diagram commutes.
f
c a≈
f˜0
˜≈
a0
f0
≈d
a0≈
(c) We say that(L, L)fulfils theS-2-arrow composition condition if the following holds.
(2accompS ) S-2-arrow composition condition. Given S-2-arrows(f1, a1),(f2, a2),(g1, b1),(g2, b2)inC with (Lf1)(La1)−1(Lg2)(Lb2)−1= (Lg1)(Lb1)−1(Lf2)(La2)−1
in L, there exist an S-2-arrow ( ˜f2,˜a2) and morphisms ˜g2,˜b2 in C such that the following diagram commutes.
f1
g1 g˜2 g2 a1≈
f˜2
˜≈
a2
f2
≈b1 ˜b2
a2≈
≈b2
If the category with denominatorsCin definition (2.31) is S-semisaturated, then the morphism˜b2in part (c) of loc. cit. is automatically a denominator, so we have an S-2-arrow(˜g2,˜b2).
(2.32) Remark. We suppose given a multiplicative category with denominatorsC, a categoryL and a func-torL:C → Lsuch thatLdis invertible in Lfor every denominatordin C.
(a) If(L, L)fulfils the S-2-arrow representative condition, thenLis surjective on the objects.
(b) If(L, L)fulfils the S-2-arrow equality condition, thenL is injective on the objects.
Proof.
(a) We suppose that(L, L)fulfils the S-2-arrow representative condition. To show thatLis surjective on the objects, we suppose given an object Xˆ in L. By the S-2-arrow representative condition, there exists an S-2-arrow(f, a) :X →Y˜ ←Y in Cwith1Xˆ = (Lf)(La)−1. We get
Xˆ = Source 1Xˆ = Source((Lf)(La)−1) = SourceLf =L(Sourcef) =LX.
ThusLis surjective on the objects.
(b) We suppose that(L, L)fulfils the S-2-arrow equality condition. To show thatLis injective on the objects, we suppose given objects X,Y in C such thatLX=LY in L. Then we have
L1X = 1LX = 1LY =L1Y,
and so by the S-2-arrow equality condition we in particular have X= Source 1X = Source 1Y =Y.
ThusLis injective on the objects.
(2.33) Proposition. We suppose given a multiplicative category with denominators C, a category L and a functorL:C → Lsuch thatLdis invertible in Lfor every denominatordin C.
(a) If(L, L)fulfils the S-2-arrow equality condition, thenC fulfils the S-Ore expansibility axiom.
(b) If(L, L)fulfils the S-2-arrow representative condition and the S-2-arrow equality condition, then C fulfils the S-Ore completion axiom.
Proof.
(a) We suppose that(L, L)fulfils the S-2-arrow equality condition. To show thatCfulfils the S-Ore expansi-bility axiom, we suppose given parallel morphismsf1,f2 and a denominatordin C withdf1=df2. Then we have
(Ld)(Lf1) =L(df1) =L(df2) = (Ld)(Lf2)
and henceLf1=Lf2 sinceLdis invertible inL. As (L, L)fulfils the S-2-arrow equality condition, there exists a denominatord0 in Csuch thatf1d0 =f2d0. ThusC fulfils the S-Ore expansibility axiom.
(b) We suppose that(L, L)fulfils the S-2-arrow representative condition and the S-2-arrow equality condition.
To show thatC fulfils the S-Ore completion axiom, we suppose given a morphismf and a denominatord in C with Sourcef = Sourced. As (L, L) fulfils the S-2-arrow representative condition, there exists an S-2-arrow(g, a)inC with(Ld)−1(Lf) = (Lg)(La)−1. We get
L(f a) = (Lf)(La) = (Ld)(Lg) =L(dg),
and so as (L, L) fulfils the S-2-arrow equality condition, there exists a denominator e in C such that f ae=dge.
g e≈
f
≈d ≈a
We setf0 :=geandd0:=ae, so thatf d0=df0. Moreover,d0 is a denominator inC by multiplicativity.
g f0
e≈
f
≈d ≈a
≈d0
ThusC fulfils the S-Ore completion axiom.
(2.34) Proposition. We suppose given a multiplicative category with denominators C, a category L and a functorL:C → Lsuch thatLdis invertible in Lfor every denominatordin C.
(a) If(L, L)fulfils the S-2-arrow composite condition, then it also fulfils the S-2-arrow equality condition.
(b) If (L, L)fulfils the S-2-arrow representative condition and the S-2-arrow equality condition, then it also fulfils the S-2-arrow composition condition.
Proof.
(b) We suppose that(L, L)fulfils the S-2-arrow representative condition and the S-2-arrow equality condition.
To show that (L, L) fulfils the S-2-arrow composition condition, we suppose given S-2-arrows (f1, a1), (f2, a2),(g1, b1),(g2, b2)inC with
(Lf1)(La1)−1(Lg2)(Lb2)−1= (Lg1)(Lb1)−1(Lf2)(La2)−1
in L. By proposition (2.33)(b), we know thatC fulfils the S-Ore completion axiom. In particular, there exist an S-Ore completion (a02, b02) for a2 and b2, an S-Ore completion (f20, b01) for f2b02 and b1, and an S-Ore completion(g20, a01)forg2a02b01anda1.
f1
g1 g20
a1≈
g2
f20
a02≈
≈ a01
≈b01
f2
≈b1
b02 a2≈
≈b2
We obtain
L(f1g02) = (Lf1)(Lg20) = (Lf1)(La1)−1(Lg2)(La02)(Lb01)(La01)
= (Lf1)(La1)−1(Lg2)(Lb2)−1(La2)(Lb02)(Lb01)(La01)
= (Lg1)(Lf20)(La01) =L(g1f20a01).
So as (L, L) fulfils the S-2-arrow expansibility axiom, there exists a denominator d in C with f1g20d = g1f20a01d.
f1
g1 g20
a1≈
g2
f20a01 d≈
a02b01≈ a01
f2
≈b1 b02b01a01
a2≈
≈b2
Setting f˜2:=f20a01d,a˜2:=a02b01a01d,g˜2:=g20d,˜b2:=b02b01a01dyields f1˜g2=f1g02d=g1f20a01d=g1f˜2,
a1g˜2=a1g02d=g2a02b01a01d=g2˜a2, f2˜b2=f2b02b01a01d=b1f20a01d=b1f˜2, a2˜b2=a2b02b01a01d=b2a02b01a01d=b2˜a2.
Moreover,˜a2 is a denominator inC by multiplicativity.
f1
g1 g˜2 g2 a1≈
f˜2
˜≈
a2
f2
≈b1 ˜b2
a2≈
≈b2
Thus(L, L)fulfils the S-2-arrow composition condition.
(2.35) Theorem (S-2-arrow calculus). Given an S-fractionable categoryC, thenOreS(C)fulfils the S-2-arrow representative condition and the S-2-arrow equality condition.
Proof. Cf. [13, sec. III.2, lem. 8].
(2.36) Proposition. We suppose given a multiplicative category with denominators C, a category L and a functorL:C → Lsuch thatLdis invertible inLfor every denominatordinC. If(L, L)fulfils the S-2-arrow rep-resentative condition and the S-2-arrow equality condition, thenLbecomes a localisation ofC with localisation functorlocL=L.
Proof. By proposition (2.33), we know that C fulfils the S-Ore expansibility axiom and the S-Ore completion axiom, that is,Cis an S-fractionable category. In particular, the S-Ore localisationOreS(C)ofCis defined. By the universal property ofOreS(C), there exists a unique functorLˆ:OreS(C)→ LwithL= ˆL◦locOreS(C).
C L
OreS(C)
L
locOreS(C) Lˆ
The S-Ore localisationOreS(C)fulfils the S-2-arrow representative condition and the S-2-arrow equality condi-tionby theorem (2.35), so in particular,Lˆ is given by
LXˆ =LX
for every objectX in Cand by
L(locˆ OreS(C)(f) locOreS(C)(a)−1) = (Lf)(La)−1
for every S-2-arrow (f, a)in C. We want to show thatLˆ is an isofunctor. Indeed,Mor ˆLis surjective as(L, L) andOreS(C)fulfil the S-2-arrow representative condition, andMor ˆLis injective as(L, L)andOreS(C)fulfil the S-2-arrow equality condition. Altogether,Mor ˆLis a bijection. But this already implies thatLˆ is an isofunctor.
ThusLbecomes a localisation of CwithlocL=L.
The next theorem states that the axiomatics of an S-fractionable category is, in some precise sense, the best to obtain an S-2-arrow calculus in the sense of theorem (2.35).
(2.37) Theorem. We suppose given a multiplicative category with denominatorsC. The following conditions are equivalent.
(a) The category with denominatorsC is an S-fractionable category.
(b) There exists a localisation ofCthat fulfils the S-2-arrow representative condition and the S-2-arrow equality condition.
(c) There exists a localisation ofC that fulfils the S-2-arrow composition condition.
Proof. If condition (a) holds, that is, if C is an S-fractionable category, then by theorem (2.35), the S-Ore localisationOreS(C)fulfils the S-2-arrow representative condition and the S-2-arrow equality condition, and so condition (b) holds.
Moreover, if condition (b) holds, that is, if there exists a localisationLofC that fulfils the S-2-arrow represen-tative condition and the S-2-arrow equality condition, then this localisation also fulfils the S-2-arrow composite condition by proposition (2.34)(b).
Finally, we suppose that condition (c) holds, that is, we suppose that there exists a localisation L of C that fulfils the S-2-arrow composition condition. ThenL fulfils in particular the S-2-arrow equality condition and therefore C fulfils the S-Ore expansibility axiom by proposition (2.33)(a). To show that C fulfils the S-Ore completion axiom, we suppose given a morphismf and a denominator dinC with Sourcef = Sourced. Then we haveloc(d)−1loc(f) = loc(d)−1loc(f)inL, and so the S-2-arrow composition condition in particular yields an S-Ore completion(f0, d0)forf andd.
f0 f
d≈
f0 d0≈
f
≈d ≈d0
HenceC is an S-fractionable category, that is, condition (a) holds.
Altogether, we have shown that condition (a), condition (b) and condition (c) are equivalent.
4 Z-2-arrows
As just shown in theorem (2.37), S-fractionable categories, as introduced in definition (2.27)(a), characterise those multiplicative categories with denominators that admit an S-2-arrow calculus in the sense of theo-rem (2.35). So by contraposition, if a multiplicative category with denominators does not fulfil the axioms of an S-fractionable category, it cannot admit such a pure S-2-arrow calculus, even if we know that every mor-phism in the localisation is represented by an S-2-arrow, see definition (2.31)(a). So if we still want to work with strictly commutative diagrams as in the S-2-arrow equality condition, see definition (2.31)(b), we have to
restrict our attention to a subset of S-2-arrows that fulfils the following two requirements simultaneously. First, it must be small enough such that two S-2-arrows that are contained in the subset represent the same morphism in the localisation if and only if they may be embedded in a2-by-2 diagram as in definition (2.31)(b). Second, it must still be large enough such that every morphism in the localisation is represented by an S-2-arrow that lies in the subset.
In this section, we are going to introduce the notion of a category with Z-2-arrows, see definition (2.38)(a), that is, a category with denominators and S-denominators equipped with a distinguished subset of normal S-2-arrows, see definition (2.1)(a) and definition (2.10). Such a category with Z-2-arrows is the basic structure for our axiomatic localisation approach, but it does not yet necessarily fulfil enough axioms to construct a generalisation of the S-Ore localisation, cf. definition (2.30). Those axioms will be introduced in section 5.
After the definition of categories with Z-2-arrows, we develop some basic properties that follow from the Z-re-placement axiom. Thereafter, we introduce the Z-fraction equality, see definition (2.50), a congruence on the Z-2-arrow graph that is analogously defined to the S-fraction equality on the S-2-arrow graph resp. the normal S-fraction equality on the normal S-2-arrow graph, cf. definition (2.14).