(1.35) Definition (multiplicativity).
(a) We suppose given a categoryC. A subsetDofMorC is said to bemultiplicative (inC) if it fulfils:
(Cat) Multiplicativity. For alld, e∈D withTargetd= Sourcee, we have de∈D, and for every objectX in C, we have1X ∈D.
(b) (i) A category with denominators C is said to be multiplicative if its set of denominators DenC is a multiplicative subset ofC.
(ii) We suppose given a Grothendieck universe U. The full subcategory CatDmul = CatDmul,(U) ofCatD(U)with
ObCatDmul,(U)={C ∈ObCatD(U)| C is multiplicative}
is called thecategory of multiplicative categories with denominators (more precisely, thecategory of multiplicativeU-categories with denominators).
(1.36) Definition (isosaturatedness).
(a) We suppose given a category C. A subset D of MorC is said to contain all isomorphisms (or to be isosaturated) inC if it fulfils:
(Iso) Isosaturatedness. For every isomorphismf in C, we havef ∈D.
(b) A category with denominatorsCis said to beisosaturatedif its set of denominatorsDenCis an isosaturated subset ofC.
(1.37) Definition (semisaturatedness).
(a) We suppose given a categoryC.
(i) A subsetD ofMorCis said to be S-semisaturated (in C) if it is multiplicative and fulfils:
(2 of 3S) S-part of 2 out of 3 axiom. For all morphismsf andg inC withf, f g∈D, we also haveg∈D.
(ii) A subsetD ofMorCis said to be T-semisaturated (inC) if it is multiplicative and fulfils:
(2 of 3T) T-part of 2 out of 3 axiom. For all morphismsf andginC withg, f g∈D, we also havef ∈D.
(iii) A subsetDofMorCis said to besemisaturated (inC) (2) if it is S-semisaturated and T-semisaturated.
(b) A category with denominatorsCis said to beS-semisaturated resp.T-semisaturated resp.semisaturated if its set of denominators DenC is an S-semisaturated resp. a T-semisaturated resp. a semisaturated subset ofC.
(1.38) Definition (weak saturatedness).
(a) We suppose given a category C. A subset D of MorC is said to be weakly saturated (in C) if it is multiplicative and fulfils:
2In the literature, semisaturatedness is sometimes calledsaturatedness; and saturatedness in our sense, see definition (1.39), is sometimes calledstrong saturatedness.
(2 of 6) 2 out of 6 axiom. For all morphismsf,g,hin Cwithf g, gh∈D, we also havef, g, h, f gh∈D.
(b) A category with denominatorsCis said to beweakly saturated if its set of denominatorsDenCis a weakly saturated subset of C.
(1.39) Definition (saturatedness).
(a) We suppose given a categoryC. A subsetDofMorC is said to besaturated (inC) if it fulfils:
(Sat) Saturatedness. There exists a localisation ofC with respect toD and we haveSatD=D.
(b) A category with denominators C is said to be saturated if its set of denominators DenC is a saturated subset ofC.
(1.40) Example. We suppose given a categoryC.
(a) The discrete structureCdisc is semisaturated.
(b) The isomorphism structureCiso is saturated.
Proof.
(a) We suppose given morphisms f and g in C with Targetf = Sourceg such that two out of the three morphisms f, g, f g are denominators in Cdisc. Then these two are equal to an identity morphism and therefore all three are equal to an identity morphism inC. But this means that all three are denominators in Cdisc. Moreover,1X is a denominator inCdisc for every objectX inC, and soCdisc is semisaturated.
(b) This follows from example (1.29)(b).
(1.41) Proposition. We suppose given an isomorphism of categories with denominatorsF:C → D.
(a) The category with denominatorsC is multiplicative if and only if Dis multiplicative.
(b) The category with denominatorsC is isosaturated if and only ifDis isosaturated.
(c) The category with denominatorsC is S-semisaturated if and only ifD is S-semisaturated. The category with denominators Cis T-semisaturated if and only ifDis T-semisaturated.
(d) The category with denominatorsC is weakly saturated if and only ifDis weakly saturated.
(e) The category with denominatorsC is saturated if and only ifDis saturated.
Proof.
(a) We suppose that C is multiplicative, and we suppose given denominators e and e0 in D with Targete = Sourcee0. Then F−1e and F−1e0 are denominators in C, and since C is multiplicative, it follows that (F−1e)(F−1e0) is a denominator in C. But this implies that ee0 = F((F−1e)(F−1e0)) is a denominator in D. Moreover, given an object Y in D, we have 1Y = F(1F−1Y), and as 1F−1Y is a denominator inC, it follows that1Y is a denominator inD. Altogether,Dis multiplicative.
The other implication follows by symmetry.
(b) We suppose that C is isosaturated, and we suppose given an isomorphism g in D. Then F−1g is an isomorphism in C, and since C is isosaturated, it follows that F−1g is an isomorphism in C. But this implies thatg=F F−1g is a denominator inD.
The other implication follows by symmetry.
(c) We suppose that C is S-semisaturated, and we suppose given morphismsg andg0 in Dwith Targetg = Sourceg0 such that g and gg0 are denominators in D. Then F−1g and (F−1g)(F−1g0) = F−1(gg0) are denominators in C. Since C is S-semisaturated, it follows that(F−1g0)is a denominator inC. But then g0 = F(F−1g0) is a denominator in D. Thus D fulfils the S-part of the 2 out of 3 axiom. As D is multiplicative by (a), we conclude thatDis semisaturated.
By duality, we obtain: If Cis T-semisaturated, thenDis T-semisaturated.
The other implications follow by symmetry.
(d) We suppose that C is weakly saturated, and we suppose given morphisms g, g0, g00 in D such that gg0 and g0g00 are denominators in D. Then (F−1g)(F−1g0) = F−1(gg0) and (F−1g0)(F−1g00) = F−1(g0g00) are denominators in C, and since C is weakly saturated, it follows that F−1g, F−1g0, F−1g00, (F−1g)(F−1g0)(F−1g00) are denominators in C. But this implies that g = F(F−1g), g0 = F(F−1g0), g00 = F(F−1g00), gg0g00 = F((F−1g)(F−1g0)(F−1g00)) are denominators in D. As D is multiplicative by (a), we conclude thatDis weakly saturated.
The other implication follows by symmetry.
(e) IfCis saturated, that is, if there exists a localisation ofCand we haveSatC=C, then by proposition (1.34) there exists a localisation of Dand we have
Den SatD=F(Den SatC) =F(DenC) = DenD, that is,Dis saturated.
The other implication follows by symmetry.
We suppose given a category with denominatorsC and a category S. By remark (1.10), we may considerCS, the category ofS-commutative diagrams inC, as a category with denominators, having pointwise denominators.
The following proposition states that various notions of saturatedness are enherited to the diagram category.
(1.42) Proposition. We suppose given a category with denominatorsC and a categoryS.
(a) IfC is multiplicative, thenCS is multiplicative.
(b) IfC is isosaturated, thenCS is isosaturated.
(c) IfC is S-semisaturated, thenCS is S-semisaturated. IfC is T-semisaturated, thenCS is T-semisaturated.
(d) IfC is weakly saturated, thenCS is weakly saturated.
Proof.
(a) We suppose that C is multiplicative. Moreover, we suppose given denominators d, e in CS with Targetd= Sourcee. Thendi andei are denominators inCfor everyi∈ObS. It follows that(de)i=diei
is a denominator inCfor everyi∈ObS, that is,deis a denominator inCS. Moreover, given an objectX in CS, then (1X)i = 1Xi is a denominator inC for every i ∈ObS, whence1X is a denominator inCS. Altogether, CS is multiplicative.
(b) We suppose thatC is isosaturated. Moreover, we suppose given an isomorphismf in CS. Then fi is an isomorphism inC for every i∈ObS. The isosaturatedness ofC implies thatfi is a denominator inC for every i∈ObS, that is,f is a denominator inCS. ThusCS is isosaturated.
(c) We suppose that C is S-semisaturated. Moreover, we suppose given morphisms f, g in CS such that f and f g are denominators in CS. Then fi and figi = (f g)i are denominators in C for every i∈ObS. It follows thatgi is a denominator inCfor everyi∈ObS, that is,gis a denominator inCS. ThusCS fulfils the S-part of the 2 out of 3 axiom. AsCS is multiplicative by (a), we conclude thatCS is S-semisaturated.
The other implication follows by duality.
(d) We suppose that C is weakly saturated. Moreover, we suppose given morphisms f, g, h in CS such that f g andgh are denominators in CS. Then figi = (f g)i and gihi = (gh)i are denominators in C for every i∈ObS. It follows thatfi, gi,hi,(f gh)i =figihi are denominators inC for everyi∈ObS, that is, f, g, h, f gh are denominators in CS. Thus CS fulfils the 2 out of 6 axiom. As CS is multiplicative by (a), we conclude thatCS is weakly saturated.
The following proposition states how the different variations of the notion of saturatedness introduced in defi-nition (1.35) to defidefi-nition (1.39) are related. Cf. figure 1.
(1.43) Proposition. We suppose given a category with denominatorsC.
(a) IfC is saturated, thenC is weakly saturated.
(Cat) (Cat), (2 of 3S), (2 of 3T) (Cat), (2 of 6) (Sat)
(Iso)
Figure 1: Levels of saturatedness.
(b) IfC is weakly saturated, thenC is semisaturated and isosaturated.
(c) IfC is semisaturated, thenC is multiplicative.
Proof.
(a) We suppose that C is saturated and we let L be a localisation of C. Moreover, we suppose given mor-phismsf,g,hinCsuch thatf gandghare denominators inC. By corollary (1.22), it follows thatloc(f), loc(g), loc(h) are invertible in L, that is, f, g, h are denominators in the saturation SatC. Moreover, loc(f gh) = loc(f) loc(g) loc(h)is invertible inLas a composite of invertible morphisms, that is,f ghis a denominator in the saturationSatC. But asC is saturated, we haveSatC=C, and sof,g,h, f ghare in fact denominators inC. So we have shown thatC fulfils the 2 out of 6 axiom.
In particular, given denominatorsf, gin C, then f1 and1gare denominators inC and hencef g=f1g is a denominator in C. Moreover, given an objectX in C, the morphismloc(1X) = 1loc(X) is invertible in L and hence1X is a denominator inSatC=C. HenceC is also multiplicative and therefore is weakly saturated.
(b) We suppose thatCis weakly saturated. ThenC is in particular multiplicative.
To show thatCis semisaturated, we suppose given morphismsf andginCwithTargetf = Sourceg. Iff andf g are denominators inC, then1f andf g are denominators inC and henceg is a denominator inC by the 2 out of 6 axiom. Dually, if g and f g are denominators in C, thenf g and g1 are denominators in C and hencef is a denominator inC by the 2 out of 6 axiom. ThusC is semisaturated.
To show thatCis isosaturated, we suppose given an isomorphismf inC, so that there exists a morphismg in C with f g= 1 andgf = 1. Since in particular identities are denominators in C, it follows that f is a denominator inC by the 2 out of 6 axiom.
f g f
(c) This holds by definition.
Now we may give an example of a semisaturated category with denominators that is not weakly saturated.
(1.44) Example. We suppose given a categoryCthat contains a non-identical isomorphism. Then the discrete structureCdisc is semisaturated, but not weakly saturated.
Proof. The discrete structure Cdisc is always semisaturated by example (1.40)(a), but if there exists a non-identical isomorphism f in C, then f is not a denominator in Cdisc, and so Cdisc is not weakly saturated by proposition (1.43)(b).
(1.45) Proposition. We suppose given a categoryC, a subsetDofMorCand a localisationLofCwith respect toD. Moreover, we let
Dmul :=\
{U ⊆MorC |D⊆U andU is multiplicative}, Dssat:=\
{U ⊆MorC |D⊆U andU is semisaturated}, Dwsat:=\
{U ⊆MorC |D⊆U andU is weakly saturated}, Dsat :=\
{U ⊆MorC |D⊆U andU is saturated}.
(a) (i) The subsetDmul is the smallest multiplicative subset of MorC that containsD.
(ii) The subsetDssat is the smallest semisaturated subset ofMorC that containsD.
(iii) The subsetDwsat is the smallest weakly saturated subset ofMorC that containsD.
(iv) The subsetDsat is the smallest saturated subset ofMorC that containsD.
(b) We haveD⊆Dmul⊆Dssat⊆Dwsat⊆Dsat = SatD.
(c) The categoryL is a localisation ofC with respect to D, to Dmul, to Dssat, toDwsatand toDsat= SatD.
Proof. We set
U :={U ⊆MorC |D⊆U},
Umul :={U ⊆MorC |D⊆U andU is multiplicative}, Ussat:={U ⊆MorC |D⊆U andU is semisaturated}, Uwsat:={U ⊆MorC |D⊆U andU is weakly saturated},
Usat :={U ⊆MorC |D⊆U andU is saturated},
so thatD=TU,Dmul=TUmul, Dssat=TUssat,Dwsat=TUwsat,Dsat =TUsat.
(a) (i) We suppose givend, e∈Dmul withTargetd= Sourcee. ForU ∈ Umul, we haveDmul =T
Umul⊆U, so it follows that d, e ∈ U and therefore de ∈ U by the multiplicativity of U. Thus we have de∈T
Umul=Dmul.
Moreover, we suppose givenX ∈ObC. Then for allU ∈ Umul, we have1X ∈U by the multiplicativity ofU, and therefore1X∈T
Umul =Dmul.
Altogether, Dmul is a multiplicative subset ofMorC.
Moreover, given an arbitrary multiplicative subset U of MorC, we have Dmul ⊆ U by definition ofDmul, soDmul is in fact the smallest multiplicative subset ofMorC.
(ii) This is proven analogously to (i).
(iii) This is proven analogously to (i).
(iv) As SatD ∈ Usat, we haveDsat =TUsat ⊆SatD. Moreover, for allU ∈ Usat, we have D ⊆U and therefore SatD⊆SatU =U by proposition (1.31) and the saturatedness ofU. Thus we also have SatD⊆T
Usat =Dsat.
Altogether, we have Dsat = SatD. In particular, Dsat is a saturated subset of MorC by corol-lary (1.33).
Moreover, given an arbitrary saturated subsetU of MorC, we have Dsat ⊆U by definition ofDsat, so Dsat is in fact the smallest saturated subset of MorC.
(b) By proposition (1.43), we have U ⊇ Umul⊇ Ussat⊇ Uwsat⊇ Usat
and therefore
\U ⊆\
Umul⊆\
Ussat⊆\
Uwsat⊆\ Usat
that is,
D⊆Dmul⊆Dssat⊆Dwsat⊆Dsat = SatD.
(c) This follows from (b) and proposition (1.32).