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The terminology is partly (un)fortunate. Namely, in practical localization theory we often consider finite compositions of localization functors and call those as well iterated or consecutive localizations. If the localization functors have left or right adjoints then the compositions of localization functors are localizations as well, but in general this is not true (there are even very small counterexamples). Here the situation is different, one may even have at every but the colimit stage a localization, it seems. It takes some effort to construct examples of iterated localizations in Joyal’s sense which are themselves not localizations and that the problem is not at a finite stage; I lost my hand-written notes where I constructed some sort of an easy but somewhat artificial example, I hope to find those notes at some point.
I was thinking a bit more. It is not difficult to have examples of iterated localizations requiring infinite number of stages, but it seems to me that in order to have an iterative localization which is not a localization functor that at finite stages one already fails to have a composition to be a localization functor. So in some sense, nothing essentially new happens in the colimit.
In other words, iterated localization in Joyal’s sense is a colimit of finite compositions of (strict) localizations.
Now let B be a finitely generated subcategory of the domain C which is closed under inverses which exist in C. Define Bn+1 as the smallest subcategory of Cn+1 containing the image of Bn and closed under inverses which exist in Cn+1 and and similarly Bω be the smallest subcategory of Cω closed under inverses and containing the colimit of Bn. I conjecture that there is n0 such that the canonical map Bn0→Bω is an isomorphism of categories. This is not true for the entire Cω unless it is finitely generated itself. By finite generation we mean that there is a finite set S of morphisms so that any morphism is a composition of some sequence of composable morphisms in S.
In other words, iterated localization in Joyal’s sense is a colimit of finite compositions of (strict) localizations.
In one direction, it is trivial, as Joyal’s construction is providing a sequence of localizations with K an equivalence.
Now take any other colimit P:C→colimCSn of composition of strict localizations with the universal cocone
CQ1→CS1=C[S−11]Q2→C[S−11][S−12]Q3→…→colimCSnwith components of the cocone Pm:CSm→colimCSn and construct the corresponding Joyal’s construction
C→C1→C2→…→colimCnK→CSnout of the “composition” P=P0.
At stage 1, the inverting set Σ such that C→C1≅C[Σ−11] is such that S1⊂Σ1, hence one has a unique functor r1:C[S−11]→C[Σ−11] such that r1∘Q1 is the localization C→C[Σ−11] and by construction P=F1∘r1∘Q1=P1∘Q1 hence by universal property P1=F1∘r1. Now start again the same procedure for P1 and so on. We need to show that K is an equivalence. The canonical functor colimCSn→colimCm is K. We claim that the inverse is simply colimrn; the equality ri∘Fi=Pi after taking the colimit implies that this is the inverse.
Therefore, P is an iterated localization in the sense of Joyal.
Ad 4: no, I was wrong about the case with finite generation – it seems, that even in a noncommutative monoid, this may be wrong. Take a free monoid on two letters, a and b and localize at ab. Then the expression c=b(ab)−1a is of course not the identity e as we have no 2-sided inverses of a and b. It holds cb=b, ac=a and cc=c but so what, as c=ce≠e. In fact, a has a right inverse b(ab)−1 and b has a left inverse (ab)−1a and that is it.
Now one proceeds with inverting c and then c1=ac−1b, then c2=bc−11a, c3=ac−12b and so on c2n+1=ac−12nb, c2n=bc−12n−1a, and take the colimit. All these expressions do not appear at any previous stages. Thus the colimit is not achieved at any finite stage either. Moreover, it seems that the colimit is not finitely generated.
I think we should define iterated localization simply as at most countable (in the sense of a colimit) composition of localization functors, period. This is equivalent to the Joyal’s more complicated definition by 5.
The Joyal’s construction how it appears in the context of a factorization system is a justification of its importance and a rather special case of its transfinite composition presentation.
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