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    • CommentRowNumber1.
    • CommentAuthormattecapu
    • CommentTimeAug 26th 2024

    definition

    diff, v2, current

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeAug 28th 2024
    • (edited Aug 28th 2024)

    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.

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeAug 28th 2024

    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.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeAug 28th 2024

    In other words, iterated localization in Joyal’s sense is a colimit of finite compositions of (strict) localizations.

    Now let BB be a finitely generated subcategory of the domain CC which is closed under inverses which exist in CC. Define B n+1B_{n+1} as the smallest subcategory of C n+1C_{n+1} containing the image of B nB_n and closed under inverses which exist in C n+1C_{n+1} and and similarly B ωB_\omega be the smallest subcategory of C ωC_\omega closed under inverses and containing the colimit of B nB_n. I conjecture that there is n 0n_0 such that the canonical map B n 0B ωB_{n_0}\to B_\omega is an isomorphism of categories. This is not true for the entire C ωC_\omega unless it is finitely generated itself. By finite generation we mean that there is a finite set SS of morphisms so that any morphism is a composition of some sequence of composable morphisms in SS.

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeAug 28th 2024
    • (edited Aug 28th 2024)

    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 KK an equivalence.

    Now take any other colimit P:CcolimC n SP:C\to colim C^S_n of composition of strict localizations with the universal cocone

    CQ 1C 1 S=C[S 1 1]Q 2C[S 1 1][S 2 1]Q 3colimC n SC\stackrel{Q_1}\to C^S_1 = C[S^{-1}_1]\stackrel{Q_2}\to C[S^{-1}_1][S^{-1}_2]\stackrel{Q_3}\to\ldots\to colim C^S_n

    with components of the cocone P m:C m ScolimC n SP_m:C^S_m \to colim C^S_n and construct the corresponding Joyal’s construction

    CC 1C 2colimC nKC n SC\to C_1\to C_2\to\ldots \to colim C_n\stackrel{K}\to C^S_n

    out of the “composition” P=P 0P = P_0.

    At stage 1, the inverting set Σ\Sigma such that CC 1C[Σ 1 1]C\to C_1\cong C[\Sigma_1^{-1}] is such that S 1Σ 1S_1\subset\Sigma_1, hence one has a unique functor r 1:C[S 1 1]C[Σ 1 1]r_1: C[S^{-1}_1]\to C[\Sigma_1^{-1}] such that r 1Q 1r_1\circ Q_1 is the localization CC[Σ 1 1]C\to C[\Sigma_1^{-1}] and by construction P=F 1r 1Q 1=P 1Q 1P = F_1\circ r_1\circ Q_1 = P_1\circ Q_1 hence by universal property P 1=F 1r 1P_1 = F_1\circ r_1. Now start again the same procedure for P 1P_1 and so on. We need to show that KK is an equivalence. The canonical functor colimC n ScolimC mcolim C^S_n\to colim C_m is KK. We claim that the inverse is simply colimr ncolim r_n; the equality r iF i=P ir_i\circ F_i = P_i after taking the colimit implies that this is the inverse.

    Therefore, PP is an iterated localization in the sense of Joyal.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeAug 28th 2024
    • (edited Aug 28th 2024)

    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, aa and bb and localize at aba b. Then the expression c=b(ab) 1ac = b(a b)^{-1} a is of course not the identity ee as we have no 2-sided inverses of aa and bb. It holds cb=bc b = b, ac=aa c = a and cc=cc c = c but so what, as c=ceec = c e \neq e. In fact, aa has a right inverse b(ab) 1b (a b)^{-1} and bb has a left inverse (ab) 1a(a b)^{-1} a and that is it.

    Now one proceeds with inverting cc and then c 1=ac 1bc_1 = a c^{-1} b, then c 2=bc 1 1ac_2 = b c^{-1}_1 a, c 3=ac 2 1bc_3 = a c^{-1}_2 b and so on c 2n+1=ac 2n 1bc_{2 n +1 } = a c_{2 n}^{-1} b, c 2n=bc 2n1 1ac_{2 n} = b c_{2 n -1}^{-1} a, 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.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeAug 29th 2024

    Have been making various cosmetic edits to the entry (already so on Aug 26)

    diff, v6, current

    • CommentRowNumber8.
    • CommentAuthorHurkyl
    • CommentTimeAug 29th 2024

    Corrected typo in definition (“of” -> “if”)

    diff, v7, current

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeAug 30th 2024
    • (edited Aug 30th 2024)

    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.