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1. • CommentRowNumber1.
   • CommentAuthormattecapu
   • CommentTimeAug 26th 2024
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   Author: mattecapu
   Format: MarkdownItexdefinition <a href="https://ncatlab.org/nlab/revision/diff/iterated+localization/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/iterated+localization/2">v2</a>, <a href="https://ncatlab.org/nlab/show/iterated+localization">current</a>

   definition

   diff, v2, current

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeAug 28th 2024
   • (edited Aug 28th 2024)
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   Author: zskoda
   Format: MarkdownItexThe 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeAug 28th 2024
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   Author: zskoda
   Format: MarkdownItexI 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.

   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.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeAug 28th 2024
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   Author: zskoda
   Format: MarkdownItexIn 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 $B_{n+1}$ as the smallest subcategory of $C_{n+1}$ containing the image of $B_n$ and closed under inverses which exist in $C_{n+1}$ and and similarly $B_\omega$ be the smallest subcategory of $C_\omega$ closed under inverses and containing the colimit of $B_n$. I conjecture that there is $n_0$ such that the canonical map $B_{n_0}\to B_\omega$ is an isomorphism of categories. This is not true for the entire $C_\omega$ 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.

   Now let $B$ be a finitely generated subcategory of the domain $C$ which is closed under inverses which exist in $C$. Define $B_{n+1}$ as the smallest subcategory of $C_{n+1}$ containing the image of $B_n$ and closed under inverses which exist in $C_{n+1}$ and and similarly $B_\omega$ be the smallest subcategory of $C_\omega$ closed under inverses and containing the colimit of $B_n$. I conjecture that there is $n_0$ such that the canonical map $B_{n_0}\to B_\omega$ is an isomorphism of categories. This is not true for the entire $C_\omega$ 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 $S$.

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeAug 28th 2024
   • (edited Aug 28th 2024)
   • PermaLink
   Author: zskoda
   Format: MarkdownItex> 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 $P:C\to colim C^S_n$ of composition of strict localizations with the universal cocone $$C\stackrel{Q_1}\to C^S_1 = C[S^{-1}_1]\stackrel{Q_2}\to C[S^{-1}_1][S^{-1}_2]\stackrel{Q_3}\to\ldots\to colim C^S_n$$ with components of the cocone $P_m:C^S_m \to colim C^S_n$ and construct the corresponding Joyal's construction $$C\to C_1\to C_2\to\ldots \to colim C_n\stackrel{K}\to C^S_n$$ out of the "composition" $P = P_0$. At stage 1, the inverting set $\Sigma$ such that $C\to C_1\cong C[\Sigma_1^{-1}]$ is such that $S_1\subset\Sigma_1$, hence one has a unique functor $r_1: C[S^{-1}_1]\to C[\Sigma_1^{-1}]$ such that $r_1\circ Q_1$ is the localization $C\to C[\Sigma_1^{-1}]$ and by construction $P = F_1\circ r_1\circ Q_1 = P_1\circ Q_1$ hence by universal property $P_1 = F_1\circ r_1$. Now start again the same procedure for $P_1$ and so on. We need to show that $K$ is an equivalence. The canonical functor $colim C^S_n\to colim C_m$ is $K$. We claim that the inverse is simply $colim r_n$; the equality $r_i\circ F_i = P_i$ after taking the colimit implies that this is the inverse. Therefore, $P$ is an iterated localization in the sense of Joyal.

   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 $P:C\to colim C^S_n$ of composition of strict localizations with the universal cocone

   $$C\stackrel{Q_1}\to C^S_1 = C[S^{-1}_1]\stackrel{Q_2}\to C[S^{-1}_1][S^{-1}_2]\stackrel{Q_3}\to\ldots\to colim C^S_n$$

   with components of the cocone $P_m:C^S_m \to colim C^S_n$ and construct the corresponding Joyal’s construction

   $$C\to C_1\to C_2\to\ldots \to colim C_n\stackrel{K}\to C^S_n$$

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

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

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

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeAug 28th 2024
   • (edited Aug 28th 2024)
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   Author: zskoda
   Format: MarkdownItexAd 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 $a b$. Then the expression $c = b(a b)^{-1} a$ is of course not the identity $e$ as we have no 2-sided inverses of $a$ and $b$. It holds $c b = b$, $a c = a$ and $c c = c$ but so what, as $c = c e \neq e$. In fact, $a$ has a right inverse $b (a b)^{-1}$ and $b$ has a left inverse $(a b)^{-1} a$ and that is it. Now one proceeds with inverting $c$ and then $c_1 = a c^{-1} b$, then $c_2 = b c^{-1}_1 a$, $c_3 = a c^{-1}_2 b$ and so on $c_{2 n +1 } = a c_{2 n}^{-1} b$, $c_{2 n} = b c_{2 n -1}^{-1} a$, 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.

   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 $a b$. Then the expression $c = b(a b)^{-1} a$ is of course not the identity $e$ as we have no 2-sided inverses of $a$ and $b$. It holds $c b = b$, $a c = a$ and $c c = c$ but so what, as $c = c e \neq e$. In fact, $a$ has a right inverse $b (a b)^{-1}$ and $b$ has a left inverse $(a b)^{-1} a$ and that is it.

   Now one proceeds with inverting $c$ and then $c_1 = a c^{-1} b$, then $c_2 = b c^{-1}_1 a$, $c_3 = a c^{-1}_2 b$ and so on $c_{2 n +1 } = a c_{2 n}^{-1} b$, $c_{2 n} = b c_{2 n -1}^{-1} a$, 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.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeAug 29th 2024
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   Author: Urs
   Format: MarkdownItexHave been making various cosmetic edits to the entry (already so on [Aug 26](https://ncatlab.org/nlab/history/iterated+localization)) <a href="https://ncatlab.org/nlab/revision/diff/iterated+localization/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/iterated+localization/6">v6</a>, <a href="https://ncatlab.org/nlab/show/iterated+localization">current</a>

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

   diff, v6, current

8. • CommentRowNumber8.
   • CommentAuthorHurkyl
   • CommentTimeAug 29th 2024
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   Author: Hurkyl
   Format: MarkdownItexCorrected typo in definition ("of" -> "if") <a href="https://ncatlab.org/nlab/revision/diff/iterated+localization/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/iterated+localization/7">v7</a>, <a href="https://ncatlab.org/nlab/show/iterated+localization">current</a>

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

   diff, v7, current

9. • CommentRowNumber9.
   • CommentAuthorzskoda
   • CommentTimeAug 30th 2024
   • (edited Aug 30th 2024)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI 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.

   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.