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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 22nd 2010
   • PermaLink
   Author: Urs
   Format: Markdowncreated [[rationalization]] even though it overlaps with the material at [[rational topological space]]

   created rationalization even though it overlaps with the material at rational topological space

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 22nd 2010
   • (edited Apr 22nd 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded in a new section <a href="http://ncatlab.org/nlab/show/rationalization#Properties">Properties</a> a proof (or what I think is a proof) that rationalization preserves homotopy pullbacks of objects of finite type. What I type there is supposed to be a more or less immediate re-packaging of a technical result due to Halperin-Thomas, which appears in Hess's review in a polished form somewhat more to the point than Halperin-Thomas's discussion. My reformulation is supposed to make it even more to the point. But check.

   added in a new section Properties a proof (or what I think is a proof) that rationalization preserves homotopy pullbacks of objects of finite type.

   What I type there is supposed to be a more or less immediate re-packaging of a technical result due to Halperin-Thomas, which appears in Hess’s review in a polished form somewhat more to the point than Halperin-Thomas’s discussion. My reformulation is supposed to make it even more to the point.

   But check.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 22nd 2010
   • (edited Apr 22nd 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded comments and links to [[rationalization]] on how Toen's theory of [[rational homotopy theory in an (infinity,1)-topos]] provides another way to regard rationalization is a localization of $\infty Grpd$/$Top$. Wanted to further expand on this, but am running out of time now...

   added comments and links to rationalization on how Toen’s theory of rational homotopy theory in an (infinity,1)-topos provides another way to regard rationalization is a localization of $\infty Grpd$/$Top$.

   Wanted to further expand on this, but am running out of time now…

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeAug 23rd 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to * [[Aldridge Bousfield]], [[Victor Gugenheim]], Def. 11.1 in: _[[On PL deRham theory and rational homotopy type]]_, Memoirs of the AMS, vol. 179 (1976) ([ams:memo-8-179](https://bookstore.ams.org/memo-8-179)) <a href="https://ncatlab.org/nlab/revision/diff/rationalization/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/rationalization/14">v14</a>, <a href="https://ncatlab.org/nlab/show/rationalization">current</a>

   added pointer to

   • Aldridge Bousfield, Victor Gugenheim, Def. 11.1 in: On PL deRham theory and rational homotopy type, Memoirs of the AMS, vol. 179 (1976) (ams:memo-8-179)

   diff, v14, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeAug 23rd 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded statement of rationalization via PL de Rham theory, by the [[fundamental theorem of dg-algebraic rational homotopy theory]] <a href="https://ncatlab.org/nlab/revision/diff/rationalization/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/rationalization/14">v14</a>, <a href="https://ncatlab.org/nlab/show/rationalization">current</a>

   added statement of rationalization via PL de Rham theory, by the fundamental theorem of dg-algebraic rational homotopy theory

   diff, v14, current

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeAug 23rd 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Aldridge Bousfield]], [[Daniel Kan]], p. 133-140 in: _Homotopy Limits, Completions and Localizations_, Lecture Notes in Mathematics Vol. 304, Springer 1972 ([doi:10.1007/978-3-540-38117-4](https://doi.org/10.1007/978-3-540-38117-4)) <a href="https://ncatlab.org/nlab/revision/diff/rationalization/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/rationalization/14">v14</a>, <a href="https://ncatlab.org/nlab/show/rationalization">current</a>

   added pointer to:

   • Aldridge Bousfield, Daniel Kan, p. 133-140 in: Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics Vol. 304, Springer 1972 (doi:10.1007/978-3-540-38117-4)

   diff, v14, current

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJul 26th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to * [[Tyler Lawson]], Example 8.12 in: *An introduction to Bousfield localization* ([arXiv:2002.03888](https://arxiv.org/abs/2002.03888)) (Incidentally, the abstract says this is "part of an upcoming book", without further details. Might it be for *[[Stable categories and structured ring spectra ]]*?) <a href="https://ncatlab.org/nlab/revision/diff/rationalization/21">diff</a>, <a href="https://ncatlab.org/nlab/revision/rationalization/21">v21</a>, <a href="https://ncatlab.org/nlab/show/rationalization">current</a>

   added pointer to

   • Tyler Lawson, Example 8.12 in: An introduction to Bousfield localization (arXiv:2002.03888)

   (Incidentally, the abstract says this is “part of an upcoming book”, without further details. Might it be for Stable categories and structured ring spectra?)

   diff, v21, current

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJul 26th 2021
   • (edited Jul 26th 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexthanks to Charles for confirming ([here](https://nforum.ncatlab.org/discussion/13233/stable-categories-and-structured-ring-spectra/?Focus=94175#Comment_94175)) so I have expanded this out to: * [[Tyler Lawson]], Example 8.12 in: *An introduction to Bousfield localization* ([arXiv:2002.03888](https://arxiv.org/abs/2002.03888)) in: [[Andrew J. Blumberg]], [[Teena Gerhardt]], [[Michael A. Hill]] (eds.), *[[Stable categories and structured ring spectra]]* MSRI Book Series, Cambridge University Press <a href="https://ncatlab.org/nlab/revision/diff/rationalization/22">diff</a>, <a href="https://ncatlab.org/nlab/revision/rationalization/22">v22</a>, <a href="https://ncatlab.org/nlab/show/rationalization">current</a>

   thanks to Charles for confirming (here) so I have expanded this out to:

   • Tyler Lawson, Example 8.12 in: An introduction to Bousfield localization (arXiv:2002.03888) in: Andrew J. Blumberg, Teena Gerhardt, Michael A. Hill (eds.), Stable categories and structured ring spectra MSRI Book Series, Cambridge University Press

   diff, v22, current

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeJul 16th 2022
   • (edited Jul 16th 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have expanded a little at *[Rationalization of a homotopy type](https://ncatlab.org/nlab/show/rationalization#RationalizationOfAHomotopyType)*, adding the Bousfield-Kan formula for rationalization via degreewise Malcev completion of the simplicial loop group, both in the simply/connected/absolute case as well as in the general/relative case. <a href="https://ncatlab.org/nlab/revision/diff/rationalization/28">diff</a>, <a href="https://ncatlab.org/nlab/revision/rationalization/28">v28</a>, <a href="https://ncatlab.org/nlab/show/rationalization">current</a>

   I have expanded a little at Rationalization of a homotopy type, adding the Bousfield-Kan formula for rationalization via degreewise Malcev completion of the simplicial loop group, both in the simply/connected/absolute case as well as in the general/relative case.

   diff, v28, current