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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeJul 5th 2018
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Author: Urs
Format: MarkdownItexjust the evident minimum <a href="https://ncatlab.org/nlab/revision/localization+of+a+2-category/1">v1</a>, <a href="https://ncatlab.org/nlab/show/localization+of+a+2-category">current</a>

just the evident minimum

v1, current
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeJul 5th 2018
- (edited Jul 5th 2018)
- PermaLink
Author: Urs
Format: MarkdownItexextracted a digest of the first theorem of [Renaudin 06](https://ncatlab.org/nlab/show/localization+of+a+2-category#Renaudin06): *** The [[2-localization]] $$ CombModCat\big[\{QuillenEquivalences\}^{-1}\big] $$ of the [[2-category]] of [[combinatorial model categories]] at the [[Quillen equivalences]] exists. Up to [[equivalence of 2-categories]], it has the same [[objects]] as $CombModCat$ and for any $\mathcal{C}, \mathcal{D} \in CombModCat$ its [[hom-category]] is the [[localization of categories]] $$ CombModCat\big[\{QuillenEquivalences\}^{-1}\big](\mathcal{C}, \mathcal{D}) \;\simeq\; ModCat( \mathcal{C}^p, \mathcal{D}^p )\big[\{QuillenHomotpies\}^{-1}\big] $$ of the category of [[left Quillen functors]] and [[natural transformations]] between local presentations $\mathcal{C}^p$ and $\mathcal{D}^p$ at those [[natural transformation]] that on [[cofibrant objects]] have components that are [[weak equivalences]] ("Quillen homotopies"). <a href="https://ncatlab.org/nlab/revision/diff/localization+of+a+2-category/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/localization+of+a+2-category/2">v2</a>, <a href="https://ncatlab.org/nlab/show/localization+of+a+2-category">current</a>

extracted a digest of the first theorem of Renaudin 06:

The 2-localization
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>CombModCat</mi><mo maxsize="1.2em" minsize="1.2em">[</mo><mo stretchy="false">{</mo><mi>QuillenEquivalences</mi><msup><mo stretchy="false">}</mo> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo maxsize="1.2em" minsize="1.2em">]</mo></mrow><annotation encoding="application/x-tex"> CombModCat\big[\{QuillenEquivalences\}^{-1}\big] </annotation></semantics></math>$
of the 2-category of combinatorial model categories at the Quillen equivalences exists. Up to equivalence of 2-categories, it has the same objects as $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>CombModCat</mi></mrow><annotation encoding="application/x-tex">CombModCat</annotation></semantics></math>$ and for any $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>𝒞</mi><mo>,</mo><mi>𝒟</mi><mo>∈</mo><mi>CombModCat</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}, \mathcal{D} \in CombModCat</annotation></semantics></math>$ its hom-category is the localization of categories
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>CombModCat</mi><mo maxsize="1.2em" minsize="1.2em">[</mo><mo stretchy="false">{</mo><mi>QuillenEquivalences</mi><msup><mo stretchy="false">}</mo> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo maxsize="1.2em" minsize="1.2em">]</mo><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>𝒟</mi><mo stretchy="false">)</mo><mspace width="0.27778em"/><mo>≃</mo><mspace width="0.27778em"/><mi>ModCat</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mi>p</mi></msup><mo>,</mo><msup><mi>𝒟</mi> <mi>p</mi></msup><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">[</mo><mo stretchy="false">{</mo><mi>QuillenHomotpies</mi><msup><mo stretchy="false">}</mo> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo maxsize="1.2em" minsize="1.2em">]</mo></mrow><annotation encoding="application/x-tex"> CombModCat\big[\{QuillenEquivalences\}^{-1}\big](\mathcal{C}, \mathcal{D}) \;\simeq\; ModCat( \mathcal{C}^p, \mathcal{D}^p )\big[\{QuillenHomotpies\}^{-1}\big] </annotation></semantics></math>$
of the category of left Quillen functors and natural transformations between local presentations $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>𝒞</mi> <mi>p</mi></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^p</annotation></semantics></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>𝒟</mi> <mi>p</mi></msup></mrow><annotation encoding="application/x-tex">\mathcal{D}^p</annotation></semantics></math>$ at those natural transformation that on cofibrant objects have components that are weak equivalences (“Quillen homotopies”).

diff, v2, current
- CommentRowNumber3.
- CommentAuthorzskoda
- CommentTimeSep 25th 2022
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Author: zskoda
Format: MarkdownItexAdded several references. Added my own (draft of the) MR review [[zoranskoda:MR4112764]] M. E. Descotte, E. J. Dubuc, M. Szyld , _A localization of bicategories via homotopies_, Theory and Applications of Categories __35__, 2020, No. 23, 845--874, [TAC](http://tac.mta.ca/tac/volumes/35/23/35-23abs.html) Also removed word "evident" from the idea section as there are several quite different nontrivial versions and some set theoretical issues when studied carefully. Added link [[localization of an enriched category]]. <a href="https://ncatlab.org/nlab/revision/diff/localization+of+a+2-category/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/localization+of+a+2-category/8">v8</a>, <a href="https://ncatlab.org/nlab/show/localization+of+a+2-category">current</a>

Added several references.

Added my own (draft of the) MR review

MR4112764 (zoranskoda) M. E. Descotte, E. J. Dubuc, M. Szyld , A localization of bicategories via homotopies, Theory and Applications of Categories 35, 2020, No. 23, 845–874, TAC

Also removed word “evident” from the idea section as there are several quite different nontrivial versions and some set theoretical issues when studied carefully.

Added link localization of an enriched category.

diff, v8, current

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