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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJul 5th 2018

just the evident minimum

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJul 5th 2018
• (edited Jul 5th 2018)

extracted a digest of the first theorem of Renaudin 06:

$CombModCat\big[\{QuillenEquivalences\}^{-1}\big]$

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”).