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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2018

    just the evident minimum

    v1, current

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

    extracted a digest of the first theorem of Renaudin 06:

    The 2-localization

    CombModCat[{QuillenEquivalences} 1] 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 CombModCatCombModCat and for any 𝒞,𝒟CombModCat\mathcal{C}, \mathcal{D} \in CombModCat its hom-category is the localization of categories

    CombModCat[{QuillenEquivalences} 1](𝒞,𝒟)ModCat(𝒞 p,𝒟 p)[{QuillenHomotpies} 1] 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 𝒞 p\mathcal{C}^p and 𝒟 p\mathcal{D}^p at those natural transformation that on cofibrant objects have components that are weak equivalences (“Quillen homotopies”).

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeSep 25th 2022

    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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