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- W4289551012 abstract "In this paper we study a 2-dimensional version of Quillen's homotopy category construction. Given a category $mathscr{A}$ and a class of morphisms $Sigma subset mathscr{A}$ containing the identities, we construct a 2-category $mathcal{H}o(mathscr{A})$ obtained by the addition of 2-cells determined by homotopies. A salient feature here is the use of a novel notion of cylinder introduced in cite{e.d.2}. The inclusion 2-functor $mathscr{C} longrightarrow mathcal{H}o(mathscr{A})$ has a universal property which implies that it will be the 2-localization of $mathscr{A}$ at $Sigma$ as soon as the arrows of $Sigma$ become equivalences in $mathcal{H}o(mathscr{A})$. This is then used to obtain 2-localizations of a model category $mathscr{A}{C}$, with $Sigma = mathcal{W}$, the weak equivalences, and $mathscr{A} = mathscr{C}_{fc}$, the full subcategory of fibrant-cofibrant objects, as well as with $mathscr{A} = mathscr{C}$. The set of connected components of the hom categories yields Quillen's results. We follow the general lines established in cite{e.d.2}, cite{e.d.} for model bicategories. The development here is not just the examination of the general theory in a particular case. It is not concerned with and avoids the problems which arise when dealing with non invertible 2-cells. Also, the use here of functorial factorization adds further simplifications by eliminating the need of pseudofunctors. New proofs are produced which are not a mere simplified adaptation of the ones of the general case." @default.
- W4289551012 created "2022-08-03" @default.
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- W4289551012 date "2022-07-30" @default.
- W4289551012 modified "2023-09-27" @default.
- W4289551012 title "The 2-localization of a model category" @default.
- W4289551012 doi "https://doi.org/10.48550/arxiv.2208.00314" @default.
- W4289551012 hasPublicationYear "2022" @default.
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