Zoran Skoda


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

18N10 2-categories, bicategories, double categories

18E35 Localization of categories, calculus of fractions

18N40 Homotopical algebra, Quillen model categories, derivators

Given an 1-category CC and a class WW of arrows, a localization functor CC[W 1]C\to C[W^{-1}] is a universal among all functors sending elements of WW to isomorphisms. According to P. Gabriel and M. Zisman [Calculus of fractions and homotopy theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer-Verlag 1967, MR0210125], if we neglect set theoretical size issues, a localization can always be constructed using zig-zags of arrows in CC and formal inverses of arrows in WW. Localization is more manageable and well-behaved if WW admits a calculus of fractions. Motivated by relating certain bicategories of geometric examples (say, of stacks and of groupoids presenting them), D. Pronk [Etendues and stacks as bicategories of fractions, Compositio Math. 102 (1996), no. 3, 243303, MR1401424] has generalized the calculus of fractions to bicategories; the resulting localization is biinitial among pseudofunctors sending a class WW of 1-cells to equivalences.

In the article under review, another construction of a bicategorical localization is presented. For a family WW of 1-cells containing all identities in a bicategory CC, a bicategorical generalization of Quillen cylinder objects, relative to WW, is defined. Using cylinders, a notion of a left homotopy from ff to gg, where f,g:XYf,g\colon X\to Y are 1-cells in CC is defined. Given two homotopies fghf\Rightarrow g\Rightarrow h there isn’t necessarily a homotopy fhf \Rightarrow h representing their composition, hence their formal compositions are considered. Heuristically, the homotopies are thought of as entities that would be actual 2-cells if the arrows of WW were (quasi)equivalences, where f:XYf\colon X\to Y is called a quasiequivalence if for any object ZZ, precomposition C(Y,Z)f *C(X,Z)C(Y,Z)\stackrel{f^*}\to C(X,Z) and postcomposition C(Z,X)f *C(Z,Y)C(Z,X)\stackrel{f_*}\to C(Z,Y) are full and faithful. Quasiequivalences preserve and reflect invertible 2-cells. If the arrows of WW are quasiequivalences, then any homotopy HH induces a 2-cell H^\hat{H} (an instance of Definition 3.12 in the paper); each 2-cell μ\mu is induced by some homotopy. Likewise, for a pseudofunctor F:CDF\colon C\to D sending elements of WW to quasiequivalences, any homotopy HH in CC induces a 2-cell FH^\widehat{F H} in DD. The localization 2-functor i:CHo(C,W)i\colon C\to\mathrm{Ho}(C,W) tautologically sends each kk-cell in CC to its copy in the bicategory Ho(C,W)\mathrm{Ho}(C,W) extending CC by adding a 2-cell for each (new) equivalence class of formal compositions of homotopies. The equivalence is semantically defined: two formal compositions are equivalent if the compositions of induced 2-cells along any 2-functor F:CDF\colon C\to D sending elements of WW to quasiequivalences are the same. The problem of describing this relation syntactically in some important cases is considered in the appendix to the paper. At the end, it is shown that if WW satisfies the 3 for 2 property and split arrows (sections and retractions) generate WW in a suitable 2-categorical way, then i:CHo(C,W)i\colon C\to\mathrm{Ho}(C,W) is a strict localization (precomposing with ii is not only a biequivalence but an isomorphism).

The motivating example of such a localization is a homotopy bicategory of a model bicategory, studied by the same authors [Model bicategories and their homotopy bicategories, arXiv:1805.07749]. A model structure on a bicategory is a choice of 3 classes of 1-cells – fibrations, cofibrations and weak equivalences – with axioms of a model category satisfied up to iso-2-cells filling the diagrams involved. While model categories present a class of (,1)(\infty,1)-categories, model bicategories are viewed as presentations of some (,2)(\infty,2)-categories. Intuitively, (,2)(\infty,2)-categories are categories (weakly) enriched in (,1)(\infty,1)-categories, hence passing to the homotopy bicategory should involve adding (only) 2-cells.

Last revised on March 17, 2021 at 20:19:31. See the history of this page for a list of all contributions to it.