Zoran Skoda MR4112764

MR4112764

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 $C$ and a class $W$ of arrows, a localization functor $C\to C[W^{-1}]$ is a universal among all functors sending elements of $W$ 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 $C$ and formal inverses of arrows in $W$. Localization is more manageable and well-behaved if $W$ 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 $W$ of 1-cells to equivalences.

In the article under review, another construction of a bicategorical localization is presented. For a family $W$ of 1-cells containing all identities in a bicategory $C$, a bicategorical generalization of Quillen cylinder objects, relative to $W$, is defined. Using cylinders, a notion of a left homotopy from $f$ to $g$, where $f,g\colon X\to Y$ are 1-cells in $C$ is defined. Given two homotopies $f\Rightarrow g\Rightarrow h$ there isn’t necessarily a homotopy $f \Rightarrow h$ 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 $W$ were (quasi)equivalences, where $f\colon X\to Y$ is called a quasiequivalence if for any object $Z$, precomposition $C(Y,Z)\stackrel{f^*}\to C(X,Z)$ and postcomposition $C(Z,X)\stackrel{f_*}\to C(Z,Y)$ are full and faithful. Quasiequivalences preserve and reflect invertible 2-cells. If the arrows of $W$ are quasiequivalences, then any homotopy $H$ induces a 2-cell $\hat{H}$ (an instance of Definition 3.12 in the paper); each 2-cell $\mu$ is induced by some homotopy. Likewise, for a pseudofunctor $F\colon C\to D$ sending elements of $W$ to quasiequivalences, any homotopy $H$ in $C$ induces a 2-cell $\widehat{F H}$ in $D$. The localization 2-functor $i\colon C\to\mathrm{Ho}(C,W)$ tautologically sends each $k$-cell in $C$ to its copy in the bicategory $\mathrm{Ho}(C,W)$ extending $C$ 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 $F\colon C\to D$ sending elements of $W$ 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 $W$ satisfies the 3 for 2 property and split arrows (sections and retractions) generate $W$ in a suitable 2-categorical way, then $i\colon C\to\mathrm{Ho}(C,W)$ is a strict localization (precomposing with $i$ 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 $(\infty,1)$-categories, model bicategories are viewed as presentations of some $(\infty,2)$-categories. Intuitively, $(\infty,2)$-categories are categories (weakly) enriched in $(\infty,1)$-categories, hence passing to the homotopy bicategory should involve adding (only) 2-cells.