Zoran Skoda MR4483623

Dorette Pronk, Laura Scull, Bicategories of fractions revisited: towards small homs and canonical 2-cells, Theory and Applications of Categories 38:24 (2022) 913-1014 MR4483623

Motivated by homotopy theory, as well as examples from the study of Abelian categories, P. Gabriel and M. Zisman extended Ore’s method of universally forming monoids and rings of (equivalence classes of left or right) fractions with given denominator sets to categories [Calculus of fractions and homotopy theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer-Verlag 1967, MR0210125]. If a set $W$ of morphisms in a category $\mathcal{C}$ satisfies analogues of left Ore conditions, $W$ is said to admit a left calculus of fractions and a new category $W^{-1}\mathcal{C}$ is formed with a universal $W$-inverting (localization) functor $\mathcal{C}\to W^{-1}\mathcal{C}$ which is moreover exact and admits a fully faithful right adjoint. Dually, in the case of right fractions, the universal (colocalization) functor admits a fully faithful left adjoint. Bicategories of stack-like objects, like orbispaces, may be similarly obtained from simpler bicategories of objects which present stacks, for example étale groupoids, by inverting weak equivalences. Various formalisms have been employed to the effect of this inversion (butterflies, Hilsum-Skandalis maps, internal anafunctors) by B. Noohi, D. Roberts and others. In her earlier work, the first author introduced a direct generalization of the right calculus of fractions for bicategories [D. Pronk, Etendues and stacks as bicategories of fractions, Compositio Math. 102 (1996), no. 3, 243303, MR1401424]. Given a bicategory $\mathcal{B}$ and a set of 1-cells $W$ admitting a right calculus of fractions, she constructed a homomorphism of bicategories $\mathcal{B}\to \mathcal{B}[W^{-1}]$ which is biinitial among all $W$-inverting homomorphisms $\mathcal{B}\to\mathcal{D}$.

In practice, already in 1-categorical situation, and even more in bicategorical, one often needs to invert a proper class $W$ of 1-cells. While Gabriel and Zisman ignored set-theoretical issues, existence of $W^{-1}\mathcal{C}$ is well known in good cases like homotopy categories of model categories. In bicategorical case, for 1-cells one does not take equivalence classes of fractions, but naturally considers all fractions as distinct and explicitly forms 2-cells among them; the bicategory obtained is not necessariy locally small (having small Hom-categories). As we can not form equivalence classes, the definition of the composition also involves choices. {\em A priori}, these features point to the potential need for more set theoretical care than for the 1-categorical (co)localization. The paper under review refines the conditions for the bicategorical calculus of fractions and the construction of $\mathcal{B}[W^{-1}]$ to address the size issues and in related manner, when possible, to utilize more canonical choices for fractions. To this aim, the authors weaken the original axioms for the right bicategorical calculus of fractions, effectively leading to a choice of a possibly smaller subclass of $W$. The main difference in axioms is that the closedness of the class $W$ under composition is weakened: if $v,w\in W$ then they require only that $w\circ v\circ u\in W$ for some additional 1-cell $u$. Surely, if $w\circ v\circ u$ is to be inverted, then $w\circ v$ is, hence ommiting $w\circ v$ from $W$ (provided $u$ exists) does not change the saturation (the class of all 1-cells which are inverted), hence it should not essentially change the localization. It is proven that if there exists a weakly initial subset of $W$ containing all identities and closed under 2-isomorphisms, then one can choose a locally small subbicategory of the bicategory of fractions $\mathcal{B}[W^{-1}]$ for which the inclusion is a biequivalence. At the end of the paper, the construction is applied to orbispaces.