A class of morphisms \( \Sigma \) in a category \( C \) admits a calculus of fractions if it satisfies certain rules that ensure the construction of its localization \( C[\Sigma-1] \) can be carried out in a particular simple way [1]. In [2], Dorette Pronk extended the Zisman-Gabriel theory of categories of fractions to bicategories, considering equivalences instead of invertible \( 1 \)-cells. In [3], the classical calculus of fractions was generalized in another direction: in the context of order-enriched categories, left adjoint right inverses (laris) took the place of invertible morphisms; the universal functor from \( C \) to the category of (lax) fractions, \( C[\Sigma^*] \), not only maps morphisms in \( \Sigma \) to laris but also takes certain squares to Beck-Chevalley squares. In this talk, I will present a generalization of this calculus of lax fractions to \( 2 \)-categories, which also generalizes Pronk's calculus, and describe a corresponding "localization", a bicategory of lax fractions. This is joint work with Graham Manuell.

References:

[1] P. Gabriel and M. Zisman, Calculus of Fractions and Homotopy Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 2. Folge, vol. 35, Springer, 1967.

[2] D. A. Pronk. Etendues and stacks as bicategories of fractions, Compos. Math. 102 (1996), no. 3, 243-303.

[3] L. Sousa. A calculus of lax fractions. J. Pure Appl. Algebra 221 (2017), no. 2, 422-448.