Much has been said about inverting a class of morphisms in a category (see localization), and there are many different settings in which one wants to, and can, do this. Homotopical algebra is largely concerned with how to compute the homotopy category so it is locally small. One the other hand, we have simplicial localization which retains all the homotopy information and returns an -category.
If we have a 2-category with a notion of weak equivalence, one could localize the underlying 1-category in a way hopefully compatible with the 2-arrows, or extend the result fully into the 2-dimensional setting. In general this will require bicategories, and is the subject of the paper Etendues and stacks as bicategories of fractions by Dorette Pronk.
Let be a bicategory with a class of 1-cells. is said to admit a right calculus of fractions if it satisfies the following conditions
[2CF1.] contains all equivalences
a) is closed under composition
b) If and a iso-2-cell then
[2CF3.] For all , with there exists a 2-commutative square
[2CF4.] If is a 2-cell and there is a 1-cell and a 2-cell such that . Moreover: when is an iso-2-cell, we require to be an isomorphism too; when and form another such pair, there exist 1-cells such that and are in , and an iso-2-cell such that the following diagram commutes:
If is a category, then these axioms reduce to the ones of Gabriel and Zisman for a calculus of fractions.
Given such a setup, Pronk constructs the localization of at and the universal functor sending elements of to equivalences.
Let be a category with binary products and pullbacks together with a class of admissible maps .