Higher-dimensional Calculus of Fractions

In their 1967 book "Calculus of Fractions and Homotopy Theory", P.
Gabriel and M. Zisman introduced calculus of fractions as a tool for
understanding the localization of a category at a class of weak
equivalences. While powerful, the condition of calculus of fractions is
quite restrictive and it is rarely satisfied in various homotopical
settings, like model categories or Brown's categories of fibrant
objects, where one instead has *homotopy* calculus of fractions.

This talk is based on joint work with D. Carranza and Z. Lindsey, which
aims to reconcile the two. We define calculus of fractions for
quasicategories and give a workable model for the localization of a
marked quasicategory satisfying our condition. Although we have already
found several applications of this result, I would be very interested in
getting feedback from the audience and exploring new applications.