Calculus of Fractions

The purpose of abstract homotopy theory is to provide category theoretic constructions which are compatible with suitable notions of weak homotopy equivalences. One can revisit the concepts that have lead to Quillen's notion of model category as devices to compute mapping spaces of localizations in terms of Kan extensions, which, in turns provide tools to compute (co)limits in localized infinity-categories as homotopy (co)limits. From there, one can produce a perfect dictionary between (co)complete infinity-categories and their models together with a good theory of derived functors as Kan extensions, revisiting the work of Szumiło, Kapulkin and Mazel-Gee. Reformulating homotopy theory properly as suggested above, using mainly the language of Kan extensions is not only a pleasant way to revisit classical constructions (although that would be good enough), but also a way to internalize homotopy theory in any higher topos (in fact in any directed type theory). This will have applications, for instance, to formulate and prove the universal property of Morel and Voevodsky's motivic homotopy theory (possibly formulated within derived geometry, thus generalizing the contributions of Drew and Gallauer), as well as to study condensed/pyknotic mathematics (e.g. one can see the pro-étale topos of a scheme as a condensed/pyknotic presheaf topos on the the associated Galois category constructed by Barwick, Glasman and Haine).