Homotopical Patch Theory (Expanded Version) Carlo Angiuli ∗ Edward Morehouse ∗ Daniel R. Licata Carnegie Mellon University Carnegie Mellon University Wesleyan University
[email protected] [email protected] [email protected] Robert Harper ∗ Carnegie Mellon University
[email protected] Abstract In homotopy theory, one studies topological spaces by way of their Homotopy type theory is an extension of Martin-Löf type theory, points, paths (between points), homotopies (paths or continuous de- based on a correspondence with homotopy theory and higher cat- formations between paths), homotopies between homotopies (paths egory theory. In homotopy type theory, the propositional equality between paths between paths), and so on. In type theory, a space type becomes proof-relevant, and corresponds to paths in a space. corresponds to a type A. Points of a space correspond to elements This allows for a new class of datatypes, called higher inductive a; b : A. Paths in a space are represented by elements of the identity types, which are specified by constructors not only for points but type (propositional equality), which we notate p : a =A b. Homo- also for paths. In this paper, we consider a programming application topies between paths p and q correspond to elements of the iterated identity type p = q. The rules for the identity type allow one of higher inductive types. Version control systems such as Darcs are a=Ab based on the notion of patches—syntactic representations of edits to define the operations on paths that are considered in homotopy to a repository. We show how patch theory can be developed in ho- theory.