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Homotopical patch theory*


Homotopy type theory is an extension of Martin-Löf type theory, based on a correspondence with homotopy theory and higher category theory. In homotopy type theory, the propositional equality type is proof-relevant, and corresponds to paths in a space. This allows for a new class of datatypes, called higher inductive types, which are specified by constructors not only for points but also for paths. In this paper, we consider a programming application of higher inductive types. Version control systems such as Darcs are based on the notion of patches—syntactic representations of edits to a repository. We show how patch theory can be developed in homotopy type theory. Our formulation separates formal theories of patches from their interpretation as edits to repositories. A patch theory is presented as a higher inductive type. Models of a patch theory are given by maps out of that type, which, being functors, automatically preserve the structure of patches. Several standard tools of homotopy theory come into play, demonstrating the use of these methods in a practical programming context.

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This research was sponsored in part by the National Science Foundation under grant number CCF-1116703. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government or any other entity. This material is based on research sponsored in part by The United States Air Force Research Laboratory under agreement number FA9550-15-1-0053. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the United States Air Force Research Laboratory, the U.S. Government or Carnegie Mellon University.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

M. Abbott , T. Altenkirch & N. Ghani (2005) Containers: Constructing strictly positive types. Theor. Comput. Sci. 342 (1), 327.

N. Gambino & R. Garner (2008) The identity type weak factorisation system. Theor. Comput. Sci. 409 (3), 94109.

D. R. Licata , & G. Brunerie (2013) π n (Sn ) in homotopy type theory. In Proceedings of the Third International Conference on Certified Programs and Proofs. New York, NY, USA: Springer-Verlag New York, Inc., pp. 116.

D. R. Licata , & R. Harper (2011) 2-dimensional directed type theory. In Electron. Notes Theor. Comput. Sci. 276, 263289.

S. Mimram & C. Di Giusto (2013) A categorical theory of patches. Electron. Notes Theor. Comput. Sci. 298, 283307.

B. van den Berg & R. Garner (2011) Types are weak ω-groupoids. Proc. London Math. Soc. 102 (2), 370394.

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Journal of Functional Programming
  • ISSN: 0956-7968
  • EISSN: 1469-7653
  • URL: /core/journals/journal-of-functional-programming
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