Experience Implementing a Performant Category-Theory Library in Coq

  • Jason Gross
  • Adam Chlipala
  • David I. Spivak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8558)


We describe our experience implementing a broad category-theory library in Coq. Category theory and computational performance are not usually mentioned in the same breath, but we have needed substantial engineering effort to teach Coq to cope with large categorical constructions without slowing proof script processing unacceptably. In this paper, we share the lessons we have learned about how to represent very abstract mathematical objects and arguments in Coq and how future proof assistants might be designed to better support such reasoning. One particular encoding trick to which we draw attention allows category-theoretic arguments involving duality to be internalized in Coq’s logic with definitional equality. Ours may be the largest Coq development to date that uses the relatively new Coq version developed by homotopy type theorists, and we reflect on which new features were especially helpful.


Coq category theory homotopy type theory duality performance 


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Jason Gross
    • 1
  • Adam Chlipala
    • 1
  • David I. Spivak
    • 1
  1. 1.Massachusetts Institute of TechnologyCambridgeUSA

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