The identity type weak factorisation system
Theoretical Computer Science
Two-dimensional models of type theory
Mathematical Structures in Computer Science
2-Dimensional Directed Type Theory
Electronic Notes in Theoretical Computer Science (ENTCS)
Topological and Simplicial Models of Identity Types
ACM Transactions on Computational Logic (TOCL)
Canonicity for 2-dimensional type theory
POPL '12 Proceedings of the 39th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages
Inductive Types in Homotopy Type Theory
LICS '12 Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer Science
Calculating the Fundamental Group of the Circle in Homotopy Type Theory
LICS '13 Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
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Higher-dimensional categories have recently emerged as a natural context for modelling intensional type theories; this raises the question of what higher-categorical structures the syntax of type theory naturally forms. We show that for any type in Martin-Löf Intensional Type Theory, the system of terms of that type and its higher identity types forms a weak *** -category in the sense of Leinster. Precisely, we construct a contractible globular operad ${P_{\mathit{ML}^{\mathrm{Id}}}}$ of type-theoretically definable composition laws, and give an action of this operad on any type and its identity types.