Michael Shulman - Idempotents in intensional type theory

lmcs:2027 - Logical Methods in Computer Science, April 27, 2017, Volume 12, Issue 3 - https://doi.org/10.2168/LMCS-12(3:9)2016
Idempotents in intensional type theory

Authors: Michael Shulman

We study idempotents in intensional Martin-Löf type theory, and in particular the question of when and whether they split. We show that in the presence of propositional truncation and Voevodsky's univalence axiom, there exist idempotents that do not split; thus in plain MLTT not all idempotents can be proven to split. On the other hand, assuming only function extensionality, an idempotent can be split if and only if its witness of idempotency satisfies one extra coherence condition. Both proofs are inspired by parallel results of Lurie in higher category theory, showing that ideas from higher category theory and homotopy theory can have applications even in ordinary MLTT. Finally, we show that although the witness of idempotency can be recovered from a splitting, the one extra coherence condition cannot in general; and we construct "the type of fully coherent idempotents", by splitting an idempotent on the type of partially coherent ones. Our results have been formally verified in the proof assistant Coq.

Volume: Volume 12, Issue 3
Published on: April 27, 2017
Submitted on: July 24, 2015
Keywords: Mathematics - Logic,Computer Science - Programming Languages,Mathematics - Category Theory


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