文章基本信息

标题：Idempotents in intensional type theory
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作者：Michael Shulman
期刊名称：Logical Methods in Computer Science
印刷版ISSN：1860-5974
电子版ISSN：1860-5974
出版年度：2016
卷号：12
期号：3
页码：1
DOI：10.2168/LMCS-12(3:9)2016
出版社：Technical University of Braunschweig
摘要：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.
其他关键词：Martin-Lof type theory, dependent type theory, idempotent, univalence axiom