First year report
نویسنده
چکیده
2 Technical background 4 2.1 Martin-Löf Type Theory . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.2 Kinds of judgments . . . . . . . . . . . . . . . . . . . . . . 7 2.1.3 Rules for context formation and substitution . . . . . . . 8 2.1.4 Congruence rules for definitional equality . . . . . . . . . 9 2.1.5 Rules for the function type . . . . . . . . . . . . . . . . . 9 2.1.6 Rules for propositional equality . . . . . . . . . . . . . . . 11 2.1.7 Rules for Sigma . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.8 Rules for universes . . . . . . . . . . . . . . . . . . . . . . 13 2.1.9 Rules for finite types . . . . . . . . . . . . . . . . . . . . . 15 2.1.10 Rules for inductive types . . . . . . . . . . . . . . . . . . 18 2.1.11 Induction-recursion . . . . . . . . . . . . . . . . . . . . . . 22 2.1.12 Computation and metatheory . . . . . . . . . . . . . . . . 24 2.1.13 Extensionality . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1.14 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Category Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2.3 Natural transformations . . . . . . . . . . . . . . . . . . . 29 2.2.4 Adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2.5 Higher categories . . . . . . . . . . . . . . . . . . . . . . . 31 2.2.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3 Homotopy Type Theory . . . . . . . . . . . . . . . . . . . . . . . 33 2.3.1 Homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3.2 Homotopy levels . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.3 Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3.4 Higher inductive types . . . . . . . . . . . . . . . . . . . . 37 2.3.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
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تاریخ انتشار 2015