Truncation levels in homotopy type theory 1st Edition by Nicolai Kraus ISBN 157586181X 9781575861814

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ISBN 10: 157586181X 
ISBN 13: 9781575861814
Author: Nicolai Kraus

Homotopy Type Theory (HoTT) merges the insights of homotopy theory and type theory into a new foundational framework for mathematics. One of the core concepts in HoTT is truncation, which classifies types according to their homotopical complexity. These truncation levels – from mere propositions to sets, groupoids, and beyond – provide a structured hierarchy for understanding the “shape” of types and their paths.

This book explores truncation levels in depth: from formal definitions and constructions within type theory, to their homotopical interpretations and applications in higher category theory, logic, and topology. Readers are introduced to the theoretical machinery required to reason about truncated types, and guided through examples, theorems, and formal proofs illustrating their utility.

Designed for graduate students and researchers in type theory, category theory, and homotopy theory, this book assumes basic familiarity with dependent type theory and the Univalence Axiom.

Truncation levels in homotopy type theory 1st Table of contents:

1. Introduction to Homotopy Type Theory

2. Basics of Truncation

3. Truncation Levels and the Hierarchy

4. Constructing Truncations

5. Properties of Truncated Types

6. Applications in Logic and Foundations

7. Truncation in Practice

8. Open Questions and Research Directions

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