Introduction - from type theory and homotopy theory to univalent foundations
نویسندگان
چکیده
منابع مشابه
Homotopy type theory and Voevodsky's univalent foundations
Recent discoveries have been made connecting abstract homotopy theory and the field of type theory from logic and theoretical computer science. This has given rise to a new field, which has been christened “homotopy type theory”. In this direction, Vladimir Voevodsky observed that it is possible to model type theory using simplicial sets and that this model satisfies an additional property, cal...
متن کاملHomotopy Type Theory: Univalent Foundations of Mathematics
Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and beautiful"univalence axiom"implies that isomorphic structures can be identified. On the other hand,"higher inductive types"provide direct, logical descriptio...
متن کاملHomotopy Type Theory: Univalent Foundations of Mathematics
These lecture notes are based on and partly contain material from the HoTT book and are licensed under Creative Commons Attribution-ShareAlike 3.0.
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A model theory in the framework of Univalent Foundations requires a logic that allows us to define structures on homotopy (n-)types, similar to how first-order logic can define structures on sets. We define such an “n-level” logic for finite n. The syntax is based on a generalization of Makkai’s FOLDS, obtained by an operation that allows us to add equality sorts to FOLDS-signatures. We then gi...
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While working on the completion of the proof of the Bloch-Kato conjecture I have thought a lot about what to do next. Eventually I became convinced that the most interesting and important directions in current mathematics are the ones related to the transition into a new era which will be characterized by the widespread use of automated tools for proof construction and verification. I have star...
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عنوان ژورنال:
- Mathematical Structures in Computer Science
دوره 25 شماره
صفحات -
تاریخ انتشار 2015