A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory

This paper contributes to recent investigations of the use of homotopy type theory to give machine-checked proofs of constructions from homotopy theory. We present a mechanized proof of a result called the Blakers-Massey connectivity theorem, which relates the higher-dimensional loop structures of t...

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Vydáno v:LICS 2016 : proceedings of the 31st annual ACM-IEEE Symposium on Logic in Computer Science : July 5-8, 2016, New York City, USA s. 565 - 574
Hlavní autoři: Hou (Favonia), Kuen-Bang, Finster, Eric, Licata, Daniel R., Lumsdaine, Peter LeFanu
Médium: Konferenční příspěvek
Jazyk:angličtina
Vydáno: New York, NY, USA ACM 05.07.2016
Edice:ACM Conferences
Témata:
matematisk logik
Mathematical Logic
Theory of computation
Theory of computation > Logic
ISBN:9781450343916, 1450343910
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Abstract This paper contributes to recent investigations of the use of homotopy type theory to give machine-checked proofs of constructions from homotopy theory. We present a mechanized proof of a result called the Blakers-Massey connectivity theorem, which relates the higher-dimensional loop structures of two spaces sharing a common part (represented by a pushout type, which is a generalization of a disjoint sum type) to those of the common part itself. This theorem gives important information about the pushout type, and has a number of useful corollaries, including the Freudenthal suspension theorem, which was used in previous formalizations. The proof is more direct than existing ones that apply in general category-theoretic settings for homotopy theory, and its mechanization is concise and high-level, due to novel combinations of ideas from homotopy theory and from type theory.
AbstractList This paper contributes to recent investigations of the use of homotopy type theory to give machine-checked proofs of constructions from homotopy theory. We present a mechanized proof of a result called the Blakers-Massey connectivity theorem, which relates the higher-dimensional loop structures of two spaces sharing a common part (represented by a pushout type, which is a generalization of a disjoint sum type) to those of the common part itself. This theorem gives important information about the pushout type, and has a number of useful corollaries, including the Freudenthal suspension theorem, which was used in previous formalizations. The proof is more direct than existing ones that apply in general category-theoretic settings for homotopy theory, and its mechanization is concise and high-level, due to novel combinations of ideas from homotopy theory and from type theory.
This paper continues investigations in “synthetic homotopy theory”: the use of homotopy type theory to give machine-checked proofs of constructions from homotopy theory. We present a mechanized proof of the Blakers-Massey connectivity theorem, a result relating the higher-dimensional homotopy groups of a pushout type (roughly, a space constructed by gluing two spaces along a shared subspace) to those of the components of the pushout. This theorem gives important information about the pushout type, and has a number of useful corollaries, including the Freudenthal suspension theorem, which has been studied in previous formalizations. The new proof is more elementary than existing ones in abstract homotopy-theoretic settings, and the mechanization is concise and high-level, thanks to novel combinations of ideas from homotopy theory and type theory.
Author Hou (Favonia), Kuen-Bang
Licata, Daniel R.
Lumsdaine, Peter LeFanu
Finster, Eric
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  givenname: Kuen-Bang
  surname: Hou (Favonia)
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  organization: Carnegie Mellon University
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  givenname: Eric
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  givenname: Daniel R.
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  givenname: Peter LeFanu
  surname: Lumsdaine
  fullname: Lumsdaine, Peter LeFanu
  email: p.l.lumsdaine@gmail.com
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Snippet This paper contributes to recent investigations of the use of homotopy type theory to give machine-checked proofs of constructions from homotopy theory. We...
This paper continues investigations in “synthetic homotopy theory”: the use of homotopy type theory to give machine-checked proofs of constructions from...
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SubjectTerms matematisk logik
Mathematical Logic
Theory of computation
Theory of computation -- Logic
Title A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory
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