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The long exact sequence of homotopy n-groups

Published online by Cambridge University Press:  07 September 2023

Ulrik Buchholtz Open the ORCID record for Ulrik Buchholtz [Opens in a new window]  and
Egbert Rijke
Ulrik Buchholtz*
Affiliation:
Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgartenstrasse 7, 64289 Darmstadt, Germany School of Computer Science, University of Nottingham, Jubilee Campus, Wollaton Road, Nottingham NG8 1BB, UK
Egbert Rijke
Affiliation:
University of Ljubljana, Fakulteta za matematiko in fiziko, Jadranska 19, 1000 Ljubljana, Slovenia
*
Corresponding author: Ulrik Buchholtz; Email: ulrik.buchholtz@nottingham.ac.uk
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Abstract

Working in homotopy type theory, we introduce the notion of n-exactness for a short sequence $F\to E\to B$ of pointed types and show that any fiber sequence $F\hookrightarrow E \twoheadrightarrow B$ of arbitrary types induces a short sequence

that is n-exact at $\| E\|_{n-1}$. We explain how the indexing makes sense when interpreted in terms of n-groups, and we compare our definition to the existing definitions of an exact sequence of n-groups for $n=1,2$. As the main application, we obtain the long n-exact sequence of homotopy n-groups of a fiber sequence.

Keywords

Homotopy type theoryunivalent foundationshigher groupslong exact sequence
Type
Special Issue: Homotopy Type Theory 2019
Information
Mathematical Structures in Computer Science , Volume 33 , Special Issue 8: Homotopy Type Theory 2019 , September 2023 , pp. 679 - 687
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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