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A Descent Property for the Univalent Foundations

  • Egbert RijkeEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

We present a version of the descent property [4, 5] which is formulated using families rather than morphisms. By the univalence axiom [3], there is an equivalence \((\sum _{Y:\text{Type}}Y \rightarrow X) \simeq (X \rightarrow \text{Type})\) for every type X [1]. A similar equivalence will hold for the kind of families over graphs we will study here: the equifibered families. This equivalence can be used to translate our simple version of the descent property back into the usual formulation of it.

Keywords

Simple Version Usual Formulation Type Family Algebraic Topology Short Route 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The research leading to these results has received funding from the European Union’s 7-th Framework Programme under grant agreement n. 243847 (ForMath).

References

  1. 1.
    Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (Institute for Advaced Study, Princeton 2013). http://homotopytypetheory.org/book/
  2. 2.
    A. Joyal, I. Moerdijk, A completeness theorem for open maps. Ann. Pure Appl. Logic 70(1), 51–86 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    K. Kapulkin, P. Lumsdaine, V. Voevodsky, The simplicial model of univalent foundations (2012, preprint). arxiv:1211.2851Google Scholar
  4. 4.
    J. Lurie, Higher Topos Theory, vol. 170 (Princeton University Press, Princeton, 2009)zbMATHGoogle Scholar
  5. 5.
    C. Rezk, Toposes and homotopy toposes (version 0.15) (2010). http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Carnegie Mellon UniversityPittsburghUSA

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