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Geometric Realization of Simplicial Sets

  • Peter Gabriel
  • Michel Zisman
Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE2, volume 35)

Abstract

First, let us give some general remarks which will divert us a little:

For each natural integer n, the set Δ ([n], [1]) will be totally ordered by saying that fg if and only if f (i)≦g(i) for each i∈[n]; moreover, we can identify the ordered set Δ ([n], [1]) with [n+1] under the map f → card f−1(0). We define thus a functor [n] → Δ ([n], [1]) from Δ to Δ°, which will be noted II, and which can be described as follows:
$$II~[n]~=~[n~+~1],~II~\partial _{n}^{i}~=~\sigma _{n}^{i},~II~\sigma _{n}^{i}~=~\partial _{n+2}^{i+1}.$$

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Copyright information

© Springer-Verlag Berlin · Heidelberg 1967

Authors and Affiliations

  • Peter Gabriel
    • 1
  • Michel Zisman
    • 1
  1. 1.Departement de Mathématique StrasbourgUniversité de StrasbourgFrance

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