Calculus of Fractions and Homotopy Theory pp 41-56 | Cite as
Geometric Realization of Simplicial Sets
Chapter
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 f≧g 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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© Springer-Verlag Berlin · Heidelberg 1967