Elements of mathematics. General topology. Part 1 by Nicolas Bourbaki

By Nicolas Bourbaki

Bourbaki Library of Congress Catalog #66-25377 published in France 1966

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Additional resources for Elements of mathematics. General topology. Part 1

Example text

For each non-degenerate simplex x ∈ X, of dimension n, say, factor f (x) ∈ Y as ρ∗ (y), for a degeneracy operator ρ : [n] → [p] and a non-degenerate p-simplex y ∈ Y . For each face operator µ : [m] → [n], factor µ∗ (ρ) = ρµ as γ ∗ (ν) = νγ, for a degeneracy operator γ : [m] → [r] and a face operator ν : [r] → [p]. ∆m µ G ∆n ν  G ∆p γ x ¯ GX y¯  GY ρ  ∆r f Then ν ∗ (y) is non-degenerate, because Y is non-singular, and γ ∗ (ν ∗ (y)) is the Eilenberg–Zilber factorization of f (µ∗ (x)). Define a map hfx : |Sd(∆n )| → |∆n | (µ) → µ f x by taking the point corresponding to the 0-simplex (µ) of Sd(∆n ) to the pseudoγ ) of the face µ, with respect to γ, and extending affine barycenter µ fx = |µ|(βm linearly on each simplex of Sd(∆n ).

The cone on X is then defined as cone(X) = colim simpη (X) ([n], x) → cone(∆n ) . The inclusion [n] ⊂ [n] ∪ {v} of partially ordered sets induces the natural base inclusions ∆n → cone(∆n ) and i : X → cone(X) of simplicial sets. In the source of i it does not matter whether we form the colimit of ([n], x) → ∆n over simp(X) or simpη (X), because the value ∆−1 of this functor on (−1, η) is empty. 15. The q-simplices of cone(X) can be explicitly described as the pairs (µ : [p] ⊂ [q], x ∈ Xp ) for 0 ≤ p ≤ q, where µ = δ q .

9)? 13. We define the cone on ∆n to be cone(∆n ) = N ([n] ∪ {v}), where [n] ∪ {v} is ordered by adjoining a new, greatest, element v to [n]. 14) ([n], x) → cone(∆n ) defines a functor from simp(X) to simplicial sets. 7, because we want the cone on the empty space X = ∅ to be a single point. Let simpη (X) be the augmented simplex category obtained by adjoining an initial object (−1, η) to simp(X), thought of as a unique (−1)-simplex in X. 14) extends to a functor from simpη (X) to simplicial sets, taking the new object (−1, η) to the vertex point cone(∆−1 ) = N ({v}).

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