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By May P., Ponto K.

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We need a structural characterization that allows us to work concretely with such spaces. We briefly recall two well-known results that we will be generalizing before going into this. The classical result about Postnikov towers reads as follows. 1. A connected space X is simple if and only if it admits a Postnikov tower of principal fibrations. We recall what this means. We can always construct maps αn : X −→ Xn such that αn induces an isomorphism on πi for i ≤ n and πi Xn = 0 for i > n just by attaching cells inductively to kill the homotopy groups of X in dimension greater than n.

We let π1 (F, e) act on these groups by pull back along ι∗ : π1 (Fe , e) −→ π1 (E, e). By (i) and inspection, this implies that its actions are given by λ(Fe ,r) , λ(E,r) ◦ ι∗ , and the trivial action, respectively. The maps in the exact sequence are maps of π1 (E, e)-groups by (iii), (iv), and (v). By restricting the construction of λ(E,p) to loops α : I −→ Fe , we see that λ(Fb ,r) [α] = λ(E,p) [ι ◦ α]. This implies part (i), and part (ii) is immediate from the definition of the action of π1 (E, e) on π1 (B, b).

Assume given a commutative diagram X f g f′  G A′ o Y γ α β  X′ GAo g′  Y′ in which f and f ′ are fibrations. If α, β, and γ are homotopy equivalences, then so is their pullback X ×A Y −→ X ′ ×A′ Y ′ . Proof. The model category theory that we give later is self-dual in the very strong sense that results for a model category, when applied to its opposite category, give dual conclusions. This lemma is an illustrative example. 3 that we outlined above dualizes in this sense. 4, a model category is right proper if and only if the conclusion of this “cogluing lemma” holds.

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