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The Stacks project

Lemma 14.31.7. Let f : X \to Y be a homomorphism of simplicial abelian groups which is termwise surjective. Then f is a Kan fibration of simplicial sets.

Proof. Consider a commutative solid diagram

\xymatrix{ \Lambda _ k[n] \ar[r]_ a \ar[d] & X \ar[d] \\ \Delta [n] \ar[r]^ b \ar@{-->}[ru] & Y }

as in Definition 14.31.1. The map a corresponds to x_0, \ldots , \hat x_ k, \ldots , x_ n \in X_{n - 1} satisfying d_ i x_ j = d_{j - 1} x_ i for i < j, i, j \not= k. The map b corresponds to an element y \in Y_ n such that d_ iy = f(x_ i) for i \not= k. Our task is to produce an x \in X_ n such that d_ ix = x_ i for i \not= k and f(x) = y.

Since f is termwise surjective we can find x \in X_ n with f(x) = y. Replace y by 0 = y - f(x) and x_ i by x_ i - d_ ix for i \not= k. Then we see that we may assume y = 0. In particular f(x_ i) = 0. In other words, we can replace X by \mathop{\mathrm{Ker}}(f) \subset X and Y by 0. In this case the statement become Lemma 14.31.6. \square


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