Lemma 14.32.3 (01AB)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 14: Simplicial Methods
Section 14.32: A homotopy equivalence
Lemma 14.32.3 (cite)

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Lemma 14.32.3. Let $A$, $B$ be sets, and that $f : A \to B$ is a map. Consider the simplicial set $U$ with $n$-simplices

\[ A \times _ B A \times _ B \ldots \times _ B A\ (n + 1 \text{ factors)}. \]

see Example 14.3.5. If $f$ is surjective, the morphism $U \to B$ where $B$ indicates the constant simplicial set with value $B$ is a trivial Kan fibration.

Proof. Observe that $U$ fits into a cartesian square

\[ \xymatrix{ U \ar[d] \ar[r] & \text{cosk}_0(A) \ar[d] \\ B \ar[r] & \text{cosk}_0(B) } \]

Since the right vertical arrow is a trivial Kan fibration by Lemma 14.32.1, so is the left by Lemma 14.30.3. $\square$

Comments (2)

Comment #6779 by Bogdan on November 30, 2021 at 11:52

Shouldn't A and B be interchanged in the diagram?

Comment #6935 by Johan on January 20, 2022 at 18:50

Thanks and fixed here.

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