Let $S\mathcal{A}$ be the category of simplicial objects in an abelian category $\mathcal{A}$. In exercise 8.4.5 in Weibel's An Introduction to Homological algebra, it is said that $P \in S\mathcal{A}$ is projective if each $P_i$ is projective and $P$ is null-homotopic.

By Dold-Kan correspondence we can pass to the category of chain complex $C\mathcal{A}_{\ge0}$. Consider $P_{\cdot}$ s.t. $P_0$ is projective and $P_i=0$ for $i\neq 0$. Consider $B_\cdot \rightarrow A_\cdot$ which is surjective and $P_\cdot \rightarrow A_\cdot$. Obviously $B_0 \rightarrow A_0$ is surjective and we can lift $P_0 \rightarrow A_0$ to $P_0\rightarrow B_0$. This gives $P_\cdot \rightarrow B_\cdot$ which is a lifting. So such $P_\cdot$ is projective but not null-homotopic.

Am I wrong or Weibel? If Weibel is wrong, what is the proper characterisation of projective simplicial objects?



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