Return to Question

Edit: I have slightly misquoted [HA] which has led to some confusion. Lurie defines $\mathrm{DK}$ using $d_0$ instead of $d_n$; this is technically equivalent, but were I to prove normal Dold-Kan using $\mathrm{DK}$ as defined above I would have to take the Moore complex with `backwards' differential: $\partial_n=d_n-d_{n-1}+\cdots$ so that $d_n$ always has positive sign.

Edit$'$: Based on Chris's suggestion, let $$N_*\colon\mathbf{s}\mathbf{CMon}\to\mathrm{Ch}_{\geq0}(\mathbf{CMon})$$ be the corresponding functor. I (think I) can prove $N_*\circ\mathrm{DK}\cong\mathrm{Id}$ (see [HA, Remark 1.2.3.11]). I want to use this to deduce $\mathrm{DK}$ is fully faithful. It is clear if $N_*$ is faithful, but I'm not able to prove this without using $-1$. Another way to do it would be to show $N_*$ is right adjoint to $\mathrm{DK}$ as [HA, Lemma 1.2.3.12] does in the abelian category case. I'm having some trouble verifying the construction, though.

Edit 2: There is a serious flaw in this question currently: $\mathbf{CMon}$ is not semiadditive as its coproduct is the tensor product. However, there might still be the possibility to show that $\mathrm{DK}$ is fully faithful for any semiadditive category... I'm still in search of any counterexamples (maybe in $\mathbf{Rel}$ for instance).

Edit 2: There is a serious flaw in this question currently: $\mathbf{CMon}$ is not semiadditive as its coproduct is the tensor product. However, there might still be the possibility to show that $\mathrm{DK}$ is fully faithful for any semiadditive category... I'm still in search of any counterexamples (maybe in $\mathbf{Rel}$ for instance).