Is the (inverse) Dold-Kan functor fully faithful on chain complexes of commutative monoids?

Question

My question

The background/notation for all of the content of this post is in Lurie, Higher Algebra [HA], Ch. 1.2.3. Everything is purely 1-categorical. Let $\mathcal{A}$ a semiadditive category (with biproducts and 0). Then there is a Dold-Kan functor $$ \mathrm{DK}\colon\mathrm{Ch}_{\geq0}(\mathcal{A})\to\mathbf{s}\mathcal{A}, $$ where $\mathbf{s}\mathcal{A}=[\mathbf{\Delta}^{\mathrm{op}},\mathcal{A}]$ and $\mathrm{Ch}_{\geq0}(\mathcal{A})$ is non-negatively graded chain complexes in objects of $\mathcal{A}$. Take $\mathcal{A}=\mathbf{CMon}$ the category of commutative monoids. My question is whether $\mathrm{DK}$ fully faithful in this case.

Brief background

The definition of the functor is in [HA, Construction 1.2.3.5]; it is defined as a composite $\mathrm{Ch}_{\geq0}(\mathcal{A})\to[\mathbf{\Delta}_{\mathrm{inj}}^{\mathrm{op}},\mathcal{A}]\to\mathbf{s}\mathcal{A}$ with the middle category being semisimplicial $\mathcal{A}$-objects. The first functor takes a chain complex $(A_*,\partial)$ to the semisimplicial object $[n]\mapsto A_n$ and on all non-identity order preserving injections it is zero except for $d^n$, which is sent to $\partial_n$. The second functor sends a semisimplicial object $A_\bullet$ to the sum $$ [n]\mapsto\bigoplus_{\phi\colon[n]\twoheadrightarrow[k]}A_k $$ indexed by surjections in $\mathbf{\Delta}$. On maps $[n]\to[n']$ it defined using the unique factorization into a split epi + mono - the details are in [HA, Construction 1.2.3.5].

If we take $\mathcal{A}=\mathbf{Ab}$, then $\mathrm{DK}$ is an equivalence by the Dold-Kan correspondence; more generally the same is true if $\mathcal{A}$ is any additive idempotent-complete category [HA, Thm. 1.2.3.7]. If $\mathcal{A}$ is only additive, then $\mathrm{DK}$ is only fully faithful. This can be easily deduced, as Lurie does, from standard Dold-Kan.

However, visibly $\mathrm{DK}$ makes sense for any semiadditive $\mathcal{A}$. I am interested in the case of $\mathbf{CMon}$ as posed in the question because I think that using a purely categorical argument this will show $\mathrm{DK}$ is fully faithful for any semiadditive category.

What I have so far...

1. $\mathrm{DK}$ is faithful. This is clear from the definition since for each $n$ we can recover $f_n$ from $\mathrm{DK}(f)_n$ via the biproduct maps.
2. If $g\colon\mathrm{DK}(A_*)\to\mathrm{DK}(B_*)$ is a simplicial map, using the argument in (1) we can recover the unique $f$ which we hope satisfies $\mathrm{DK}(f)=g$. Because $\mathrm{DK}$ is defined using the boundary map, I was able to show that this $f$ is indeed a chain map by writing out some diagrams.
3. I think, from this, it suffices to show that each $g_n\colon\mathrm{DK}(A)_n\to\mathrm{DK}(B)_n$ is given by a "diagonal matrix" in the sense of the defining direct sum.

As it stands, I don't have a strong guess as to whether this is true or not. Unfortunately the original Dold-Kan proof doesn't help since we cannot define a chain complex from a simplicial commutative monoid - the standard construction needs $-1$.

If this isn't true, a counterexample in $\mathbf{CMon}$ would of course be perfect, but by my earlier remark a counterexample for any semiadditive category would suffice (e.g. maybe there's some obvious reason chain complexes in some $\mathcal{A}$ can't embed into simplicial objects). But this would have to be a non-additive semiadditive category...

Any ideas are appreciated!

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.

Answer 1

Have you looked at the same question in semi-abelian categories? I know that there is Dominique Bourn, Moore normalisation and Dold–Kan theorem for semi-Abelian categories, in Categories in algebra, geometry and mathematical physics , Contemp. Math. 431, 105–124, Amer. Math. Soc., Providence, RI. (2007). I am pretty sure that your question is not handled by this but the methods used may give you ideas on what you could try.

Answer 2

Thanks everyone, for completeness I'll outline the proof here. The answer is yes, over $\mathbf{CMon}$, $\mathrm{DK}$ is fully faithful. To check this we define $N_*$ in the opposite direction taking $A$ to the complex $NA_n=\bigcap_{i\geq1}\ker{d_i}$ with differential $d_0$.

Lemma $\mathrm{DK}$ is left adjoint to $N_*$.

Proof We can view $\mathrm{Ch}_{\geq0}(\mathbf{CMon})$ as a presheaf category enriched in $\mathbf{CMon}$ over the category $Ch$ described in Tim's latest comment. Now it is clear $\mathrm{DK}$ preserves colimits since we can check it pointwise and it was defined using coproducts. So $\mathrm{DK}$ has a right adjoint, say $R$; let the representables be $r_n$, then $R$ satisfies $$ R(A)_n=\mathrm{Hom}_{\mathbf{sCMon}}(\mathrm{DK}(r_n),A)\quad\text{ in }\mathbf{CMon} $$ The definition of how $\mathrm{DK}(-)$ on face maps shows that we can identify this hom-monoid with $NA_n$, thus $\mathrm{DK}\dashv N_*$.

Now the result follows from the observation that $N_*\circ\mathrm{DK}\cong\mathrm{Id}$. Indeed, that $A_n\subset N_n\mathrm{DK}(A)_n$ is easy to see, and the reverse inclusion follows from the fact that for any surjection $\phi\colon[n]\to[k]$, we can find $i\geq1$ such that $\phi\circ d^i$ is still surjective (then the condition on kernels implies only the $A_n$ submonoid remains) [HA, Remark 1.2.3.11]. We are done as such an isomorphism implies this is a fully faithful adjoint pair.