Recall the following categories:
- $\Delta$, the "simplicial bookkeeping category" of nonempty finite linearly ordered sets
- $\Delta_+$, the category of possibly empty finite linearly ordered sets obtained by adjoining an initial object to $\Delta$
- $\Delta_{-\infty}$, the category of nonempty finite linearly ordered sets and minimum-preserving maps
- $\Delta_{\pm\infty}$, the category of nonempty finite linearly ordered sets and minimum-and-maximum preserving maps
Evidently, we have faithful functors $\Delta_{\pm\infty}\to\Delta_{-\infty}\to\Delta\to\Delta_+\to \Delta_{-\infty}$, where the last map adds a new minimum to the partially ordered set.
Functors from (the opposites of) these categories to abelian groups have the following names:
- A simplicial abelian group $A_\bullet:\Delta^{op}\to \operatorname{Ab}$ determines a normalized chain complex $N_*(A_\bullet)$ (in fact, this is an equivalence of categories)
- An augmented simplicial abelian group $A_\bullet:\Delta^{op}_+\to\operatorname{Ab}$ determines a map $H_0(N_*(A_\bullet))\to A(\emptyset)$ (in fact, this is an equivalence of categories)
- A split simplicial abelian group $A_\bullet:\Delta^{op}_{-\infty}\to\operatorname{Ab}$ determines a strong chain deformation retraction $(A(\emptyset)\to N_*(A_\bullet),h:N_*(A_\bullet)\to N_{*+1}(A_\bullet))$ of the associated augmented simplicial object (this should also be an equivalence of categories).
- Finally, the category $\Delta_{\pm\infty}$ is equivalent to $\Delta_+$ by sending a function $f:[n]\to [m]$ to its "effect on gaps", i.e. the map $\widetilde f:[m-1]\to [n-1], \widetilde f(j) = \max \{i\mid f(i) \le j\}$. Thus a functor $A_\bullet:\Delta_{\pm\infty}^{op}\to\operatorname{Ab}$ is the same data as a cosimplicial abelian group, which in turn form a category equivalent to augmented cochain complexes.
Your construction takes a simplicial ableian group $A_\bullet$ and essentially produces the mapping cone of the augmentation $A_0 = B(\emptyset)\to N^*(B^\bullet)$ associated to the augmented cosimplicial group $B^\bullet$. It is straightforward to show that the "effect on gaps" construction extends to a commutative diagram
$$
\require{AMScd}
\begin{CD}
\Delta_{\pm\infty} @>{\cong}>> \Delta_+^{op}\\
@VVV @VVV\\
\Delta_{-\infty}@>{\cong}>>\Delta_{+\infty}^{op}\cong \Delta_{-\infty}^{op}
\end{CD}
$$
where $\Delta_{+\infty}$ is the subcategory of maximum-preserving monotone functions (the equivalence to $\Delta_{-\infty}$ is given by reversing the partial order.) Since $A_\bullet$ extends over $\Delta_{-\infty}$, so does $B^\bullet$, which shows that this mapping cone is equipped with a canonical chain nullhomotopy. (This is essentially the "extra degeneracy" argument of Tom Goodwillie's comment.)
In fact, this works in any abelian category, even if chain complexes with vanishing homology don't necessarily possess a chain homotopy retraction. This should show that the Whitehead torsion mentioned by Connor Malin vanishes.
(In fact)${}^2$, this works in any $\infty$-category: augmented simplicial objects are (homotopy) colimit cones under the corresponding simplicial object, and split simplicial objects are absolute (homotopy) colimit cones in the sense that the underlying augmented simplicial object gets sent to a (homotopy) colimit cone by any functor. This is shown in Lemma 6.3.1.16 of Lurie's Higher Topos Theory.
