Generalizing from 1-category theory, there's a simple definition of a "naive complex" in a stable $\infty$-category. Considering bounded positive graded chain complexes, they are a sequence of maps


$$ A\_n \xrightarrow{d\_n} A\_{n-1} \xrightarrow{d\_{n-1}} \ldots \xrightarrow{d\_1} A\_0 $$


with the property that $d\_k d\_{k+1} \simeq 0$. If I've not made any errors, then at first blush they seem to have a reasonable theory, along with a nice "realization" given by iteratively taking homotopy cofibers:


$$ |A| = \mathrm{cofib}(\mathrm{cofib}(\ldots) \to A\_0) $$


that parallels taking the total complex of a bicomplex. This realization can also be given by the colimit of the diagram


$$ A\_n \rightrightarrows^{d\_n}\_0 A\_{n-1} \rightrightarrows \ldots \rightrightarrows^{d\_1}\_0 A\_0$$




---


However, naive complexes don't seem to be studied. Instead, in Higher Algebra, Lurie proves a Dold-Kan correspondence stable $\infty$-category asserting equivalences between the categories of:


* Simplicial objects
* Filtered objects (i.e. arbitrary $\mathbb{Z}\_{\geq 0}$-indexed diagrams)
* upper-triangular array with zeroes along the diagonal, in which every square is a pushout


and it is this last sort of thing that Lurie calls a complex. (Specifically, a $\mathbb{Z}\_{\geq 0}$-complex)


It is only when looking at the homotopy category does Lurie say anything about a correspondence between simplicial objects and naive complexes.


So my question is whether this is an oversight; i.e.



> 
> Is the $\infty$-category of simplicial objects in a stable $\infty$-category equivalent to the $\infty$-category of (unbounded, positively graded) naive complexes?
> 
> 
> 


and if the answer is no, what goes wrong?



Complexes in stable categories
Here is the problem with the notion of naive complex. Suppose we have a naive complex
$$ \require{AMScd} \begin{CD}
A @>f>> B @>g>> C @>h>> D \end{CD} $$
If we propose to compute the realization iteratively, the first step would be to produce the sequence
$$ \require{AMScd} \begin{CD}
\mathrm{cofib}(f) @>>> C @>h>> D \end{CD} $$
However, this need not be a naive complex! So everything falls apart.


A more elaborate explanation is to look at the big diagram of pushout squares one can construct from the complex:


$$ \require{AMScd} \begin{CD}
A @>f>> B @>>> 0
\\ @VVV @VVV @VVV
\\ 0 @>>> \mathrm{cofib}(f) @>>> \Sigma A @>>> 0
\\ & & @VVV @VVV @VVV
\\ & & C @>>> \mathrm{cofib}(g) @>>> \mathrm{cofib}(\mathrm{cofib}(f) \to C)
\\ & & & & @VVV
\\ & & & & D
\end{CD} $$
There is no direct way to produce the expected map $\mathrm{cofib}(\mathrm{cofib}(f) \to C) \to D$. In fact, by the rightmost square, one can only possibly exist if $(\Sigma A \to D) \simeq 0$.


The map $\Sigma A \to D$ is the Toda bracket mentioned in the comments. ([wikipedia](https://en.wikipedia.org/wiki/Toda_bracket) [nlab](https://ncatlab.org/nlab/show/Toda+bracket))




---


We can give an explicit counterexample. In the $\infty$-category of chain complexes, suppose our naive complex is given by:


$$ \require{AMScd} \begin{CD}
0 @>>> 0 @>>> \mathbb{Z}/4 @>1>> \mathbb{Z}/4
\\ @VVV @VVV @VV4V @VVV
\\ \mathbb{Z} @>2>> \mathbb{Z}/4 @>2>> \mathbb{Z}/8 @>>> 0
\end{CD} $$


That this is a naive complex can be checked on the homology groups. This isn't a bicomplex as is, and the point is that it cannot be improved to become one! One can show that after replacing $A \xrightarrow{f} B$ with $\mathrm{cofib}(f)$, the diagram becomes


$$ \require{AMScd} \begin{CD}
\mathbb{Z} @>1 + 2n>> \mathbb{Z}/4 @>1>> \mathbb{Z}/4
\\ @VV2V @VV4V @VVV
\\ \mathbb{Z}/4 @>2>> \mathbb{Z}/8 @>>> 0
\end{CD} $$


where $n$ encodes the choice of chain homotopy and on homology groups, the top row does not compose to zero for any choice of $n$.




---


The fix is to redefine "naive complex" to include the parallel zeroes: i.e. so that the index 1-category is a sequence of parallel pairs of arrows such that each arrow coequalizes the pair before it.


(the second arrow in each pair is still required to be zero!)


I interpret Pavlov's comment above as saying that this redefinition of "naive complex" results in a category of complexes that is equivalent to simplicial objects, although I have not proved that myself.


