I have heard the expression recently that one should be careful when constructing cones in the homotopy category - namely, that this is not functorial. However, when working through some examples in cochain complexes, I was playing around with the following construction which I haven't been able to understand in relation to the above.


For simplicity let $k$ be a field, and let $Ch^\*\_h(k)^{\rightarrow}$ be the category with


* Objects cochain maps $f : C \to D$
* Morphisms given by triples $(\rho,\sigma,h)$ that fit into homotopy commutative squares
$$\require{AMScd}
\begin{CD}
C @>{f}>> D ;\\
@V{\rho}VV @VV{\sigma}V \\
C' @>{f'}>> D';
\end{CD}
$$ where the key point is that homotopy is specified.


The identity map is given by $(1\_C, 1\_D, 0)$ and composition of squares
$$\require{AMScd}
\begin{CD}
C @>{f}>> D ;\\
@V{\rho}VV @VV{\sigma}V \\
C' @>{f'}>> D';
\end{CD}
$$
with homotopy $h$ and
$$\require{AMScd}
\begin{CD}
C' @>{f'}>> D' ;\\
@V{\rho'}VV @VV{\sigma'}V \\
C'' @>{f''}>> D'';
\end{CD}
$$ and homotopy $h'$ is given by
$$ (\rho', \sigma', h') \circ (\rho, \sigma, h) = (\rho' \rho, \sigma' \sigma, \sigma'h + h' \rho)$$ where $\sigma'h + h' \rho$ is indeed a homotopy between $\sigma'\sigma f$ and $f'' \rho' \rho$.


By my calculations this induces a functor $\mathsf{Cone} : Ch^\*\_h(k)^{\rightarrow} \to Ch^\*(k)$ into the category of cochain complexes where


* On objects $\mathsf{Cone}(f)$ is sent to the usual cone of a morphism
* On morphisms $(\rho,\sigma,h)$ from $f$ to $f'$ the map is given by the cochain map
$$\begin{bmatrix} \rho & 0 \\ h & \sigma \end{bmatrix} : \mathsf{Cone}(f) \to \mathsf{Cone}(f').$$


That this agrees with composition follows from matrix multiplication
$$\begin{bmatrix} \rho' \rho & 0 \\ \sigma'h + h' \rho & \sigma' \sigma \end{bmatrix} = \begin{bmatrix} \rho' & 0 \\ h' & \sigma' \end{bmatrix} \begin{bmatrix} \rho & 0 \\ h & \sigma \end{bmatrix} : \mathsf{Cone}(f) \to \mathsf{Cone}(f') \to \mathsf{Cone}(f'').$$


Now let $K(Ch^\*(k))$ be the homotopy category of cochain complexes. If I have understood correctly, the difference between this construction and the one on the homotopy category is that the objects in $K(Ch^\*(k))^{\rightarrow}$ are homotopy classes of morphisms and representative homotopies are not specified.


Taking the Kan extension of $\mathsf{Cone} : Ch^\*\_h(k)^{\rightarrow} \to Ch\_\*(k)$ along the functor $\Pi : Ch^\*\_h(Q)^{\rightarrow} \to K(Ch^\*(k))$ into the homotopy category (by sending everything to its homotopy class) yields a functor
$$ L : K(Ch^\*(k))^{\rightarrow} \to Ch^\*(k).$$
where there is a natural transformation from $L \Pi$ to $\mathsf{Cone}$, and we get a square
$$\require{AMScd}
\begin{CD}
Ch\_h^\*(k)^{\rightarrow} @>{\mathsf{Cone}}>> Ch^\*(k) ;\\
@V{\Pi}VV @VV{\Pi'}V \\
K(Ch^\*(k))^{\rightarrow} @>{\Pi' L}>> K(Ch^\*(k));
\end{CD}$$ which commutes up to natural transformation and where $\Pi'$ sends morphisms to their homotopy classes.


My question: this functor $\Pi' L$ looks like a good candidate for a functorial cone. My intuition is that something must go wrong but I've been unable to figure out what it is so far. I also want to know if this kind of thing is what people are really asking for when they ask for a functorial cone, or whether they mean something different.



Why does this construction not give a functorial cone in the homotopy category of cochain complexes?
The main issue you run into is that the projection
$\Pi: Ch^\*\_h(k)^{\to} \to K(Ch^\*(k))^{\to}$
is not fully faithful, and so the Kan extension $L$ is not automatically an "extension": it does not satisfy $L \Pi(f) \simeq \mathrm{Cone}(f)$.


(This occurs prominently when the map $f$ is the map $0 \to k$. If $g = k[-1] \to 0$, there are several maps $g \to f$ in $Ch^\*\_h(k)^{\to}$ but only one map in the homotopy category, and they all induce different maps $\mathrm{Cone}(g) \to \mathrm{Cone}(f)$.)


Another small issue is that $K(Ch^\*(k))^{\to}$ is a large category, and so you need some work to ensure that such a Kan extension exists.


