TITLE: Cylinder object in the model category of chain complexes
QUESTION [9 upvotes]: Let $\text{Ch}⁺(R)$ be the category of non-negative chain complexes of $R$-modules where $R$ is a commutative ring. What is a cylinder object, in the sense of model categories, for a given complex $M^\bullet$? Thinking about algebraic topology (a circle and an annulus) I was thinking of tensoring $M^\bullet$ by $R$ (over $R$!) but this is quite trivial and surely I am making a mistake. Could somebody help me?
Thank you!

REPLY [2 votes]: You can see the cylinder $Cyl(f)$ defined as in Zhen's answer as the object in your category s.t.  the following short  sequence 
$0\rightarrow X \rightarrow Cyl(f) \rightarrow C(f) \rightarrow 0$ 
is exact, for any morphism $f: X\rightarrow Y$ and denoting by $C(f)$ the cone of $f$. 
Alternatively you can see the above cylinder as the cone of the induced morphism $C(f)[-1]\rightarrow X$. 
The tricky part in dealing with cylinders and cones is that the 'innocent looking" short exact sequence above is not split, but rather semisplit in the sense of Gelfand Manin's "Methods of  Homological Algebra' textbook.
In other words, the natural splittings $r: Cyl(f)\rightarrow X$ and $s:C(f)\rightarrow Cyl(f)$ are not morphisms in your category of complexes (they do not commute with the differentials, due to the twisted differentials on the cylinder and cone).
This  fact is strongly used in defining the triangulated structure on homotopy categories and  derived categories.
For a related discussion, please check Split-Lemma for chain complexes