TITLE: Cofibrations in a category of fibrant objects
QUESTION [8 upvotes]: There is an obvious (?) notion of cofibration in a category of fibrant objects, namely a morphism which satisfies the left lifting property with respect to all trivial fibrations.  I don't seem to be able to find a place in the literature where this concept is discussed. 
For a concrete question: suppose $A$ is a cofibrant object in a category of fibrant objects $\mathcal{C}$, and $X_\bullet\to X$ is a simplicial resolution of the object $X$ of $\mathcal{C}$.  Does the simplicial set $X_\bullet(A)$ represent the homotopy type of the mapping space $\mathop{Hom}(A,X)$ in the simplicial localization of $\mathcal{C}$?

REPLY [2 votes]: Let $\mathcal{C}$ be a category. A cylinder, $\mathbf{I}$, on $\mathcal{C}$ is a functor (cylinder functor)
$$I:\mathcal{C} \longrightarrow \mathcal{C}$$
together with three natural transformations
$$e^{0}: 1_{\mathcal{C}} \Longrightarrow I , e^{1}: 1_{\mathcal{C}} \Longrightarrow I,  \sigma: I \Longrightarrow 1_{\mathcal{C,}}$$
such that $\sigma  e^{0}= \sigma  e^{1}= 1,$ with $1: 1_{\mathcal{C}} \Longrightarrow 1_{\mathcal{C}}.$
A morphism $i:A\rightarrow X$ of Category with cylinder $\equiv$  $(\mathcal{C}, I,e^{0},e^{1},\sigma)$ is a  cofibration  if and only  if the diagram
$\require{AMScd}$
\begin{CD}
     A  @>>e^{0}_{A}> I(A)\\
    @V i V V @VV I(i) V\\
    X @>>e^{0}_{X}> I(X)
\end{CD}
is weak pushout in $\mathcal{C}.$
K. H. Kamps and T. Porter, Abstract Homotopy and Simple Homotopy Theory.