Cofibrations in a category of fibrant objects

Question

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 $\mathcal{C}$?

Answer 1

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.