I am reading Lefevre-Hasegawa's thesis on $A_\infty$-categories. It makes reference to this paper of V. Hinich, Homological Algebra of Homotopy Algebras. I am trying to show the following (notation should be clear from the paper or after the explanation to come):
If $f : M\to A^\#$ is a map of complexes where $M$ is degreewise $k$-free and has zero differential, the canonical map $A\to A(M,f)$ is a cofibration.
If $f: M\to A^\#$ is a map of complexes where $M$ is degreewise $k$-free and contractible, the canonical map $A\to A(M,f)$ is an acyclic cofibration.
which are stated without proof in the paper.
Let $k$ be some commutative ring an write $\mathsf{Ch}_k$ for the category of (all, possibly unbounded) complexes of $k$-modules, and let $\mathsf C$ be some other category (perhaps the category of DGA-algebras, or something similar.)
Assume we have a pair of adjoint functors $\# : \mathsf{C} \iff\mathsf{Ch}_k : F$ ($F$ the left adjoint, and $\#$ should be thought of as a forgetful functor, say from DGA-algebras to complexes, $F$ being the free algebra functor.)
Assume further that $\mathsf{C}$ has finite limits and arbitrary colimits, and that $U$ preserves filtered colimits, and that if $D(n)$ is the disk complex (i.e. an isomorphism $k\to k$ from degree $n$ to degree $n-1$), then the canonical map $A \to A \coprod F(D^n)$ is a quasi-isomorphism under $U$.
Given a map of complexes $f : M\to A^\#$, let $C_f$ be its mapping cone, and define $A(M,f)$ to be the pushout of the diagram $A\leftarrow F(A^\#) \rightarrow FC_f$. We then have a canonical map $$i : A\to A(M,f).$$ The (acyclic) fibrations of $\mathsf{C}$ are the maps $f$ so that $f^\#$ is a surjective (quasi-isomorphism.), and I want to show $i$ is an (acyclic) cofibration in the cases $1.$ and $2.$ above, i.e. show they are left rothogonal to acyclic fibrations.
I can do $1.$: let $\{m_i\}$ be a basis of $M$. A map $f:M\to A^\#$ chooses a corresponding set of cocycles $\{z_i\}$ in $A^\#$. Consider a diagram
$\require{AMScd}$ \begin{CD} A @>f>> B\\ @V i V V@VV g V\\ A(M,f) @>>h> C \end{CD}
In $A(M,f)$ we have $dm_i = z_i$. Pick $b_i\in UB$ such that $h^\#(m_i) = g^\#(b_i)$, possible since $g^\#$ is onto, and now note that $$g^\#(f^\#(z_i)-db_i) = 0.$$ Since $g^\#$ is quasi-isomorphism we now have $f^\#(z_i) = dB_i$ for some $B_i$. The recipe $a(m_i) = B_i$ and $a\mid_A = f$ then defines the suitable map $a : A(M,f) \to B$ that lifts the diagram. Strictly speaking, $a$ is induced from the maps $ f : A\to B$ and $F : C_f\to B^\#$ such that $F(a,m_i) = f^\#(a) + B_i$.
I have been trying to prove $2.$ in a very similar fashion, but I am not getting anywhere, either on showing that $i$ is a cofibration (which whould be similar to the above) and on showing $i^\#$ is a quasi-isomorphism.
I think $i^\#$ should be a quasi-isomorphism on account on the axiom on the disc $D(n)$ and $F$, and since $A^\# \to C_f$ is a quasi-isomorphism, on account on $M$ being contractible. I would guess I should be able to show $i$ is a cofibration by imitating the proof for $1.$, but I couldn't get anywhere. Any help would be appreciated.
Add To show the maps $i$ are cofibrations it suffices to show the following is true: in $\mathsf{Ch}_k$ the degree-wise injections with $k$-free contractible cokernel (resp. $k$-free cokernel with zero differential) have the left lifting property with respect to degree-wise surjective quasi-isomorphisms.
This is true in $\mathsf{Ch}_k^+$. In fact, there is a model structure in this category where the cofibrations are the degree-wise injections with degree-wise projective cokernel, the fibrations are the degree-wise surjections and the weak equivalences are the quasi-isomorphisms.
Partial answer. To show that $i$ is a cofibration, it suffices the prove the following: let $h_2^\flat$ denote the image of the map $h_2 : FC_f\to C$ under the adjunction $(F,\#)$, where $h_2 = hp$ and $p : FC_f \to A(M,f)$ is the map of the pushout. Then there is a lift of the following diagram:
$$\begin{CD} A^\# @>f^\#>> B^\#\\ @V j V V@VV g^\# V\\ C_f@>>h_2^\flat> C^\# \end{CD}$$
In Hovey's book it is shown there is a model structure on $\mathsf{Ch}_k$ where the cofibrations are the degree-wise split injections with cofibrant cokernel, the fibrations are the degree-wise surjective maps, and the weak equivalences are the quasi-isomorphisms. It suffices to show that $k$-free contractible chain complexes and $k$-free complexes with zero differentials are cofibrant, and this is easy.