$\require{AMScd}$
I want to understand how the morphism category (arrow category) of $\operatorname{Mod}(A)$ works. The material I have found online have not really helped me that much with proper definitions, so I have made some guesses as to how different concepts might be defined.
Denote the category by $\operatorname{Mor}(A)$ and an object $M \xrightarrow{f} N$ as $Z_{f}$, where obviously $M,N$ are $A$-modules. Morphisms in $\operatorname{Mor}(A)$ are given by commutative squares.
Subobjects:
Let $Z_{f}, Z_{g}$ be two objects in $\operatorname{Mor}(A)$ given by $M \xrightarrow{f} N$ and $M' \xrightarrow{g} N'$ such that $M'$ is a submodule of $M$ and $N'$ is a submodule of $N$, and $f|_{M'} = g$. Then we have a commutative diagram where the vertical maps are inclusions of modules.
\begin{CD}
M' @>{g}>> N'\\
@Vi_{M'}VV @VVi_{N'}V\\
M @>{f}>> N
\end{CD}
Is this the correct way of defining subobjects?
Factors:
Consider the commutative square above, then $\operatorname{Cok}(i_{M'}) = M/M'$ and $\operatorname{Cok}(i_{N'}) = N/N'$. Then we have a commutative square
\begin{CD}
M @>{f}>> N\\
@V\operatorname{Cok}(i_{M'})VV @VV\operatorname{Cok}(i_{N'})V\\
M/M' @>{\overline{f}}>> N/N'
\end{CD}
where $\overline{f}(m+M') = f(m) + N'$
Again, is this the correct way of defining factors?
Two subcategories:
The following two subcategories are defined in Universal localizations via silting by F. Marks and J. Stovicek
\begin{align*}
\mathcal{L} &:= \operatorname{Mor}(\operatorname{proj}(A)) = \{ Z_{\sigma} \: | \: \sigma \in \operatorname{proj}(A) \} \\
\\
\mathcal{BL} &:= \operatorname{Mor}(\operatorname{Proj}(A)) = \{ Z_{\sigma} \: | \: \sigma \in \operatorname{Proj}(A) \}
\end{align*}
He claims that for any $P \in \operatorname{Proj}(A)$, the objects $Z_{id_{P}}$ and $Z_{(0 \rightarrow P)}$ are projective in $\operatorname{Mor}(A)$ and $\mathcal{BL}$, while the objects $Z_{id_{P}}$ and $Z_{(P \rightarrow 0)}$ are injective in $\mathcal{BL}$ but usually not in $\operatorname{Mor}(A)$.
It's pretty straightforward why the two projective objects are in fact projective, however the injective case has me stumped. Take an injective map $\phi : Z_{f} \rightarrow Z_{g}$ between objects in $\mathcal{BL}$, where $P' \xrightarrow{f'} Q'$ and $P'' \xrightarrow{f''} Q''$. Then checking that $Z_{(P \rightarrow 0)}$ is injective reduces to making the following diagram commute
\begin{CD}
P' @>{h}>> P''\\
@V\alpha VV @VV1V\\
P @<\text{show existence}<< P''
\end{CD}
which I can't seem to do. Does anyone know how this works?
Best Answer
I agree with your definitions of subobjects and quotients in $\mathrm{Mor}(A)$.
However, the paper seems to treat $\mathcal{L}$ and $\mathcal{BL}$ as "exact category where conflations [i.e. short exact sequences] are defined as degreewise exact sequences". So a short exact sequence in the context of $\mathcal{L}$ or $\mathcal{BL}$ is an exact sequence in $\mathrm{Mor}(A)$ such that all three terms are in $\mathcal{L}$, resp. in $\mathcal{BL}$.
In particular, since the terms are formed by projectives, these short exact sequences split degree-wise, and injectivity and projectivity is (probably) then meant with respect to these admissible monics and admissible epi's.
(Note that these are still not necessarily split epimorphisms or split monomorphisms in $\mathcal{L}$ or $\mathcal{BL}$, so it is not a vague statement to say that something is injective/projective in these categories.)