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So I have a couple of questions:

1- Formally speaking, what is a "quotient of a chain complex" of $R$-modules?

2- I want to show that any chain complexes of $R$-modules $C_\bullet $ is the quotient of some projective chain complex.

Although I am not sure about the quotient thing , I'd like to provide my two cents here:

$\underline{\textbf{My attempt}}$

I know that I can find a projective resolution for each $R$-module $C_k$ to get something like this

$\cdots \rightarrow P^k_n \rightarrow P_{n-1}^k \rightarrow \cdots \rightarrow P_0^k \rightarrow C_k \rightarrow 0$.

Now I get several such exact sequences with the $P_i^j $ projective.

I know that I can connect these exact sequences by the comparison theorem,

and I get a collection of projective (?!) complexes $\left\lbrace P_\bullet^j\right\rbrace_j $ for which we have the exact sequence:

$\cdots \rightarrow P^k_\bullet \rightarrow P_\bullet^{k-1} \rightarrow \cdots \rightarrow P_\bullet^0 \rightarrow C_\bullet \rightarrow 0$

Does this make sense?!

And does this mean that $C_\bullet $ is the quotient of a projective complex?

$\underline{\textbf{Another approach}}$

I also thought of writing each $C_i$ as a quotient of a free $R$-module and continuing from there ... But this brings me back to my first question.... What is the quotient of a chain complex?!

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For the first question: Quotients are defined in arbitrary categories. They are represented by epimorphisms. In abelian categories (and chain complexes form an abelian category) epimorphisms come from quotients by subobjects. Here, epi = degreewise epi, and mono = degreewise mono. –  Martin Brandenburg Jan 2 at 19:47
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Also, projective objects are defined in any category. But I doubt that every chain complex is a quotient of a projective one ... but rather of one which is degreewise projective. Search for Cartan-Eilenberg resolution. –  Martin Brandenburg Jan 2 at 20:00

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