This question might be a bit dumb but I'm tired right now and this is just going over my head at the moment, in "The homology of Banach and topological algebras" Helemskii said that relative homological algebra can be used to define certain homological concepts in categories of locally convex modules, he refers to this paper by Taylor (http://bit.ly/1BeS6q5), I was reading this paper but there are a couple of things I was confused about:
There's a map $\epsilon: C \rightarrow A$ defined on page 141, but what is $C$?
I went to page 146 where Taylor says:
Note that our projective modules are not projective objects in the category of all $A$-modules. They are, however, relative projectives in the sense of chapter 13 of "Homology" by Sanders Mac Lane, if the class of allowable short exact sequences is taken to be the class of $C$-split short exact sequences.
In the context of relative homological algebra a relative projective is a $\square$-allowable projective objet $P$ in some abelian category $\mathcal{A}$, where $\square$ is a covariant functor between two abelian categories $\mathcal{A}$ and $\mathcal{B}$ which is additive, exact and faitful, if in this case $\mathcal{A}$ is the category of all $A$-modules, What is the abelian category $\mathcal{B}$?
Regarding your question on notation in Taylor's paper $C$ means the field of complex numbers and $\epsilon$ stands for the natural embedding of $C$ into unital algebra $A$.
As for the deinition of relative homology you must understand that you can't transfer Mac Lane's definition word by word to the realm of topological modules for one simple reason - their categories are not abelian. But you don't need to do so!
In fact you don't need to put any restrictions on $\mathcal{A}$ and $\mathcal{B}$ in order for this definition to work. For example if you take $\mathcal{A}$ as category of compact Hausdorff topological spaces, $\mathcal{B}$ the category of all sets and $\square:\mathcal{A}\to\mathcal{B}$ as obvious forgetful functor, then $\square$-allowable projective Husdorff topological spaces are extremelly disconnected compact spaces. This result is due to Rainwater, but, of course, he hasn't used such fancy language.