I am thinking about K-theory. The first idea I had was about a construction which seems similar${}^{**}$. The similar idea featured some kind of cofibrant replacement of R-mod (as an infinity category). Here cofibrant could be interpreted in terms of the three anodyne model structures spelled out in Lurie's HTT on page 53.
In this approach, we form a cofibrant resolution of R-mod in ∞-Cat, afterwards performing some operation like [-,Y].
My question is, how does this construction relate to $K_n(R)$?
Another question I have is whether the category of projective R-modules is related to a cofibrant replacement of the category of R-modules. This would relate "internally projective" and "externally projective".
${}^{**}$ The question is intended in some universe polymorphic type system, in which the computer often has to be supplied with the information of which objects get a one universe advantage.