Let $\mathscr{C}$ be a DG category (not much will be lost if you assume that $\mathscr{C}$ has one object, i.e. is a DG algebra). One way to construct the unbounded derived category of $\mathscr{C}$-modules is by using semi-free resolutions. Recall that a DG module $F$ is *free* if it is a sum of shifts of corepresentable modules, or *semi-free* if it has an exhaustive filtration $0 = F\_0 \subset F\_1 \subset \cdots \subset F$ such that the subquotients $F\_{n+1}/F\_n$ are free.


I've seen the existence of semi-free resolutions stated as follows: for any DG module $M$, there is a semi-free module $F$ and a surjective quasi-isomorphism $F \to M$. But how does one actually construct $F$? I'm especially confused about how to make $F \to M$ surjective, since generators of free modules are cycles.


**Edit:** I found a construction of $F \to M$ in Drinfeld's paper "DG quotients of DG categories." Start by choosing a free module $F\_1$ and a morphism $F\_1 \to M$ which is surjective on cohomology. Now induct: given $F\_n \to M$, let $C\_n = \text{Cone}(F\_n \to M)$ then find a free module $P\_n$ and a morphism $P\_n \to C\_n$ which is surjective on cohomology. Using the map $C\_n \to F\_n[1]$ we get $P\_n \to F\_n[1]$, and Drinfeld takes $F\_{n+1} = \text{Cone}(P\_n[-1] \to F\_n)$. Of course $F$ is the direct limit of the $F\_n$.


This is nice, but I still don't understand how to make $F \to M$ surjective. Does this happen automatically for the construction given above, or do we have to do something extra?



Semi-free resolutions
There is a recent paper of Tobi Barthel, Emily Riehl, and myself
that answers this question in a model categorical framework. We
were lazy and only considered the one object case, although we
believe our work generalizes to DG categories. In another respect we
were not lazy: we work over a general commutative ring $R$, not just
a field, and this introduces interesting subtleties. The reference is


<https://arxiv.org/abs/1310.1159>


The standard model structure on (unbounded) DG modules over a DG
$R$-algebra $A$ takes quasi-isomorphisms as weak equivalences.
Cofibrant approximations are more general than semi-free resolutions,
but they are retracts of semi-free resolutions given by model theoretic
cellular DG modules. There is an early construction by my adviser, John Moore,
in the 1959-60 Cartan seminar, which we modernize. This uses bicomplexes,
as usual in differential homological algebra. There is another construction,
originally due to Gugenheim and myself, dating from 1974, which we also
modernize. Qiaochu, you will be interested that we drop surjectivity in that
construction, meaning that we do not have model theoretic fibrations. We use
multicomplexes, which are bigraded but have differentials that are sums of
pieces that mimic differentials in spectral sequences. The drop of
surjectivity and the use of multicomplexes gives a great gain of
computability, as we illustrate by modernizing 1974 applications to the
computation of the cohomology of many homogeneous spaces.


We also explain the role of the bar construction. When R is a field it
constructs cofibrant approximations in the standard model structure. In
general, it constructs cofibrant approximations in a relative model
structure for which the weak equivalences are the maps of DG $A$-modules
which are chain homotopy equivalences of underlying $R$-modules, and in
that generality it is not semi-free.


