I'm fairly new to thinking about homological algebra and chain complexes in their own right, *i.e* outside of isolated examples such as for constructing simplicial homology, or for computing $Ext$ groups for some Hopf algebroid.

Given an abelian category $\mathscr{A}$ with a category of chain complexes $Ch(\mathscr{A})$, the homotopy category
$K(\mathscr{A})$ is defined as the naive homotopy category (*i.e.* replace chain maps with chain homotopy classes of chain maps) but with quasi-isomorphisms inverted (these are the maps which induce isomorphisms on homology). This is obviously the result of placing some kind of model structure on $Ch(\mathscr{A})$. This leads me to consider a few obvious questions.

- Are there alternate, interesting model structures for $Ch(\mathscr{A})$?
- (How) has the advancement of model category theory impacted the study of things such as chain complexes, and perverse sheaves for example?
- Are there any good treatments of homological algebra which makes use of the rich theory of model categories?