Although dealing with this in one or other form for many years, to my shame this question only struck me now.
One of the most radical differences between categories of "algebraic" and "topological" kind is absence/presence of nontrivial bimorphisms, i. e. non-invertible morphisms which are both mono and epi. Such guys exist in e. g. distributive lattices or rings, posets or topological spaces; they are absent in e. g. groups or compact Hausdorff spaces. Important case is that of quasitoposes: presence of nontrivial bimorphism is what distinguishes them from toposes.
Now, in homotopical algebra one inverts weak equivalences, and "morally" it very much looks like inverting bimorphisms, although there is no rigorous reason at all that weak equivalences be related to bimorphisms at all, and this is the first thing I don't quite understand:
Is there any connection between inverting weak equivalences and inverting bimorphisms?
Does one obtain any recognizable category by inverting bimorphisms of, say, posets or commutative rings?
Independently whether weak equivalences are "like bimorphisms" or not - ok but what are they anyway? That is, in what way do we improve a, say, model category when we invert weak equivalences? Another example where my ignorance is especially apparent is the case of homotopy categories of complexes versus derived categories. In the former we identify homotopic maps, in the latter - additionally invert quasiisomorphisms. In both cases we obtain triangulated categories. I have no idea how the latter (i. e. the derived category) is a "better" category than the former (i. e. the homotopy category). They both are triangulated, which basically means that all "nice" category-theoretic properties related to limits, etc. are destroyed anyway. So what I don't understand here is this -
Are there some abstract category-theoretic properties which are improved by passing from a homotopy category of chain complexes to the derived category?