Homotopy in Exact Categories

In this monograph we develop various aspects of the homotopy theory of exact
categories. We introduce different notions of compactness and generation in
exact categories $E$, and use these to study model structures on categories of
chain complexes $Ch_{*}(E)$ which are induced by cotorsion pairs on $E$. As a
special case we show that under very general conditions the categories
$Ch_{+}(E)$, $Ch_{\ge0}(E)$, and $Ch(E)$ are equipped with the projective model
structure, and that a generalisation of the Dold-Kan correspondence holds. We
also establish conditions under which categories of filtered objects in exact
categories are equipped with natural model structures. When $E$ is monoidal we
also examine when these model structures are monoidal and conclude by studying
some homotopical algebra in such categories. In particular we provide
conditions under which $Ch(E)$ and $Ch_{\ge0}(E)$ are homotopical algebra
contexts, thus making them suitable settings for derived geometry.