Homotopy in Exact Categories

Abstract

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 $C{h}_{\ast }(E)$ which are induced by cotorsion pairs on $E$. As a special case we show that under very general conditions the categories $C{h}_{+}(E)$, $C{h}_{\ge 0}(E)$, 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 $C{h}_{\ge 0}(E)$ are homotopical algebra contexts, thus making them suitable settings for derived geometry.

arXiv comment131 pages; major revision based on referee comments: new material on complete and homotopically complete filtered objects, model structures on algebras over operads, and both Dold-Kan and cosimplicial Dold-Kan equivalences for algebras

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Cited by in corpus (1)

• On the Sheafyness Property of Spectra of Banach Rings