| Summary: | <p>Recent work of Bambozzi, Ben-Bassat, and Kremnitzer suggests that derived analytic geometry over a valued field <em>k</em> can be modelled as geometry relative to the quasi-abelian category of Banach spaces, or rather its completion <em>Ind</em>(<em>Ban<sub>k</sub></em>). In this thesis we develop a robust theory of homotopical algebra in <em>Ch</em>(<em>E</em>) for <em>E</em> any sufficiently 'nice' quasi-abelian, or even exact, category.</p> <p>Firstly we provide sufficient conditions on weakly idempotent complete exact categories <em>E</em> such that various categories of chain complexes in <em>E</em> are equipped with projective model structures. In particular we show that as soon as <em>E</em> has enough projectives, the category <em>Ch<sub>+</sub></em>(<em>E</em>) of bounded below complexes is equipped with a projective model structure. In the case that <em>E</em> also admits all kernels we show that it is also true of <em>Ch<sub>≥0</sub></em>(<em>E</em>), and that a generalisation of the Dold-Kan correspondence holds. Supplementing the existence of kernels with a condition on the existence and exactness of certain direct limit functors guarantees that the category of unbounded chain complexes <em>Ch</em>(<em>E</em>) also admits a projective model structure. When <em>E</em> is monoidal we also examine when these model structures are monoidal.</p> <p>We then develop the homotopy theory of algebras in <em>Ch</em>(<em>E</em>). In particular we show, under very general conditions, that categories of operadic algebras in <em>Ch</em>(<em>E</em>) can be equipped with transferred model structures. Specialising to quasi-abelian categories we prove our main theorem, which is a vast generalisation of Koszul duality. We conclude by defining analytic extensions of the Koszul dual of a Lie algebra in <em>Ind</em>(<em>Ban<sub>k</sub></em>).</p>
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