The main theme of this thesis is the parallel between results in topos theory and the theory of additive functor categories. In chapter 2, we provide a general overview of the topics used in the rest of the thesis. Locally finitely presentable categories are introduced, and their expression as essentially algebraic categories is explained. The theory of localization for toposes and abelian categories is introduced, and it is shown how these localizations correspond to theories in appropriate logics. In chapter 3, we look at conditions under which the category of modules for a ring object R in a topos E is locally finitely presented, or locally coherent. We show that if E is locally finitely presented, then the category of modules is also; however we show that far stronger conditions are required for the category of modules to be locally coherent. In chapter 4, we show that the Krull-Gabriel dimension of a locally coherent abelian category C is equal to the socle length of the lattice of regular localizations of C. This is used to make an analogous definition of Krull-Gabriel dimension for regular toposes, and the value of this dimension is calculated for the classifying topos of the theory of G-sets, where G is a cyclic group admitting no elements of square order. In chapter 5, we introduce a notion of strong flatness for algebraic categories (in the sense studied by Adamek, Rosickey and Vitale). We show that for a monoid M of finite geometric type, or more generally a small category C with the corresponding condition, the category of M-acts, or more generally the category of set-valued functors on C, has strongly flat covers.

Recent work of Bambozzi, Ben-Bassat, and Kremnitzer suggests that derived analytic geometry over a valued field k can be modelled as geometry relative to the quasi-abelian category of Banach spaces, or rather its completion Ind(Ban_k). In this thesis we develop a robust theory of homotopical algebra in Ch(E) for E any sufficiently 'nice' quasi-abelian, or even exact, category. Firstly we provide sufficient conditions on weakly idempotent complete exact categories E such that various categories of chain complexes in E are equipped with projective model structures. In particular we show that as soon as E has enough projectives, the category Ch₊(E) of bounded below complexes is equipped with a projective model structure. In the case that E also admits all kernels we show that it is also true of Ch≥0(E), 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 Ch(E) also admits a projective model structure. When E is monoidal we also examine when these model structures are monoidal. We then develop the homotopy theory of algebras in Ch(E). In particular we show, under very general conditions, that categories of operadic algebras in Ch(E) 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 Ind(Ban_k).