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).

Thesis (Ph. D.)--University of Washington, 2003. / Vita. Includes bibliographical references (p. 79-80).

Thesis (Ph. D.)--University of Oregon, 2002. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 118-120). Also available for download via the World Wide Web; free to University of Oregon users.

Thesis (Ph. D.)--Duke University, 2007. / Includes bibliographical references.

Thesis (M.Sc.) -- Memorial University of Newfoundland. 1973. / Typescript. Bibliography : leaves 45. Also available online.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1991. / Includes bibliographical references (p. 96-98).

Thesis (Ph. D.)--University of Washington, 2005. / Vita. Includes bibliographical references (p. 101-104 ).

Thesis (doctoral)--Bonn, 1997. / On t.p. x̲ and y̲ are superscript. Includes bibliographical references (p. 117-119).

We construct categories of spectra for two model categories. The first is the category of small categories with the canonical model structure, and the second is the category of directed graphs with the Bisson-Tsemo model structure. In both cases, the category of spectra is homotopically trivial. This implies that the Goodwillie derivatives of the identity functor in each category, if they exist, are weakly equivalent to the zero spectrum. Finally, we give an infinite family of model structures on the category of small categories.

This thesis explores connections between homotopy sheaves, manifold calculus of functors and operad theory. We argue that there is a deep overlap between these, and as evidence we give a new operadic description of the homotopy theoretical obstructions to deforming a smooth immersion into a smooth embedding. We then discuss an application which improves on some aspects of recent results of Arone-Turchin and Dwyer-Hess concerning spaces of long knots and high-dimensional variants. Along the way, we define fibrewise complete Segal spaces, a mild generalisation of Rezk's notion of complete Segal spaces. Also in the context of Segal spaces, we define right fibrations and prove a Grothendieck construction theorem for presheaves with values in spaces. Finally, we prove a result of independent interest which states that weakly k-reduced operads (those with contractible space of operations in arity j ? k) can be strictified when k = 0, 1.