Descent for n-Bundles, Tate Objects in Exact Categories, the Index Map and Reciprocity Laws

In Chapter 1, we consider the following. Given a Lie group G, one constructs a principal G-bundle on a manifold X by taking a cover U→X, specifying a transition cocycle on the cover, and descending the trivialized bundle UxG along the cover. We demonstrate the existence of an analogous construction for local n-bundles for general n. We establish analogues for simplicial Lie groups of Moore's results on simplicial groups; these imply that bundles for strict Lie n-groups arise from local n-bundles. Our construction leads to simple finite dimensional models of Lie 2-groups such as String(n).;Chapter 2 and Chapter 3 are logically independent of Chapter 1 and represent joint work with Oliver Braunling and Michael Groechenig. In Chapter 2, we develop Beilinson's theory of Tate objects in an exact category. We develop properties needed for applications to K-theory and we tailor our approach so as to admit geometric examples such as the adeles of a complex curve. We include a comparison of Beilinson's approach to Tate modules over a ring with Drinfeld's. We study the Sato Grassmannian of an elementary Tate object in an idempotent complete exact category. We close by introducing n-Tate objects and establishing a number of their properties.;In Chapter 3, we construct an analogue, in algebraic K-theory, of the weak equivalence, due to Atiyah and Jaenich, between the space of Fredholm operators and the classifying space of topological complex K-theory. This takes the form of an algebraic index map from the looping of the K-theory of Tate objects in an idempotent complete exact category C to the K-theory of C. We relate the index map to higher tame symbols, and we use this to give a homotopy theoretic proof of Parshin--Kato reciprocity laws in close analogy with a proof of Weil reciprocity.