The purpose of this paper is to give the algebraic topological background for elliptic cohomology theory and to pose some number theoretic problems suggested by these concepts. Algebraic topologists study functors from the category of spaces to various algebraic categories. In particular there are functors to the category of graded rings called multiplicative generalized cohomology theories. (A...

We describe a fundamental additive functor Efund on the orbit category of a group. We prove that any isomorphism conjecture valid for Efund also holds for all additive functors, like K-theory, (topological) Hochschild or cyclic homology, etc. Finally, we reduce this universal isomorphism conjecture to K-theoretic ones, at the price of introducing some coefficients.

We carry out a detailed comparison of the two topological dualities for distributive meet-semilattices studied by Celani [3] and by Bezhanishvili and Jansana [2]. We carry out such comparison, that was already sketched in [2], by defining the functors involved in the equivalence of both dual categories of distributive meet-semilattices.

0BQ6 Contents 1. Introduction 1 2. Schemesétale over a point 1 3. Galois categories 2 4. Functors and homomorphisms 9 5. Finité etale morphisms 11 6. Fundamental groups 14 7. Topological invariance of the fundamental group 15 8. Finité etale covers of proper schemes 17 9. Local connectedness 19 10. Fundamental groups of normal schemes 24 11. Group actions and integral closure 26 12.

Let A be a sub-uncountable scalar. Recent developments in formal K-theory [20] have raised the question of whether every ordered topological space is additive. We show that 1 0 6= 1. Every student is aware that there exists a nonnegative and anti-injective factor. We wish to extend the results of [20] to Weyl functors.

1. Categories and functors 2. Standard (boring) examples 3. Initial and final objects 4. Categories of diagrams: products and coproducts 5. Example: sets 6. Example: topological spaces 7. Example: products of groups 8. Example: coproducts of abelian groups 9. Example: vectorspaces and duality 10. Limits 11. Colimits 12. Example: nested intersections of sets 13. Example: ascending unions of sets...

Given a group, we construct a fundamental additive functor on its orbit category. We prove that any isomorphism conjecture valid for this fundamental additive functor holds for all additive functors, like K-theory, cyclic homology, topological Hochschild homology, etc. Finally, we reduce this fundamental isomorphism conjecture to K-theoretic ones.

This thesis contains three papers. Paper A and Paper B deal with homotopy theory and Paper C deals with Topological Data Analysis. All three papers are written from a categorical point of view. In Paper A we construct categories of short hammocks and show that their weak homotopy type is that of mapping spaces. While doing this we tackle the problem of applying the nerve to large categories wit...