A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in The Faculty of Graduate Studies Department of Mathematics. LetMk be the category of algebras over a unique factorization domain k, and let ind−Affk denote the category of pro-representable functions from Mk to the category E of sets. It is shown that ind−Affk is a closed model category in such...

Let $A$ be a nite dimensional $k-$algebra and $R$ be a locally bounded category such that $R rightarrow R/G = A$ is a Galois covering dened by the action of a torsion-free group of automorphisms of $R$. Following [30], we provide criteria on the convex subcategories of a strongly simply connected category R in order to be a cycle- nite category and describe the module category of $A$. We p...

Let k be a field and let D be a k-linear algebraic triangulated category with split idempotents. Let Σ be the suspension functor of D and let s be a 2-spherical object of D, that is, the morphism space D(s,Σs) is k for i = 0 and i = 2 and vanishes otherwise. Assume that s classically generates D, that is, each object of D can be built from s using (de)suspensions, direct sums, direct summands, ...

LetM be a tensor category with coefficients in a fieldK of characteristic 0, that is, a K-linear pseudo-abelian symmetric monoidal category such that the tensor product ⊗ of M is bilinear. Then symmetric and exterior powers of an object M ∈ M make sense, by using appropriate projectors relative to the action of the symmetric groups on tensor powers of M . One may therefore introduce the zeta fu...

Let L be the function field of a projective space P n k over an algebraically closed field k of characteristic zero, and H be the group of projective transformations. An H-sheaf V on P n k is a collection of isomorphisms V −→ g * V for each g ∈ H satisfying the chain rule. We construct, for any n > 1, a fully faithful functor from the category of finite-dimensional L-semi-linear representations...

Let L be the function field of a projective space P n k over an algebraically closed field k of characteristic zero, and H be the group of projective transformations. An H-sheaf V on P n k is a collection of isomorphisms V −→ g * V for each g ∈ H satisfying the chain rule. We construct, for any n > 1, a fully faithful functor from the category of finite-dimensional L-semi-linear representations...

| In the late 50s Slepian introduced a structure theory for linear, binary codes and proved that every such code was uniquely the sum of indecomposable codes. He had hoped to produce a canonical form for the generator matrix of an indecompos-able code so that he might read oo the properties of the code from such a matrix, but such a program proved impossible. We here work over an arbitrary eld ...

This paper presents an introduction to category theory with an emphasis on those aspects relevant to the analysis of the model theory of linear logic. With this in mind, we focus on the basic definitions of category theory and categorical logic. An analysis of cartesian and cartesian closed categories and their relation to intuitionistic logic is followed by a consideration of symmetric monoida...

Let k be a field of characteristic p. We introduce a formalism of mixed sheaves with coefficients in k and apply it in representation theory. We construct a system of k-linear triangulated category of motives on schemes over Fp, which has a six functor formalism and computes higher Chow groups. Indeed, it behaves similarly to other categories of mixed sheaves that one is used to. We attempt to ...