No description available.

For each abelian category A, there is a category D(A), called the derived category of A, whose objects are complexes of objects of A, and whose morphisms are formal fractions of homotopy classes of complex morphisms having as denominators homotopy classes inducing isomorphisms in cohomology. If F : A →B is an additive functor between abelian categories, then under suitable conditions on A, there is a functor RF : D(A) → D(B) with the property that if objects X of A are considered as complexes concentrated at degree 0, then there are isomorphisms [formula omitted] for all n, where [formula omitted] is the ordinary [formula omitted] right derived functor of F. RF is called the derived functor of F, and one may look upon it as a kind of extension of F. / Science, Faculty of / Mathematics, Department of / Graduate

Abstract Not Provided / Thesis / Master of Science (MSc)

No description available.

The purpose of this paper is to examine some basic topics in category theory. A category consists of a class of mathematical objects along with a morphism class having an associative composition. The paper is divided into two chapters. Chapter I deals with intrinsic properties of categories. Various "sub-objects" and properties of morphisms are defined and examples are given. Chapter II deals with morphisms between categories called functors and the natural transformations between functors. Special types of functors are defined and examples are given.

Polynomial functors were introduced by Professors Eilenberg and Mac Lane in 1954, who used them to study certain homology rings. Strict polynomial functors were invented by Professors Friedlander and Suslin in 1997, in order to develop the theory of group schemes. The first real investigation of their intrinsic properties was performed in 1988, when Professor Pirashvili showed that polynomial functors are equivalent to modules over a certain ring. A similar study was conducted on strict polynomial functors in 2003 by Dr. Salomonsson in his doctoral thesis. A radically different method of attack was initiated by Dr. Dreckman and Professors Pirashvili, Franjou, and Baues in the year 2000. Their approach was to combinatorially encode polynomial functors, and utilised for this purpose the category of sets and surjections. Dr. Salomonsson would later repeat the feat for strict polynomial functors, employing instead the category of multi-sets. This thesis proposes the following: 1:o. To generalise the notion of polynomial functor to more general base rings than Z, so that it smoothly agree with the existing definition of strict polynomial functor, allowing for easy comparison. This results in the definition of numerical functors. 2:o. To make an extensive study of numerical maps of modules, to see how they fit into Professor Roby's framework of strict polynomial maps. 3:o. To conduct a survey of numerical rings. 4:o. To develop the theories of numerical and strict polynomial functors so that they run in parallel. 5:o. To show how also numerical functors may be interpreted as modules over a certain ring. 6:o. To expound the theory of mazes, which will be seen to vastly generalise the category of surjections employed by Professor Pirashvili et al., since they turn out to encode, not only polynomial or numerical functors, but all module functors over any base ring. 7:o. To simplify Dr. Salomonsson's construction involving multi-sets, making it more amenable to a comparison with mazes. 8:o. To prove comparison theorems interrelating numerical and strict polynomial functors. 9:o. And, finally, to indicate how polynomial functors may be used to extend the operad concept.

Thesis (Ph. D.)--University of Washington, 2001. / Vita. Includes bibliographical references (p. 177-179).

Note: / Butler's Theorems, with one minor exception, are resolved: in a 2-categorical setting. His Adjointness theorems are all proved correct, after one tiny modification. Then, using a condition on adjoint squares, twenty-two of his Tripleability theorems are proved correct; th~ee are proved false. The other theorem is still unresolved, but it is of very minor importance. / Les theoremes de Butler, a l'exception d'un seul de peu d'importance, sont resulus dans un contexte 2-categorique.Tous ses theoremes d'adjonction sont demontres etre valides apres une modification minime. Ensuit~ utilisant une condition de carres adjoints, vingt-deux de ses Theoremes de monadicite sont demontres et trois autres sont refutes. La validation ou refutation d'un seul de ses +heoremes, de peu d'importance, demeure en suspense.

Butler's Theorems, with one minor exception, are resolved: in a 2-categorical setting. His Adjointness theorems are all proved correct, after one tiny modification. Then, using a condition on adjoint squares, twenty-two of his Tripleability theorems are proved correct; three are proved false. The other theorem is still unresolved, but it is of very minor importance.

Thesis (Ph. D.)--University of Chicago, Dept. of Statistics, August 2000. / Includes bibliographical references. Also available on the Internet.