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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

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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.

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.

Bibliography: pages 90-94. / Ordered spaces were introduced by Leopoldo Nachbin [1948 a, b, c, 1950, 1965]. We will be primarily concerned with completely regular ordered spaces, because they are precisely those ordered spaces which admit quasi-uniform structures. A recent and convenient study of these spaces is in the book by P. Fletcher and W.F. Lindgren [1982]. In this thesis we consider functorial quasi-uniformities over (partially) ordered spaces. The functorial methods which we use were developed by Brummer [1971, 1977, 1979, 1982] and Brummer and Hager [1984, 1987] in the context of functorial uniformities over completely regular topological spaces, and of functorial quasi-uniformities over pairwise. completely regular bitopological spaces. We obtain results which are to a large extent analogous to results in those papers. We also introduce some functors which relate our functorial quasi-uniformities to the structures studied by Brummer and others (e.g. Salbany [1984]).

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