Categorical Framework for the Study of Singular Spaces

Download or read book Categorical Framework for the Study of Singular Spaces written by William Fulton and published by American Mathematical Soc.. This book was released on 1981 with total page 174 pages. Available in PDF, EPUB and Kindle. Book excerpt: In several areas of geometry and topology it has become apparent that the traditional covariant and contravariant functors are insufficient, particularly for dealing with geometric questions about singular spaces. We develop here a new formalism called bivariant theories. These are simultaneous generalizations of covariant group valued "homology-like" theories and contravariant ring valued "cohomology-like" theories. Most traditional pairs of covariant and contravariant theories turn out to extend to bivariant theories. A bivariant theory assigns a group not to an object but to a morphism of the original category; it has products compatible with composition of morphisms. We will also define transformations from one bivariant theory to another, called Grothendieck transformations, which generalize ordinary natural transformations. A number of standard natural transformations turn out to extend to Grothendieck transformations, and this extension has deep consequences.