Homological and homotopical constructions for functors on ordered groupoids

Abstract

The main topic of this thesis is the generalization to ordered groupoids of some results and constructions that have arisen in groupoid theory and its applications in homological and homotopical algebra. We study fibrations of ordered groupoids, and show that the covering homotopy property and star-surjectivity are not equivalent properties. We establish some formal properties of functors having these properties, and define a new quotient construction for ordered groupoids that leads to a factorization of any functor of ordered groupoids as a star-surjective followed by a star-injective functor. We give a direct proof of Ehresmann’s Maximum Enlargement Theorem. Coupled with our quotient construction, The Maximum Enlargement Theorem gives a universal factorization of any functor of ordered groupoids as a fibration followed by an enlargement followed by a covering. We construct the mapping cocylinder M of an ordered functor : G ! H, and show directly that the morphism M ! H has the covering homotopy property. We construct the derived module D of an ordered functor and use it to study two adjoint functors between the category of ordered crossed complexes and the category of ordered chain complexes. Finally, we consider the groupoid of derivations of crossed modules of groups and of ordered groupoids, and in the latter case we use semiregular crossed modules to derive results on homotopies and endomorphisms. (Mathematical symbols not available - please refer to the PDF).