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We define morphism (E, M)-structures in an abstract category, develop their basic properties and present some examples. We also consider the existence of such factorization structures, and find conditions under which they can be extended to factorization structures for certain classes of sources. There is a Galois correspondence between the collection of all subclasses of X-morphisms and the collection of all subclasses of X-objects. A-epimorphisms diagonalize over A-regular morphisms. Given an (E, M)-factorization structure on a finitely complete category, E-separated objects are those for which diagonal morphisms lie in M. Other characterizations of E-separated objects are given. We give a bijective correspondence between the class of all (E, M)factorization structures with M contained in the class of all X-embeddings and the class of all strong limit operators. We study M-preserving morphisms, M-perfect morphisms and M-compact objects in a morphism (E, M)-hereditary construct, and prove some of their properties which are analogous to the topological ones. / Mathematical Sciences / M. Sc. (Mathematics)

We develop some basic homological theory of hopfological algebra as defined by Khovanov. A simplicial bar resolution for an arbitrary hopfological module is constructed, and some derived analogue of Morita theory is established. We also discuss about some special classes of examples that appear naturally in categorification.

The compact symmetric multicategories (CSMs) introduced by Joyal and Kock in their 2011 note 'Feynman Graphs, and Nerve Theorem for Compact Symmetric Multicategories' [JK11] directly generalise a number of unital operad types, such as wheeled properads, that admit a contraction operation as well as an operadic multiplication. These structures are known to exhibit strange behaviour related to the contraction of units, and this is problematic for [JK11]. In this thesis, I modify the construction of [JK11] to obtain non unital (coloured) modular operads as algebras for a monad defined in terms of connected graphs, and use this as a foundation for a new construction of CSMs based on special graph morphisms. A corresponding nerve theorem characterises CSMs in terms of a Segal condition. This construction sheds light, and provides some control, on the behaviour of the contracted units.

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Thesis (Ph.D.)--Tufts University, 2004. / Adviser: Loring W. Tu. Submitted to the Dept. of Mathematics. Includes bibliographical references (leaves 43-45). Access restricted to members of the Tufts University community. Also available via the World Wide Web;

Thesis (Ph. D.)--University of Canterbury, 2007. / Typescript (photocopy). Includes bibliographical references (p. 127-135). Also available via the World Wide Web.