Department of Mathematical Sciences

Mathematics

Category Theory

Category Theory brings understanding of pure mathematics at a deep foundational level. In some sense it is a next step after Set Theory in organizing mathematics. The main areas of interest of the Stellenbosch University Category Theory research group lie within General Category Theory, as well as Category Theory specifically arising in General Topology, Universal Algebra, and in the interaction between Geometry and Physics.

Members of this group are actively involved in international collaborations, as well as being part of the local Category Theory community (which also includes the University of Cape Town and the University of Western Cape).

Prof David Holgate’s particular interest is in Category Theory and Topology. Current topics being researched by himself along with graduate students and collaborators include:

  • Lax algebras and enriched categories – categorical structures which provide a good foundation for topology, and in particular the study of uniform topology.
  • Closure operators – categorical tools inspired by topology and inspiring the application of topological methods in other branches of mathematics.
  • Frame or locale theory – lattice theoretic approach to studying topology.
    The recent book M.C. Peddichio & W. Tholen (Editors). Categorical foundations, Encyclopedia Math. Appl., 97, Cambridge Univ. Press, Cambridge, 2004, provides a collection of articles with an overview of the subject area.

Recent publications showing Prof Holgate’s research interests include:

  1. G.C.L. Brümmer, E. Giuli and D. Holgate. Splitting closure operators. Acta Math. Hungar. 106(1-2), 2005, 1-16;
  2. D. Holgate and M. Sioen. Approach structures and measures of connectedness. Quaestiones Mathematicae, 30, 2007, 321–334;
  3. D. Holgate. A generalisation of the functional approach to compactness. Topology and its Applications, 2009.doi:10.1016/j.topol.2009.03.032;
  4. G. Boxall and D. Holgate. Connected and disconnected maps. To appear in Applied Categorical Structures.

Dr Zurab Janelidze’s particular interest is in Categorical Algebra, which has strong connections with Universal Algebra. The areas of his current research include:

  • The study of subtractive categories in the sense of Z. Janelidze, which generalize pointed subtractive varieties of universal algebras in the sense of A. Ursini; among these categories are many important categories studied in categorical algebra such as pointed Mal’tsev categories, pointed protomodular categories, and additive categories.
  • Relativisation of topics from algebra with respect to cover relations, which are closely related with closure operators mentioned above.
  • Categorical term conditions, i.e. investigation of general methods of lifting certain universal-algebraic conditions (such as Mal’tsev conditions), and results / arguments involving these conditions to the level of general categories.

Recent publications showing Dr Janelidze’s research interests include:

  1. Z. Janelidze, Closedness properties of internal relations V: Linear Mal’tsev conditions, Algebra Universalis 58, 2008, 105-117;
  2. D. Bourn and Z. Janelidze, Pointed protomodularity via natural imaginary subtractions, Journal of Pure and Applied Algebra, 2009, doi:10.1016/j.jpaa.2009.02.005;
  3. Z. Janelidze and A. Ursini, Split short five lemma for clots and subtractive categories, Applied Categorical Structures, 2009, doi:10.1007/s10485-009-9192-5.

Dr. Bruce Bartlett’s particular research interest is in Category Theory arising from the interaction between Geometry and Physics (see the section Quantum Topology for more information).

Recent publications showing Dr. B. Bartlett’s research interests include:

  1.  B. Bartlett, The Geometry of Unitary 2-Representations of Finite Groups and their 2-Characters, Applied Categorical Structures, doi:10.1007/s10485-009-9189-0;
  2. B. Bartlett, On unitary 2-representations of finite groups and topological quantum field theory, PhD thesis, University of Sheffield (2008). Available at arxiv:0901.3975.