Category Theory

Category theory looks at mathematics on a large scale: objects and the relations between them, in the abstract. The aim is to strip away inessential details and get to the essence of things. By doing this one finds fundamental concepts - "category" and "functor" being well-known examples - that are very general and therefore invite comparisons between apparently unrelated parts of mathematics. Put another way, if you screw up your eyes then you can sometimes see the similarity between objects that you had previously thought quite different.

Much of modern mathematics is, literally, near-unthinkable without the organizing principles of category theory. This is especially true of algebraic geometry, topology, homological algebra, logic, and theoretical computer science, and increasingly many parts of the mathematical sciences (physics, particularly) are finding categorical ways of thinking to be useful.

Dr Tom Leinster works mainly on higher-dimensional algebra. Naively, this is algebra that cannot be expressed naturally by writing along one-dimensional lines in the customary way; practically, it is the study of structures such as n-categories, operads, and multicategories. These structures are officially algebraic, but have a very high geometric content: naively again, it is almost impossible to understand them without drawing some pictures; at a more sophisticated level, there appear to be intimate connections between higher categorical structures and both homotopy theory and topological quantum field theory. An informal survey of such connections is "Topology and higher-dimensional category theory: the rough idea".

Recently he has finished writing the first book on higher-dimensional algebra, "Higher Operads, Higher Categories". He is currently working on a general theory of self-similar objects (geometrical or not). This has produced in particular the surprising result that every compact metrizable space is, in a certain precise sense, self-similar; the classical result that every compact metrizable space is a continuous image of the Cantor set follows as a corollary.

Richard Steiner works on n-categories, in particular on connections with algebraic and geometrical objects. On the algebraic side, he is investigating relationships between n-categories and chain complexes; see Omega-categories and chain complexes, Homotopy, Homology and Applications 6(1)(2004), 175-200. On the geometric side, he has studied relationships between n-categories and cubes; see Multiple categories: the equivalence of a globular and a cubical approach (with F. A. Al-Agl and R. Brown), Advances in Mathematics 170 (2002), 71-118 . The cubical version of n-categories has been applied to concurrency theory in computing science.