Higher dimensional categories

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The standard foundations of mathematics are provided by Set Theory, with the Zermelo-Fraenkel axiomatization. Category Theory initially arose as a tool to organize certain branches of mathematics such as Algebra, Topology and Algebraic Geometry. Starting in the 1960's, the realization has been growing that Category Theory may replace Set Theory as the foundations of mathematics as a whole. The decisive discovery in this regard was that of the concept of elementary topos by F. W. Lawvere and M. Tierney in 1970. A topos is a category that serves as a universe in which mathematics can be developed. The objects of the topos are regarded as abstract sets; they are related to each other by the morphisms, which are abstract functions, and by the constructions that are available by the axioms of a topos. A topos is limited in that the usual large categories themselves are not objects of it. One needs another universe, which, however, is no longer a category, but a 2-dimensional category, to accomodate categories as objects. The need for ever higher dimensional categories will not stop, and the universe has to be taken as consisting of all the  n-categories, for various  n. Another aspect of the new foundational theory is the realization that entities of the universe will have varying concepts of identity, depending on their types. This fact gives rise to the so-called weak n-categories whose precise determination is the main present task of the theory of Higher Dimensional Categories (HDC's). HDc's are also important for varios mathematical purposes, and also in Theoretical Physics. Although the applcant's work is focused on the foundational aspects of HDC's rather than the more mathematical ones, he uses all the available techniques of Category Theory to clarify and validate his approach.