The Algebra and the Logic of
Higher Dimensional Categories

M. Makkai

Higher dimensional categories have recently received much attention, from the quarters of topology, mathematical physics, quantum groups, etc. The present author's interest in them is motivated by foundational considerations. A foundational system, called Structuralist Foundation for Abstract Mathematics (SFAM), is envisaged in which n-dimensional categories, for each non-negative natural number n, are basic (primitive) concepts, replacing the sets and classes of the traditional set-theoretic foundation of mathematics. SFAM is not in a final form yet; however, several aspects of it already have precise mathematical formulations.

As in the case of any foundational system, there are two aspects of SFAM. One is the proposed answer to the question "what does the universe consist of"; the other is the formal language of the system. In the case of SFAM, the two are especially tightly related. The main guide-line is the requirement that any grammatically correct predicate of SFAM should be invariant in the sense appropriate for the argument(s) of the predicate; e.g., any SFAM-property of a variable group must be invariant under isomorphism of groups, any SFAM-property of a variable category must be invariant under equivalence of categories, etc. This requirement of invariance necessitates the exclusion of certain ingredients usually taken for granted. In particular, there is no global equality of all things in SFAM; in fact, the only kind of equality present is the one for elements of a fixed set; in particular, no equality of elements of different sets is present.

The very concept of n-dimensional category we need is not completely formulated as yet. Recently, a breakthrough was achieved by John Baez and James Dolan [1] in this regard; the notion they define is also called "weak n-category". Even more recently, J. Power, C. Hermida and the author have made alterations on the concept, with the intention of simplifying it. Motivated by foundational considerations, the present author's [2] gives a revision of some basic concepts of category theory in low dimensions; in particular, the concept of functor is replaced by "(saturated) anafunctor". The combination of [1] and [2] provides for some of the necessary elements of the global structure formed by higher dimensional categories, e.g. the n+1-dimensional category of all n-dimensional categories.

The language of SFAM is approached in [3] and [4]. Both First Order Logic with Dependent Sorts (FOLDS) of [3], and the syntax of [4] based on (generalized) sketches are intended as "invariant" languages. Both are given detailed meta-theories, although they have not yet been applied to a final form of SFAM.

The talk will give an introduction to the program of SFAM.