Michael Makkai, McGill

Some (possibly weakly) closed categories part 2

I am interested in the so-called "weak" higher dimensional categories (HDC's) as universes in which to do category theory, and intuitionistic set theory as a part of category theory. The adjective "weak" refers to a type-dependent replacement of (Fregean, logical) equality by a "coherence" structure. On the lowest level, this means replacing equality of sets, and more generally, equality of objects in a category, by isomorphisms -- following Bourbaki and Lawvere. The totally-weak HDC's, for instance tricategories, and more generally the Batanin-type n-categories, are ideal from a conceptual point of view, but very difficult to work with, or in. Therefore a coherence theorem, such as the one that establishes that a certain "semi-strict" (or "semi-weak") concept called "Gray category" is "equivalent" to "tricategory" is a welcome excuse to concentrate on the semi-strict concept. To motivate the technical work on Gray categories, I will show how "weak" versions of the usual categorical concepts of pullback and discrete fibration give intuitively convincing access to set-theoretic concepts such as the power-set, differently from topos theory. To begin the mathematics of Gray categories, I will define, for Gray categories X and A, an internal hom-object [X,A], itself a Gray category, and show that it is the basis for an -- at least "weakly" -- closed structure in the sense of Eilenberg and Kelly.