Avoiding the axiom of choice in general category theory

• Published in: Journal of pure and applied algebra Vol. 108; no. 2; pp. 109 - 173
• Main Author: Makkai, M.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 22.04.1996
• ISSN: 0022-4049; 1873-1376
• DOI: 10.1016/0022-4049(95)00029-1

The notion of anafunctor is introduced. An anafunctor is, roughly, a “functor defined up to isomorphism”. Anafunctors have a general theory paralleling that of ordinary functors; they have natural transformations, they form categories, they can be composed, etc. Anafunctors can be saturated, to ensure that any object isomorphic to a possible value of the anafunctor is also a possible value at the same argument object. The existence of anafunctors in situations when ordinarily one would use choice is ensured without choice; e.g., for a category which has binary products, but not specified binary products, the anaversion of the product functor is canonically definable, unlike the ordinary product functor that needs the axiom of choice. When the composition functors in a bicategory are changed into anafunctors, one obtains anabicategorics. In the standard definitions of bicategories such as the monoidal category of modules over a ring, or the bicategory of spans in a category with pullbacks, and many others, one uses choice; the anaversions of these bicategories have canonical definitions. The overall effect is an elimination of the axiom of choice, and of non-canonical choices, in large parts of general category theory. To ensure the Cartesian closed character of the bicategory of small categories, with anafunctors as 1-cells, one uses a weak version of the axiom of choice, which is related to A. Blass' axiom of Small Violations of Choice (1979).