Avoiding the axiom of choice in general category theory
Category-theorists are accustomed to speak of such things as `the limit of a diagram' or `the adjoint of a functor', neglecting the fact that such things are only defined up to (canonical) isomorphism; in practice, specifying a particular adjoint of a given functor involves making a proper class of arbitrary choices. (Of course, in many particular cases the structure of the categories involved contains enough data to enable us to make the choices in a sensible way.) A number of previous authors (including \textit{G. M. Kelly} in 1964 and \textit{R. Paré} in 1975) have speculated about the possibility of refounding category theory on a concept which, instead of forcing one to make a particular choice for each object of the domain category, allows the possibility of making several choices at once. However, the present paper is (to the best of the reviewer's knowledge) the first in which this possibility has been seriously developed. The author calls these generalized functors `anafunctors', following a suggestion of \textit{D. Pavlović}; the ones which correspond to making all possible choices play a special rôle in the theory, and they are called saturated. The author shows how much of basic category theory can be developed using anafunctors in place of functors, and thereby avoiding arbitrary choices; however, problems arise when we wish to study (ana)functor categories, the basic one being that the category of all anafunctors \(X \to A\) is not small even if both \(X\) and \(A\) are small. But he shows that, under a `weak choice principle' related to \textit{A. Blass}' axiom of `small violations of choice', one can replace this category with an equivalent small category, thus obtaining a cartesian closed bicategory of small categories and anafunctors. Further developments of the theory are promised in two papers in preparation, one of them joint with \textit{R. Paré}.
cartesian closed bicategory of small categories and anafunctors
scientific article; zbMATH DE number 897485
scientific article