Avoiding the axiom of choice in general category theory
type: Item
instance of: scholarly article
title: Avoiding the axiom of choice in general category theory
zbMATH Open document ID: 0859.18001
DOI: 10.1016/0022-4049(95)00029-1
published in: Journal of Pure and Applied Algebra
publication date: 1997-04-09T00:00:00Z
review text: 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}.
Mathematics Subject Classification ID: 18A05
Mathematics Subject Classification ID: 18A22
zbMATH DE Number: 897485
zbMATH Keywords: weak choice principle
zbMATH Keywords: anafunctors
zbMATH Keywords: cartesian closed bicategory of small categories and anafunctors
Wikidata QID: Q105836937
author: Michael Makkai
reviewed by: Peter Johnstone
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00029-1
OpenAlex ID: W2038364868
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description: scientific article; zbMATH DE number 897485