Axiomatizing a Category of Categories

Elementary axioms describe a category of categories. Theorems of category theory follow, including some on adjunctions and triples. A new result is that associativity of composition in categories follows from Cartesian closedness of the category of categories. The axioms plus an axiom of infinity are consistent iff the axioms for a well-pointed topos with separation axiom and natural numbers are. The theory is not finitely axiomatizable. Each axiom is independent of the others. Further independence and definability results are proved. Relations between categories and sets, the latter defined as discrete categories, are described, and applications to foundations are discussed. ?0. Introduction. The following elementary axioms produce the major theorems of general category theory, with adjunctions, tripleability and the central theorems of topos theory as test cases. The axioms plus an axiom of infinity are equiconsistent with the axioms for a well-pointed topos with natural number object plus the axiom of separation. The axioms are independent of one another, and several theorems are shown independent of other theorems plus all but one of the axioms. We show which categories can be proved to exist by these axioms plus one further assumption. All are finite. It follows that the categories with all finite products which can be proved to exist are equivalent to lattices, and no nontrivial topos can be proved to exist. It is consistent with the axioms for the coequalizer of the two functors from 1 to 2 to be any cyclic group or the group of integers. This paper is heavily indebted to Lawvere [6], but that pioneering work had technical flaws. Blanc and Donnadieu [2] rectified the flaws by relying completely on sets in the form of discrete categories. Their axioms make categories and functors definable in terms of discrete categories, and all the work is actually done with discretes. The present axioms seem not to imply that every category A has a maximal discrete subcategory (i.e. a set of all objects of A), and we prove that they do not imply every internally defined set of arrows and objects equipped with a category structure corresponds to an actual category. One or both of these nontheorems may be important to future efforts to describe a category of categories with an actual category of all the categories in it. There cannot be an actual category of all categories if sets form a topos, every category has a set of objects, and every set of objects and arrows with a category structure corresponds to a category. Received December 6, 1989; revised November 15, 1990. ? 1991, Association for Symbolic Logic 0022-4812/91/5604-0007/$02.80