Cohomology detects failures of the axiom of choice

Abstract

We propose that failures of the axiom of choice, that is, surjective functions admitting no sections, can be reasonably classified by means of invariants borrowed from algebraic topology. We show that cohomology, when defined so that its usual exactness properties hold even in the absence of the axiom of choice, is adequate for detecting failures of this axiom in the following sense. If a set X, viewed as a discrete space, has trivial first cohomology for all coefficient groups, then every X-indexed family of nonempty sets has a choice function. We also obtain related results when the coefficient groups are required to be abelian or well-orderable. In particular, we show that, if all discrete spaces have trivial first cohomology for all abelian coefficient groups, then the axiom of choice holds. Introduction. The axiom of choice, in one of its many equivalent forms, asserts that, for any surjective function p, from a set Y onto a set X, there exists a section, i.e., a map s: X -Y with p o s = idx. A formally similar concept, the existence (or, more often, nonexistence) of continuous sections for continuous surjections, is one of the central concerns of algebraic topology, and topologists have created an impressive arsen4l of sophisticated tools for analyzing it. It therefore seems reasonable to try to use these tools to describe the ways in which the axiom of choice can fail. The sets and functions that are relevant to the axiom of choice can be viewed as topological spaces and continuous functions, and thus brought formally within the domain of algebraic topology, by simply giving all of the sets the discrete topology. An obvious difficulty with this project is that the homotopy, homology, and cohomology groups of discrete spaces all vanish, under the usual definitions, in all positive dimensions, whether or not the axiom of choice holds. One can circumvent this difficulty by using more exotic invariants, such as K-theory, but we shall adopt a more radical approach. We shall argue, in ?1, that the usual definitions of cohomology are appropriate only in the presence of the axiom of choice; in its absence they fail to satisfy some simple exactness conditions. We therefore propose to adopt a definition of cohomology due to Giraud [3] which has the expected exactness properties even if the axiom of choice is false. With this definition, or indeed with any definition having a certain (rather tiny) amount of exactness, we can at least make a start on the project of using cohomology to describe failures of the axiom of Received by the editors October 7, 1981 and, in revised form, September 15, 1982. 1980 Mathematics Subject Classification. Primary 03E25, 55N99.