Category Theory is a Contentful Theory

Abstract

In (Linnebo & Pettigrew, 2011), some objections to category theory as an autonomous foundation are presented. The authors of that paper do a commendable job making clear several distinct senses of “autonomous” as it occurs in the phrase “autonomous foundation.” Unfortunately, the paper seems to treat the “categorist” perspective rather unfairly. Several infelicities of this sort were addressed by (McLarty, 2012). I wish in this note to address yet another apparent infelicity. The subject of this paper is the comments in (Linnebo & Pettigrew, 2011) concerning the contentfulness of William Lawvere’s axiomatic system CCAF. For details of this system itself the reader is encouraged to consult (Lawvere, 1966) or (Lawvere, 1963). No technical details from these expositions will be needed, however. The authors of (Linnebo & Pettigrew, 2011) are willing to admit “CCAF asserts the existence of certain categories and describes some of the functors between them.” However, as they correctly point out, this by itself is insufficient for CCAF to serve as a foundation of mathematics. As the authors point out, a necessary condition for CCAF to serve as a foundation of mathematics is that be contentful, and, “if it is to make a contentful assertion, we need to be able to identify its subject matter – namely, categories – independently of the theory.” The authors provide us an illustrative example of a non-contentful theory to help make this objection more clear. “[A lack of content] would be the problem, for instance, with the theory that consists of the following sentence: ‘the mome raths outgrabe’. The reason that this theory lacks content is that there is no way of identifying, independently of the theory, what mome raths are, or what it is to outgrabe. Without identifying the subject matter of the theory in such a way, the statements of the theory do not make contentful assertions.” (Linnebo & Pettigrew, 2011, p. 231) Presumably (and the authors are certainly correct here) a theory that runs afoul this objection (which I will call the Contentful Theory Objection [CTO]) cannot serve as an autonomous foundation for mathematics. Of course, a proponent of CCAF can respond to CTO by simply providing a CCAF-free identification of the subject matter of category theory. The problem, the authors hold, is that this cannot be accomplished without running afoul what they call the Logical Dependence Objection [LDO]: if any theory T is to provide an alternative foundation for mathematics to the foundation provided by set theory, it must not be the case that T “depend[s] logically on a prior theory of classes and functions in order to ground [its] existential assertions.” That is, if one cannot identify This work was supported by the National Science Foundation under Grant No. 00006595. A side note: CCAF is often assumed to abbreviate either “category of categories and functors” or “category of categories as foundation.” In Lawvere’s thesis, however, the system normally called CCAF appears under the heading “category of categories and adjoint functors”.