Categories with New Foundations

While the interaction between set theory and category theory has been studied extensively, the set theories considered have remained almost entirely within the Zermelo family. Quine’s New Foundations has received limited attention, despite being the one-sorted version of a theory mentioned as a possible foundation for Category Theory by Mac Lane and Eilenberg in their seminal paper on the subject. The lack of attention given to NF is not without justification. The category of NF sets is not cartesian closed and the failure of choice is a theorem of NF. But those results should not obscure the aspects of NF that have foundational appeal, nor the value of studying category theory in the context of a universal set. The present research is not intended to “advocate” for the use of NF as a practical foundation for category theory. Instead, the work presents a broad survey of the interaction between the set theory and category theory of NF, examining the relationship in both directions. The abstract structure, of which both type restriction (in the category of NF sets) and size restriction (in the category of all categories) are specific cases, appears to be the study of relative algebra. In a number of cases, the existence of relative algebraic structures in NF can be proven more generally for a class of relative adjoints, (pseudo)monads, etc. Thus, where it seems appropriate to do so, this thesis seeks to contribute to the broader study of relative algebra