Category Theory with Stratified Set Theory

TL;DR

This paper explores the categorical structure of stratified set theories like NF and KF, introducing new functorial concepts, a stratified Yoneda Lemma, and a novel smallness notion, expanding their foundational understanding.

Contribution

It develops a categorical framework for stratified set theories, including a functorial T-operation, a stratified Yoneda Lemma, and a new axiom SCU for NF without Choice.

Findings

Categorical properties of NF and KF are characterized.

A stratified Yoneda Lemma is established.

NF + SCU exhibits specific properties without Choice.

Abstract

This paper examines the category theory of stratified set theory (NF and KF). We work out the properties of the relevant categories of sets, and introduce a functorial analogue to Specker's T-operation. Such a development leads one to consider the appropriate notion of "elementary topos" for stratified set theories. In addition to considering the categorical properties of a generic model of NF set theory, we identify a stratified Yoneda Lemma and show NF encodes itself as a full internal subcategory. Finally, our desire to examine NF in the context of category theory motivates a more precise examination of strongly cantorian as an appropriate notion of smallness, replacing it with the notion of fibrewise strongly cantorian. In the absence of Choice, we introduce a new axiom (SCU) to NF, and examine some properties of NF + SCU.