# Concrete Universals in Category Theory

This old essay deals with a connection between a relatively recent (1940s and 1950s) field of mathematics, category theory, and a hitherto vague notion of philosophical logic usually associated with Plato, the self-predicative universal or concrete universal.  While category theory can be quite forbidding to the non-specialist, simple examples can be used based on inclusion between sets.  Given two sets A and B, consider the property of being a set X that is contained in A and is contained in B.  In other words, the property is the property of being both a subset of A and a subset of B.  In this example, the “participation” relation is the subset relation.  There is a set, namely the intersection, meet, or overlap of A and B, denoted $A\cap B$, that has the property (so it is a “concrete” instance of the property), and it is universal in the sense that any other set has the property if and only if it participates in the universal example:

concreteness: $A\cap B$ is a subset of both A and B, and

universality:          X is a subset of $A\cap B$ if and only if X is contained in both A and B.

Thus the intersection $A\cap B$ is the concrete universal for the property of being a subset of A and a subset of B.  I argue that category theory is relevant to foundations as the theory of concrete universals.  Category theory provides the framework to identify the concrete universals in mathematics, the concrete instances of a mathematical property that exemplify the property is such a perfect and paradigmatic way that all other instances have the property by virtue of participating in the concrete universal.

This paper is also an updated version of a chapter in the Intellectual Trespassing book which can be downloaded by clicking on the title.