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 , 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: is a subset of both A and B, and

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

Thus the intersection 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.

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