# On Self-Predicative Universals in Category Theory

25 Pages Posted: 12 May 2015

Date Written: May 11, 2015

### Abstract

1. This paper shows how the universals of category theory in mathematics provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in Hegel and similar ideas of paradigmatic exemplars in ordinary thought.

2. The paper also shows how the always-self-predicative universals of category theory provide the "opposite bookend" to the never-self-predicative universals of iterative set theory and thus that the paradoxes arose from having one theory (e.g., Frege's Paradise) where universals could be either self-predicative or non-self-predicative (instead of being always one or the other).

3. Moreover the paper considers one of the most important examples of self-predicative universals in pure mathematics, namely adjoint functors or adjunctions. It gives a parsing of adjunctions into two halves (left and right semi-adjunctions) using the heterodox notion of heteromorphisms, and then shows that the parts can be recombined in a new way to define the cognate-to-adjoints notion of a brain functor that provides an abstract conceptual model of a brain.

4. Finally the paper argues that at least one way category theory has foundational relevance is that it isolates the universal concepts and structures that are important throughout mathematics.

**Keywords:** category theory, universals, Platonic philosophy, logic, set theory paradoxes

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