On coalgebra based on classes

TL;DR

This paper proves that every endofunctor on the category of classes is set-based, ensuring the existence of a final coalgebra and other fundamental properties, advancing coalgebra theory in class-based contexts.

Contribution

It establishes that all endofunctors on classes are set-based, leading to the existence of final coalgebras and free completely iterative theories.

Findings

Every endofunctor of classes is set-based.

Existence of a final coalgebra for these endofunctors.

Proof of a free completely iterative theory.

Abstract

Every endofunctor of the category of classes is proved to be set-based in the sense of Aczel and Mendler, therefore, it has a final coalgebra. Other basic properties of these endofunctors are proved, e.g. the existence of a free completely iterative theory.