Category Free Category Theory and Its Philosophical Implications

TL;DR

This paper explores the possibility of formulating category theory without relying on objects, focusing instead on morphisms and functors, and discusses the philosophical implications of such an objectless approach.

Contribution

It introduces the concept of objectless and categoryless formulations of category theory, emphasizing the role of morphisms and identities in philosophical and mathematical contexts.

Findings

Objectless categories can be defined solely in terms of morphisms and identity morphisms.

Category theory can be reformulated entirely using functors and identity functors.

The approach highlights a shift from set-based to arrow-based philosophical ontologies.

Abstract

There exists a dispute in philosophy, going back at least to Leibniz, whether is it possible to view the world as a network of relations and relations between relations with the role of objects, between which these relations hold, entirely eliminated. Category theory seems to be the correct mathematical theory for clarifying conceptual possibilities in this respect. In this theory, objects acquire their identity either by definition, when in defining category we postulate the existence of objects, or formally by the existence of identity morphisms. We show that it is perfectly possible to get rid of the identity of objects by definition, but the formal identity of objects remains as an essential element of the theory. This can be achieved by defining category exclusively in terms of morphisms and identity morphisms (objectless, or object free, category) and, analogously, by defining category theory entirely in terms of functors and identity functors (categoryless, or category free, category theory). With objects and categories eliminated, we focus on the "philosophy of arrows" and the roles various identities play in it (identities as such, identities up to isomorphism, identities up to natural isomorphism...). This perspective elucides a contrast between "set ontology" and "categorical ontology".