Category Free Category Theory and Its Philosophical Implications

• Authors: Heller Michael
• Languages of publication: EN

Abstracts

EN

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 elucidates a contrast between “set ontology” and “categorical ontology”.

Keywords

EN

category theory; identity; individuality; “philosophy of arrows”

• Publisher: Uniwersytet Mikołaja Kopernika (Nicolaus Copernicus University in Toruń)
• Journal: Logic and Logical Philosophy
• Year: 2016
• Volume: 25
• Issue: 4
• Pages: 447-459

Dates

online

2016-06-14

Contributors

author

Heller Michael

• Copernicus Center for Interdisciplinary Studies, ul. Sławkowska 17, 31-016 Kraków, Poland
• The Pontifical University of John Paul II, Faculty of Philosophy, ul. Kanonicza 9, 31-002 Kraków, Poland

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Identifiers

DOI

10.12775/LLP.2016.013