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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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SCImago Journal Rank (SJR) 2015: 0.134
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ISSN
1898-9934
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Volume 21, Issue 3 (Oct 2013)

Issues

Object-Free Definition of Categories

Marco Riccardi
  • Via del Pero 102 54038 Montignoso Italy

Summary

Category theory was formalized in Mizar with two different approaches [7], [18] that correspond to those most commonly used [16], [5]. Since there is a one-to-one correspondence between objects and identity morphisms, some authors have used an approach that does not refer to objects as elements of the theory, and are usually indicated as object-free category [1] or as arrowsonly category [16]. In this article is proposed a new definition of an object-free category, introducing the two properties: left composable and right composable, and a simplification of the notation through a symbol, a binary relation between morphisms, that indicates whether the composition is defined. In the final part we define two functions that allow to switch from the two definitions, with and without objects, and it is shown that their composition produces isomorphic categories.

Keywords: object-free category; correspondence between different approaches to category

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About the article

Published in Print: 2013-10-01


Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.2478/forma-2013-0021. Export Citation

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