# Object-Free Definition of Categories

Formalized Mathematics (2013)

• Volume: 21, Issue: 3, page 193-205
• ISSN: 1426-2630

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## Abstract

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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.

## How to cite

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Marco Riccardi. "Object-Free Definition of Categories." Formalized Mathematics 21.3 (2013): 193-205. <http://eudml.org/doc/267014>.

@article{MarcoRiccardi2013,
abstract = {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.},
author = {Marco Riccardi},
journal = {Formalized Mathematics},
keywords = {object-free category; correspondence between different approaches to category},
language = {eng},
number = {3},
pages = {193-205},
title = {Object-Free Definition of Categories},
url = {http://eudml.org/doc/267014},
volume = {21},
year = {2013},
}

TY - JOUR
AU - Marco Riccardi
TI - Object-Free Definition of Categories
JO - Formalized Mathematics
PY - 2013
VL - 21
IS - 3
SP - 193
EP - 205
AB - 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.
LA - eng
KW - object-free category; correspondence between different approaches to category
UR - http://eudml.org/doc/267014
ER -

## References

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