Object-Free Definition of Categories

Marco Riccardi

Formalized Mathematics (2013)

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

Abstract

top
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

top

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

top
  1. [1] Jiri Adamek, Horst Herrlich, and George E. Strecker. Abstract and Concrete Categories: The Joy of Cats. Dover Publication, New York, 2009. Zbl0695.18001
  2. [2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990. 
  3. [3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990. 
  4. [4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990. 
  5. [5] Francis Borceaux. Handbook of Categorical Algebra I. Basic Category Theory, volume 50 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1994. 
  6. [6] Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990. 
  7. [7] Czesław Bylinski. Introduction to categories and functors. Formalized Mathematics, 1 (2):409-420, 1990. 
  8. [8] Czesław Bylinski. Subcategories and products of categories. Formalized Mathematics, 1 (4):725-732, 1990. 
  9. [9] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990. 
  10. [10] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990. 
  11. [11] Czesław Bylinski. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521-527, 1990. 
  12. [12] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990. 
  13. [13] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990. 
  14. [14] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990. 
  15. [15] Krzysztof Hryniewiecki. Graphs. Formalized Mathematics, 2(3):365-370, 1991. 
  16. [16] Saunders Mac Lane. Categories for the Working Mathematician, volume 5 of Graduate Texts in Mathematics. Springer Verlag, New York, Heidelberg, Berlin, 1971. 
  17. [17] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990. 
  18. [18] Andrzej Trybulec. Categories without uniqueness of cod and dom. Formalized Mathematics, 5(2):259-267, 1996. 
  19. [19] Andrzej Trybulec. Isomorphisms of categories. Formalized Mathematics, 2(5):629-634, 1991. 
  20. [20] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990. 
  21. [21] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990. 
  22. [22] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.