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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year

SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

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Volume 21, Issue 3


Object-Free Definition of Categories

Marco Riccardi


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

  • [1] Jiri Adamek, Horst Herrlich, and George E. Strecker. Abstract and Concrete Categories: The Joy of Cats. Dover Publication, New York, 2009.Google Scholar

  • [2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Google Scholar

  • [3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Google Scholar

  • [4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Google Scholar

  • [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.Google Scholar

  • [6] Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Google Scholar

  • [7] Czesław Bylinski. Introduction to categories and functors. Formalized Mathematics, 1 (2):409-420, 1990.Google Scholar

  • [8] Czesław Bylinski. Subcategories and products of categories. Formalized Mathematics, 1 (4):725-732, 1990.Google Scholar

  • [9] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Google Scholar

  • [10] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Google Scholar

  • [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.Google Scholar

  • [12] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Google Scholar

  • [13] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Google Scholar

  • [14] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Google Scholar

  • [15] Krzysztof Hryniewiecki. Graphs. Formalized Mathematics, 2(3):365-370, 1991.Google Scholar

  • [16] Saunders Mac Lane. Categories for the Working Mathematician, volume 5 of Graduate Texts in Mathematics. Springer Verlag, New York, Heidelberg, Berlin, 1971.Google Scholar

  • [17] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Google Scholar

  • [18] Andrzej Trybulec. Categories without uniqueness of cod and dom. Formalized Mathematics, 5(2):259-267, 1996.Google Scholar

  • [19] Andrzej Trybulec. Isomorphisms of categories. Formalized Mathematics, 2(5):629-634, 1991.Google Scholar

  • [20] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Google Scholar

  • [21] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Google Scholar

  • [22] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990. Google Scholar

About the article

Published in Print: 2013-10-01

Citation Information: Formalized Mathematics, Volume 21, Issue 3, Pages 193–205, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.2478/forma-2013-0021.

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