Object-Free Definition of Categories

• Autorzy: Marco Riccardi
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

object-free category; correspondence between different approaches to category

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2013
• Tom: 21
• Numer: 3
• Strony: 193-205

Daty

wydano

2013-10-01

Twórcy

autor

Marco Riccardi

• Via del Pero 102 54038 Montignoso Italy

Bibliografia

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

Identyfikatory

DOI

10.2478/forma-2013-0021