Exponential Objects

• Autorzy: Marco Riccardi
• Języki publikacji: EN

Abstrakty

EN

In the first part of this article we formalize the concepts of terminal and initial object, categorical product [4] and natural transformation within a free-object category [1]. In particular, we show that this definition of natural transformation is equivalent to the standard definition [13]. Then we introduce the exponential object using its universal property and we show the isomorphism between the exponential object of categories and the functor category [12].

Słowa kluczowe

EN

exponential objects; functor category; natural transformation

Kategorie tematyczne

• 03B35: Mechanization of proofs and logical operations
• 18A99: None of the above, but in this section
• 18A25: Functor categories, comma categories

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2015
• Tom: 23
• Numer: 4
• Strony: 351-369

Daty

wydano

2015-12-01

otrzymano

2015-08-15

online

2016-03-25

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] Francis Borceaux. Handbook of Categorical Algebra I. Basic Category Theory, volume 50 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1994.
• [5] Czesław Byliński. Introduction to categories and functors. Formalized Mathematics, 1 (2):409–420, 1990.
• [6] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.
• [7] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.
• [8] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.
• [9] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.
• [10] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.
• [11] Krzysztof Hryniewiecki. Graphs. Formalized Mathematics, 2(3):365–370, 1991.
• [12] F. William Lawvere. Functorial semantics of algebraic theories and some algebraic problems in the context of functorial semantics of algebraic theories. Reprints in Theory and Applications of Categories, 5:1–121, 2004.
• [13] Saunders Mac Lane. Categories for the Working Mathematician, volume 5 of Graduate Texts in Mathematics. Springer Verlag, New York, Heidelberg, Berlin, 1971.
• [14] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.
• [15] Marco Riccardi. Object-free definition of categories. Formalized Mathematics, 21(3): 193–205, 2013. doi:10.2478/forma-2013-0021.[Crossref]
• [16] Marco Riccardi. Categorical pullbacks. Formalized Mathematics, 23(1):1–14, 2015. doi:10.2478/forma-2015-0001.[Crossref]
• [17] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.
• [18] Andrzej Trybulec. Isomorphisms of categories. Formalized Mathematics, 2(5):629–634, 1991.
• [19] Andrzej Trybulec. Natural transformations. Discrete categories. Formalized Mathematics, 2(4):467–474, 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

10.1515/forma-2015-0028

bwmeta1.id-class.MML

CAT 8