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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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SCImago Journal Rank (SJR) 2015: 0.134
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1898-9934
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In This Section
Volume 23, Issue 1 (Mar 2015)

Issues

Categorical Pullbacks

Marco Riccardi
  • Via del Pero 102, 54038 Montignoso, Italy
Published Online: 2015-03-31 | DOI: https://doi.org/10.2478/forma-2015-0001

Summary

The main purpose of this article is to introduce the categorical concept of pullback in Mizar. In the first part of this article we redefine homsets, monomorphisms, epimorpshisms and isomorphisms [7] within a free-object category [1] and it is shown there that ordinal numbers can be considered as categories. Then the pullback is introduced in terms of its universal property and the Pullback Lemma is formalized [15]. In the last part of the article we formalize the pullback of functors [14] and it is also shown that it is not possible to write an equivalent definition in the context of the previous Mizar formalization of category theory [8].

MSC: 18A30; 03B35

Keywords: category pullback; pullback lemma

MML: identifier: CAT 7; version: 8.1.03 5.29.1227

References

  • [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. The well ordering relations. Formalized Mathematics, 1(1):123–129, 1990.

  • [5] Grzegorz Bancerek. Zermelo theorem and axiom of choice. Formalized Mathematics, 1 (2):265–267, 1990.

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

  • [7] Francis Borceaux. Handbook of Categorical Algebra I. Basic Category Theory, volume 50 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1994.

  • [8] Czesław Byliński. Introduction to categories and functors. Formalized Mathematics, 1 (2):409–420, 1990.

  • [9] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.

  • [10] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

  • [11] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

  • [12] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

  • [13] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

  • [14] 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.

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

  • [16] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.

  • [17] Marco Riccardi. Object-free definition of categories. Formalized Mathematics, 21(3): 193–205, 2013. doi:10.2478/forma-2013-0021. [Crossref]

  • [18] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.

  • [19] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

  • [20] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.

  • [21] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

About the article

Received: 2014-12-31

Published Online: 2015-03-31

Published in Print: 2015-03-01



Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, DOI: https://doi.org/10.2478/forma-2015-0001. Export Citation

© Marco Riccardi. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. (CC BY-SA 3.0)

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