Categorical Pullbacks

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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].

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  • [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.

  • [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.

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