# Categorical Pullbacks

Formalized Mathematics (2015)

- Volume: 23, Issue: 1, page 1-14
- ISSN: 1426-2630

## Access Full Article

top## Abstract

top## How to cite

topMarco Riccardi. "Categorical Pullbacks." Formalized Mathematics 23.1 (2015): 1-14. <http://eudml.org/doc/270913>.

@article{MarcoRiccardi2015,

abstract = {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].},

author = {Marco Riccardi},

journal = {Formalized Mathematics},

keywords = {category pullback; pullback lemma},

language = {eng},

number = {1},

pages = {1-14},

title = {Categorical Pullbacks},

url = {http://eudml.org/doc/270913},

volume = {23},

year = {2015},

}

TY - JOUR

AU - Marco Riccardi

TI - Categorical Pullbacks

JO - Formalized Mathematics

PY - 2015

VL - 23

IS - 1

SP - 1

EP - 14

AB - 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].

LA - eng

KW - category pullback; pullback lemma

UR - http://eudml.org/doc/270913

ER -

## References

top- [1] Jiri Adamek, Horst Herrlich, and George E. Strecker. Abstract and Concrete Categories: The Joy of Cats. Dover Publication, New York, 2009. Zbl0695.18001
- [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. Zbl1062.18004
- [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. Zbl1298.18001
- [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.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.