Categorical Pullbacks

Marco Riccardi

Formalized Mathematics (2015)

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

Abstract

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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].

How to cite

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Marco 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

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  14. [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
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  16. [16] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990. 
  17. [17] Marco Riccardi. Object-free definition of categories. Formalized Mathematics, 21(3): 193–205, 2013. doi:10.2478/forma-2013-0021. Zbl1298.18001
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