Journal of Formalized Mathematics
Volume 3, 1991
University of Bialystok
Copyright (c) 1991
Association of Mizar Users
Natural transformations.
Discrete categories
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Andrzej Trybulec
-
Warsaw University, Bialystok
Summary.
-
We present well known concepts of category theory: natural transformations
and functor categories, and prove propositions related to. Because of
the formalization it proved to be convenient to introduce some auxiliary
notions, for instance: transformations. We mean by a transformation of
a functor $F$ to a functor $G$, both covariant functors from $A$ to $B$,
a function mapping the objects of $A$ to the morphisms of $B$ and assigning
to an object $a$ of $A$ an element of $\mathop{\rm Hom}(F(a),G(a))$.
The material included roughly corresponds to that presented on pages
18,129-130,137-138 of the monography ([9]).
We also introduce discrete categories and prove some propositions
to illustrate the concepts introduced.
The terminology and notation used in this paper have been
introduced in the following articles
[10]
[5]
[12]
[11]
[8]
[13]
[2]
[3]
[6]
[1]
[4]
[7]
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Preliminaries
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Application of a functor to a morphism
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Transformations
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Natural transformations
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Functor category
-
Discrete categories
Bibliography
- [1]
Grzegorz Bancerek.
Cardinal numbers.
Journal of Formalized Mathematics,
1, 1989.
- [2]
Czeslaw Bylinski.
Functions and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
- [3]
Czeslaw Bylinski.
Functions from a set to a set.
Journal of Formalized Mathematics,
1, 1989.
- [4]
Czeslaw Bylinski.
Introduction to categories and functors.
Journal of Formalized Mathematics,
1, 1989.
- [5]
Czeslaw Bylinski.
Some basic properties of sets.
Journal of Formalized Mathematics,
1, 1989.
- [6]
Czeslaw Bylinski.
The modification of a function by a function and the iteration of the composition of a function.
Journal of Formalized Mathematics,
2, 1990.
- [7]
Czeslaw Bylinski.
Subcategories and products of categories.
Journal of Formalized Mathematics,
2, 1990.
- [8]
Agata Darmochwal.
Finite sets.
Journal of Formalized Mathematics,
1, 1989.
- [9]
Zbigniew Semadeni and Antoni Wiweger.
\em Wst\c ep do teorii kategorii i funktorow, volume 45 of
\em Biblioteka Matematyczna.
PWN, Warszawa, 1978.
- [10]
Andrzej Trybulec.
Tarski Grothendieck set theory.
Journal of Formalized Mathematics,
Axiomatics, 1989.
- [11]
Andrzej Trybulec.
Tuples, projections and Cartesian products.
Journal of Formalized Mathematics,
1, 1989.
- [12]
Zinaida Trybulec.
Properties of subsets.
Journal of Formalized Mathematics,
1, 1989.
- [13]
Edmund Woronowicz.
Relations and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
Received May 15, 1991
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