Volume 3, 1991

University of Bialystok

Copyright (c) 1991 Association of Mizar Users

**Andrzej Trybulec**- Warsaw University, Bialystok

- 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.

- Preliminaries
- Application of a functor to a morphism
- Transformations
- Natural transformations
- Functor category
- Discrete categories

- [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.

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