Journal of Formalized Mathematics
Volume 3, 1991
University of Bialystok
Copyright (c) 1991 Association of Mizar Users

Natural transformations. Discrete categories


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.

MML Identifier: NATTRA_1

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]

Contents (PDF format)

  1. Preliminaries
  2. Application of a functor to a morphism
  3. Transformations
  4. Natural transformations
  5. Functor category
  6. 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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