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

Category Ens


Czeslaw Bylinski
Warsaw University, Bialystok

Summary.

If $V$ is any non-empty set of sets, we define $\hbox{\bf Ens}_V$ to be the category with the objects of all sets $X \in V$, morphisms of all mappings from $X$ into $Y$, with the usual composition of mappings. By a mapping we mean a triple $\langle X,Y,f \rangle$ where $f$ is a function from $X$ into $Y$. The notations and concepts included corresponds to that presented in [12], [10]. We also introduce representable functors to illustrate properties of the category {\bf Ens}.

MML Identifier: ENS_1

The terminology and notation used in this paper have been introduced in the following articles [15] [6] [18] [16] [14] [19] [2] [3] [5] [7] [1] [17] [11] [13] [4] [8] [9]

Contents (PDF format)

  1. Mappings
  2. Category Ens
  3. Representable Functors

Acknowledgments

I would like to thank Andrzej Trybulec for his useful suggestions and valuable comments.

Bibliography

[1] Grzegorz Bancerek. Curried and uncurried functions. Journal of Formalized Mathematics, 2, 1990.
[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. Partial functions. Journal of Formalized Mathematics, 1, 1989.
[6] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[7] 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.
[8] Czeslaw Bylinski. Subcategories and products of categories. Journal of Formalized Mathematics, 2, 1990.
[9] Czeslaw Bylinski. Opposite categories and contravariant functors. Journal of Formalized Mathematics, 3, 1991.
[10] Saunders Mac Lane. \em Categories for the Working Mathematician, volume 5 of \em Graduate Texts in Mathematics. Springer Verlag, New York, Heidelberg, Berlin, 1971.
[11] Bogdan Nowak and Grzegorz Bancerek. Universal classes. Journal of Formalized Mathematics, 2, 1990.
[12] Zbigniew Semadeni and Antoni Wiweger. \em Wst\c ep do teorii kategorii i funktorow, volume 45 of \em Biblioteka Matematyczna. PWN, Warszawa, 1978.
[13] Andrzej Trybulec. Binary operations applied to functions. Journal of Formalized Mathematics, 1, 1989.
[14] Andrzej Trybulec. Domains and their Cartesian products. Journal of Formalized Mathematics, 1, 1989.
[15] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[16] Andrzej Trybulec. Tuples, projections and Cartesian products. Journal of Formalized Mathematics, 1, 1989.
[17] Andrzej Trybulec. Function domains and Fr\aenkel operator. Journal of Formalized Mathematics, 2, 1990.
[18] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[19] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.

Received August 1, 1991


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