Journal of Formalized Mathematics
Volume 7, 1995
University of Bialystok
Copyright (c) 1995 Association of Mizar Users

## Indexed Category

Grzegorz Bancerek
Institute of Mathematics, Polish Academy of Sciences

### Summary.

The concept of indexing of a category (a part of indexed category, see [14]) is introduced as a pair formed by a many sorted category and a many sorted functor. The indexing of a category $C$ against to [14] is not a functor but it can be treated as a functor from $C$ into some categorial category (see [1]). The goal of the article is to work out the notation necessary to define institutions (see [11]).

#### MML Identifier: INDEX_1

The terminology and notation used in this paper have been introduced in the following articles [15] [8] [20] [16] [21] [4] [5] [7] [18] [17] [19] [12] [3] [6] [9] [10] [2] [13] [1]

#### Contents (PDF format)

1. Category-yielding Functions
2. Pairs of Many Sorted Sets
3. Indexing
4. Indexing vs Functors
5. Composing Indexings and Functors

#### Bibliography

[1] Grzegorz Bancerek. Categorial categories and slice categories. Journal of Formalized Mathematics, 6, 1994.
[2] Grzegorz Bancerek and Agata Darmochwal. Comma category. Journal of Formalized Mathematics, 4, 1992.
[3] Grzegorz Bancerek and Piotr Rudnicki. On defining functions on trees. Journal of Formalized Mathematics, 5, 1993.
[4] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[5] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[6] Czeslaw Bylinski. Introduction to categories and functors. Journal of Formalized Mathematics, 1, 1989.
[7] Czeslaw Bylinski. Partial functions. Journal of Formalized Mathematics, 1, 1989.
[8] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[9] Czeslaw Bylinski. Subcategories and products of categories. Journal of Formalized Mathematics, 2, 1990.
[10] Czeslaw Bylinski. Opposite categories and contravariant functors. Journal of Formalized Mathematics, 3, 1991.
[11] Joseph A. Goguen and Rod M. Burstall. Introducing institutions. \em Lecture Notes in Computer Science, 164:221--256, 1984.
[12] Beata Madras. Product of family of universal algebras. Journal of Formalized Mathematics, 5, 1993.
[13] Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Preliminaries to circuits, I. Journal of Formalized Mathematics, 6, 1994.
[14] Andrzej Tarlecki, Rod M. Burstall, and A. Goguen, Joseph. Some fundamental algebraic tools for the semantics of computation: Part 3. indexed categories. \em Theoretical Computer Science, 91:239--264, 1991.
[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] Andrzej Trybulec. Many-sorted sets. Journal of Formalized Mathematics, 5, 1993.
[19] Andrzej Trybulec. Many sorted algebras. Journal of Formalized Mathematics, 6, 1994.
[20] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[21] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.