Journal of Formalized Mathematics
Volume 2, 1990
University of Bialystok
Copyright (c) 1990 Association of Mizar Users

## Subcategories and Products of Categories

Czeslaw Bylinski
Warsaw University, Bialystok
Supported by RPBP.III-24.C1.

### Summary.

The {\it subcategory} of a category and product of categories is defined. The {\it inclusion functor} is the injection (inclusion) map $E \atop \hookrightarrow$ which sends each object and each arrow of a Subcategory $E$ of a category $C$ to itself (in $C$). The inclusion functor is faithful. {\it Full subcategories} of $C$, that is, those subcategories $E$ of $C$ such that $\hbox{Hom}_E(a,b) = \hbox{Hom}_C(b,b)$ for any objects $a,b$ of $E$, are defined. A subcategory $E$ of $C$ is full when the inclusion functor $E \atop \hookrightarrow$ is full. The proposition that a full subcategory is determined by giving the set of objects of a category is proved. The product of two categories $B$ and $C$ is constructed in the usual way. Moreover, some simple facts on $bifunctors$ (functors from a product category) are proved. The final notions in this article are that of projection functors and product of two functors ({\it complex} functors and {\it product} functors).

#### MML Identifier: CAT_2

The terminology and notation used in this paper have been introduced in the following articles [10] [7] [12] [9] [13] [3] [4] [6] [2] [8] [1] [11] [5]

Contents (PDF format)

#### Bibliography

[1] Grzegorz Bancerek. Curried and uncurried functions. Journal of Formalized Mathematics, 2, 1990.
[2] Czeslaw Bylinski. Basic functions and operations on functions. Journal of Formalized Mathematics, 1, 1989.
[3] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[4] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[5] Czeslaw Bylinski. Introduction to categories and functors. Journal of Formalized Mathematics, 1, 1989.
[6] Czeslaw Bylinski. Partial functions. Journal of Formalized Mathematics, 1, 1989.
[7] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[8] 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.
[9] Andrzej Trybulec. Domains and their Cartesian products. Journal of Formalized Mathematics, 1, 1989.
[10] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[11] Andrzej Trybulec. Function domains and Fr\aenkel operator. Journal of Formalized Mathematics, 2, 1990.
[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 31, 1990