Journal of Formalized Mathematics
Volume 1, 1989
University of Bialystok
Copyright (c) 1989 Association of Mizar Users

## Introduction to Categories and Functors

Czeslaw Bylinski
Warsaw University, Bialystok

### Summary.

The category is introduced as an ordered 5-tuple of the form $\langle O, M, dom, cod, \cdot, id \rangle$ where $O$ (objects) and $M$ (morphisms) are arbitrary nonempty sets, $dom$ and $cod$ map $M$ onto $O$ and assign to a morphism domain and codomain, $\cdot$ is a partial binary map from $M \times M$ to $M$ (composition of morphisms), $id$ applied to an object yields the identity morphism. We define the basic notions of the category theory such as hom, monic, epi, invertible. We next define functors, the composition of functors, faithfulness and fullness of functors, isomorphism between categories and the identity functor.

#### MML Identifier: CAT_1

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

Contents (PDF format)

#### Bibliography

[1] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[2] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[3] Czeslaw Bylinski. Partial functions. Journal of Formalized Mathematics, 1, 1989.
[4] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[5] Andrzej Trybulec. Binary operations applied to functions. Journal of Formalized Mathematics, 1, 1989.
[6] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[7] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[8] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.