Volume 8, 1996

University of Bialystok

Copyright (c) 1996 Association of Mizar Users

**Andrzej Trybulec**- Warsaw University, Bialystok

- An attempt to define the concept of a functor covering both cases (covariant and contravariant) resulted in a structure consisting of two fields: the object map and the morphism map, the first one mapping the Cartesian squares of the set of objects rather than the set of objects. We start with an auxiliary notion of {\em bifunction}, i.e. a function mapping the Cartesian square of a set $A$ into the Cartesian square of a set $B$. A {\em bifunction} $f$ is said to be {\em covariant} if there is a function $g$ from $A$ into $B$ that $f$ is the Cartesian square of $g$ and $f$ is {\em contravariant} if there is a function $g$ such that $f(o_1,o_2) = \langle g(o_2),g(o_1) \rangle$. The term {\em transformation}, another auxiliary notion, might be misleading. It is not related to natural transformations. A transformation from a many sorted set $A$ indexed by $I$ into a many sorted set $B$ indexed by $J$ w.r.t. a function $f$ from $I$ into $J$ is a (many sorted) function from $A$ into $B \cdot f$. Eventually, the morphism map of a functor from $C_1$ into $C_2$ is a transformation from the arrows of the category $C_1$ into the composition of the object map of the functor and the arrows of $C_2$.\par Several kinds of functor structures have been defined: one-to-one, faithful, onto, full and id-preserving. We were pressed to split property that the composition be preserved into two: comp-preserving (for covariant functors) and comp-reversing (for contravariant functors). We defined also some operation on functors, e.g. the composition, the inverse functor. In the last section it is defined what is meant that two categories are isomorphic (anti-isomorphic).

- Preliminaries
- Functions Between Cartesian Squares
- Unary Transformations
- Functors
- The Composition of Functors
- The Inverse Functor

- [1]
Czeslaw Bylinski.
Basic functions and operations on functions.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Czeslaw Bylinski.
Binary operations.
*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.
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]
Malgorzata Korolkiewicz.
Homomorphisms of many sorted algebras.
*Journal of Formalized Mathematics*, 6, 1994. - [9]
Beata Madras.
Product of family of universal algebras.
*Journal of Formalized Mathematics*, 5, 1993. - [10]
Andrzej Trybulec.
Binary operations applied to functions.
*Journal of Formalized Mathematics*, 1, 1989. - [11]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [12]
Andrzej Trybulec.
Tuples, projections and Cartesian products.
*Journal of Formalized Mathematics*, 1, 1989. - [13]
Andrzej Trybulec.
Many-sorted sets.
*Journal of Formalized Mathematics*, 5, 1993. - [14]
Andrzej Trybulec.
Many sorted algebras.
*Journal of Formalized Mathematics*, 6, 1994. - [15]
Andrzej Trybulec.
Categories without uniqueness of \rm cod and \rm dom.
*Journal of Formalized Mathematics*, 7, 1995. - [16]
Andrzej Trybulec.
Examples of category structures.
*Journal of Formalized Mathematics*, 8, 1996. - [17]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [18]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [19]
Edmund Woronowicz.
Relations defined on sets.
*Journal of Formalized Mathematics*, 1, 1989.

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