Journal of Formalized Mathematics
Volume 8, 1996
University of Bialystok
Copyright (c) 1996 Association of Mizar Users

## Functors for Alternative Categories

Andrzej Trybulec
Warsaw University, Bialystok

### Summary.

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).

#### MML Identifier: FUNCTOR0

The terminology and notation used in this paper have been introduced in the following articles [11] [6] [17] [18] [19] [12] [3] [5] [4] [2] [10] [1] [7] [13] [9] [14] [8] [15] [16]

#### Contents (PDF format)

1. Preliminaries
2. Functions Between Cartesian Squares
3. Unary Transformations
4. Functors
5. The Composition of Functors
6. The Inverse Functor

#### Bibliography

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