Functors for Alternative Categories

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 bifunction, i.e. a function mapping the Cartesian square of a set A into the Cartesian square of a set B. A bifunction f is said to be covariant if there is a function g from A into B that f is the Cartesian square of g and f is contravariant if there is a function g such that f (o1,o2) = 〈g(o2),g(o1)〉. The term 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 · f . Eventually, the morphism map of a functor from C1 into C2 is a transformation from the arrows of the category C1 into the composition of the object map of the functor and the arrows of C2. 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).