In my research aroused a new kind of structures which I call categories with star-morphisms. In this blog post I define categories with star-morphisms. For sample usages of star categories see this draft article.
Definition 1 A pre-category with star-morphisms consists of
- a pre-category (the base pre-category);
- a set (star-morphisms);
- a function “” defined on (how many objects are connected by this multimorphism);
- a function defined for every ;
- a function (star composition) defined for and being an -indexed family of morphisms of such that ( is the source object of the morphism ) such that
such that it holds:
- (associativiy law)
(Here by definition .)
The meaning of the set is an extension of having as morphisms things with arbitrary (possibly infinite) indexed set of objects, not just two objects as morphims of have only source and destination.
Definition 2 A star category is a star pre-category whose base is a category and the following equality (the law of composition with identity) holds for every multimorphism :