Each star category gives rise to a category (abrupt category, see a remark below why I call it “abrupt”), as described below. Below for simplicity I assume that the set and the set of our indexed families of functions are disjoint. The general case (when they are not necessarily disjoint) may be easily elaborated by the reader.
- Objects are indexed (by for some ) families of objects of the category and an (arbitrarily choosen) object not in this set
- There are the following disjoint sets of morphims:
- indexed (by for some ) families of morphisms of
- elements of
- the identity morphism on
- Source and destination of morphims are defined by the formulas:
- Compositions of morphisms are defined by the formulas:
- Identity morphisms for an object are:
We need to prove it is really a category.
Proof We need to prove:
- Composition is associative
- Composition with identities complies with the identity law.
- ; ; .
- ; .
Remark I call the above defined category abrupt category because (excluding identity morphisms) it allows composition with an only on the left (not on the right) so that the morphism is “abrupt” on the right.