**Definition 1**. Let be a set partially ordered by (we only need to be reflexive and transitive) and let be a category. Let be a family of objects in and suppose that for every with there exists a morphism such that

1) for all

2) whenever

Then the objects together with the morphisms is called a **direct sysetem** in We will use the notation for this direct system.

If we change the direction of the arrows in the above definition, we’ll get the definition of a inverse system , i.e.

**Definition 2**. Let be a set partially ordered by (we only need to be reflexive and transitive) and let be a category. Let be a family of objects in and suppose that for every with there exists a morphism such that

1)

2) whenever

Then the objects together with the morphisms is called an **inverse sysetem** in We will use the notation for this inverse system.

It is good definition but you have not example. so, add some examples in this note.