Direct and inverse systems; definition

Posted: January 12, 2011 in Direct and Inverse Limit, Noncommutative Ring Theory Notes
Tags: ,

Definition 1. Let I be a set partially ordered by \leq (we only need \leq to be reflexive and transitive) and let \mathcal{C} be a category. Let \{M_i \}_{i \in I} be a family of objects in \mathcal{C} and suppose that for every i, j \in I with i \leq j there exists a morphism f_{ij} : M_i \longrightarrow M_j such that

1) f_{ii}=\text{id}_{M_i}, for all i \in I,

2) f_{jk}f_{ij}=f_{ik} whenever i \leq j \leq k.

Then the objects M_i together with the morphisms f_{ij} is called a direct sysetem in \mathcal{C}. We will use the notation \{M_i, f_{ij} \} 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 I be a set partially ordered by \leq (we only need \leq to be reflexive and transitive) and let \mathcal{C} be a category. Let \{M_i \}_{i \in I} be a family of objects in \mathcal{C} and suppose that for every i, j \in I with i \leq j there exists a morphism f_{ij} : M_j \longrightarrow M_i such that

1) f_{ii}=\text{id}_{M_i},

2) f_{ij}f_{jk}=f_{ik} whenever i \leq j \leq k.

Then the objects M_i together with the morphisms f_{ij} is called an inverse sysetem in \mathcal{C}. We will use the notation \{M_i, f_{ij} \} for this inverse system.

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Comments
  1. solomon says:

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

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