## Direct and inverse systems; definition

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