# System (in a category)

direct and inverse system in a category $C$

A direct system $\{ Y ^ \alpha , f _ \alpha ^ { \beta } \}$ in $C$ consists of a collection of objects $\{ Y ^ \alpha \}$, indexed by a directed set $\Lambda = \{ \alpha \}$, and a collection of morphisms $\{ f _ \alpha ^ { \beta } : Y ^ \alpha \rightarrow Y ^ \beta \}$ in $C$, for $\alpha \leq \beta$ in $\Lambda$, such that

a) $f _ \alpha ^ { \alpha } = 1 _ {Y ^ \alpha }$ for $\alpha \in \Lambda$;

b) $f _ \alpha ^ { \gamma } = f _ \beta ^ { \gamma } f _ \alpha ^ { \beta } : Y ^ \alpha \rightarrow Y ^ \gamma$ for $\alpha \leq \beta \leq \gamma$ in $\Lambda$.

There exists a category, $\mathop{\rm dir} \{ Y ^ \alpha , f _ \alpha ^ { \beta } \}$, whose objects are indexed collections of morphisms $\{ g _ \alpha : Y ^ \alpha \rightarrow Z \} _ {\alpha \in \Lambda }$ such that $g _ \alpha = g _ \beta f _ \alpha ^ { \beta }$ if $\alpha \leq \beta$ in $\Lambda$ and whose morphisms with domain $\{ g _ \alpha : Y ^ \alpha \rightarrow Z \}$ and range $\{ g _ \alpha ^ \prime : Y ^ \alpha \rightarrow Z ^ \prime \}$ are morphisms $h: Z \rightarrow Z ^ \prime$ such that $hg _ \alpha = g _ \alpha ^ \prime$ for $\alpha \in \Lambda$. An initial object of $\mathop{\rm dir} \{ Y ^ \alpha , f _ \alpha ^ { \beta } \}$ is called a direct limit of the direct system $\{ Y ^ \alpha , f _ \alpha ^ { \beta } \}$. The direct limits of sets, topological spaces, groups, and $R$- modules are examples of direct limits in their respective categories.

Dually, an inverse system $\{ Y _ \alpha , f _ \alpha ^ { \beta } \}$ in $C$ consists of a collection of objects $\{ Y _ \alpha \}$, indexed by a directed set $\Lambda = \{ \alpha \}$, and a collection of morphisms $\{ f _ \alpha ^ { \beta } : Y _ \beta \rightarrow Y _ \alpha \}$ in $C$, for $\alpha \leq \beta$ in $\Lambda$, such that

a ${} ^ \prime$) $f _ \alpha ^ { \alpha } = 1 _ {Y _ \alpha }$ for $\alpha \in \Lambda$;

b ${} ^ \prime$) $f _ \alpha ^ { \gamma } = f _ \alpha ^ { \beta } f _ \beta ^ { \gamma } : Y _ \gamma \rightarrow Y _ \alpha$ for $\alpha \leq \beta \leq \gamma$ in $\Lambda$.

There exists a category, $\mathop{\rm inv} \{ Y _ \alpha , f _ \alpha ^ { \beta } \}$, whose objects are indexed collections of morphisms $\{ g _ \alpha : X \rightarrow Y _ \alpha \} _ {\alpha \in \Lambda }$ such that $g _ \alpha = f _ \alpha ^ { \beta } g _ \beta$ if $\alpha \leq \beta$ in $\Lambda$ and whose morphisms with domain $\{ g _ \alpha : X \rightarrow Y _ \alpha \}$ and range $\{ g _ \alpha ^ \prime : X ^ \prime \rightarrow Y _ \alpha \}$ are morphisms $h: X \rightarrow X ^ \prime$ of $C$ such that $g _ \alpha ^ \prime h = g _ \alpha$ for $\alpha \in \Lambda$. A terminal object of $\mathop{\rm inv} \{ Y _ \alpha , f _ \alpha ^ { \beta } \}$ is called an inverse limit of the inverse system $\{ Y _ \alpha , f _ \alpha ^ { \beta } \}$. The inverse limits of sets, topological spaces, groups, and $R$- modules are examples of inverse limits in their respective categories.

The concept of an inverse limit is a categorical generalization of the topological concept of a projective limit.

#### References

 [1] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)