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 $\mathrm{\Lambda }=\{\alpha \}$, and a collection of morphisms $\{f_{\alpha }^{\beta }:Y^{\alpha }\to Y^{\beta }\}$ in $C$, for $\alpha \le \beta$ in $\mathrm{\Lambda }$, such that

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

b) $f_{\alpha }^{\gamma }=f_{\beta }^{\gamma }{f}_{\alpha }^{\beta }:Y^{\alpha }\to Y^{\gamma }$ for $\alpha \le \beta \le \gamma$ in $\mathrm{\Lambda }$.

There exists a category, $\mathrm{dir}\{Y^{\alpha },f_{\alpha }^{\beta }\}$, whose objects are indexed collections of morphisms $\{g_{\alpha }:Y^{\alpha }\to Z{\}}_{\alpha \in \mathrm{\Lambda }}$ such that $g_{\alpha }=g_{\beta }{f}_{\alpha }^{\beta }$ if $\alpha \le \beta$ in $\mathrm{\Lambda }$ and whose morphisms with domain $\{g_{\alpha }:Y^{\alpha }\to Z\}$ and range $\{g_{\alpha }^{\prime }:Y^{\alpha }\to Z^{\prime }\}$ are morphisms $h:Z\to Z^{\prime }$ such that $h{g}_{\alpha }=g_{\alpha }^{\prime }$ for $\alpha \in \mathrm{\Lambda }$. An initial object of $\mathrm{dir}\{Y^{\alpha },f_{\alpha }^{\beta }\}$ 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 $\mathrm{\Lambda }=\{\alpha \}$, and a collection of morphisms $\{f_{\alpha }^{\beta }:Y_{\beta }\to Y_{\alpha }\}$ in $C$, for $\alpha \le \beta$ in $\mathrm{\Lambda }$, such that

a ${}^{\prime }$) $f_{\alpha }^{\alpha }=1_{Y_{\alpha }}$ for $\alpha \in \mathrm{\Lambda }$;

b ${}^{\prime }$) $f_{\alpha }^{\gamma }=f_{\alpha }^{\beta }{f}_{\beta }^{\gamma }:Y_{\gamma }\to Y_{\alpha }$ for $\alpha \le \beta \le \gamma$ in $\mathrm{\Lambda }$.

There exists a category, $\mathrm{inv}\{Y_{\alpha },f_{\alpha }^{\beta }\}$, whose objects are indexed collections of morphisms $\{g_{\alpha }:X\to Y_{\alpha }{\}}_{\alpha \in \mathrm{\Lambda }}$ such that $g_{\alpha }=f_{\alpha }^{\beta }{g}_{\beta }$ if $\alpha \le \beta$ in $\mathrm{\Lambda }$ and whose morphisms with domain $\{g_{\alpha }:X\to Y_{\alpha }\}$ and range $\{g_{\alpha }^{\prime }:X^{\prime }\to Y_{\alpha }\}$ are morphisms $h:X\to X^{\prime }$ of $C$ such that $g_{\alpha }^{\prime }h=g_{\alpha }$ for $\alpha \in \mathrm{\Lambda }$. A terminal object of $\mathrm{inv}\{Y_{\alpha },f_{\alpha }^{\beta }\}$ 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[edit]

Comments[edit]

There is a competing terminology, with "direct limit" replaced by "colimit" , and "inverse limit" by "limit" .

References[edit]