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[edit]

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

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#### Comments[edit]

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

#### References[edit]

[1a] |  B. Mitchell, "Theory of categories" , Acad. Press (1965)

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