Limit (category theory)
Definition

Limits and colimits in a category C {\displaystyle C} are defined by means of diagrams in C {\displaystyle C}. Formally, a diagram of shape J {\displaystyle J} in C {\displaystyle C} is a functor from J {\displaystyle J} to C {\displaystyle C}:

The category J {\displaystyle J} is thought of as an index category, and the diagram F {\displaystyle F} is thought of as indexing a collection of objects and morphisms in C {\displaystyle C} patterned on J {\displaystyle J}.

One is most often interested in the case where the category J {\displaystyle J} is a small or even finite category. A diagram is said to be small or finite whenever J {\displaystyle J} is.
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