Limits and colimits

Let D be a small category and let C be any category. A D-shaped diagram in C is a functor F : D −→ C . A morphism F −→ F ′ of D-shaped diagrams is a natural transformation, and we have the category D [C ] of D-shaped diagrams in C . Any object C of C determines the constant diagram C that sends each object of D to C and sends each morphism of D to the identity morphism of C. The colimit, colimF , of a D-shaped diagram F is an object of C together with a morphism of diagrams ι : F −→ colim F that is initial among all such morphisms. This means that if η : F −→ A is a morphism of diagrams, then there is a unique map η̃ : colim F −→ A in C such that η̃ ◦ ι = η. Diagrammatically, this property is expressed by the assertion that, for each map d : D −→ D in D , we have a commutative diagram