A cocone under a diagram is an object equipped with morphisms from each vertex of the diagram into it, such that all new diagrams arising this way commute.
A cocone which is universal is a colimit.
The dual notion is cone .
Let and be categories; we generally assume that is small. Let be a functor (called a diagram in this situation). Then a cocone (or inductive cone) over is a pair of an object and a natural transformation (where is the constant diagram , , ). In other words, a diagram as follows, together with a natural transformation going south west to north east.
Note that a cocone in is precisely a cone in the opposite category .
Terminology for natural transformations can also be applied to cocones. For example, a component of a cocone is a component of the natural transformation ; that is, the component for each object of is the morphism .
A morphism of cocones is a morphism in such that for all objects in (symbolically ); the composition being the composition of underlying morphisms in . Thus cocones form a category whose initial object if it exists is a colimit of .
Last revised on June 20, 2020 at 09:30:37. See the history of this page for a list of all contributions to it.