Cones and Limits
A cone with vertex N of a diagram D : J → C is a morphism from the constant diagram Δ(N) to D. The constant diagram is the diagram which sends every object of J to an object N of C and every morphism to the identity morphism on N.
The limit of a diagram D is a universal cone to D. That is, a cone through which all other cones uniquely factor. If the limit exists in a category C for all diagrams of type J one obtains a functor
- lim : CJ → C
which sends each diagram to its limit.
Dually, the colimit of diagram D is a universal cone from D. If the colimit exists for all diagrams of type J one has a functor
- colim : CJ → C
which sends each diagram to its colimit.
Read more about this topic: Diagram (category Theory)
Famous quotes containing the words cones and, cones and/or limits:
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—Martha Shelley, U.S. author and social activist. As quoted in Making History, part 3, by Eric Marcus (1992)
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—Henry David Thoreau (1817–1862)
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—Ron Taffel (20th century)