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:
“...there was the annual Fourth of July picketing at Independence Hall in Philadelphia. ...I thought it was ridiculous to have to go there in a skirt. But I did it anyway because it was something that might possibly have an effect. I remember walking around in my little white blouse and skirt and tourists standing there eating their ice cream cones and watching us like the zoo had opened.”
—Martha Shelley, U.S. author and social activist. As quoted in Making History, part 3, by Eric Marcus (1992)
“...there was the annual Fourth of July picketing at Independence Hall in Philadelphia. ...I thought it was ridiculous to have to go there in a skirt. But I did it anyway because it was something that might possibly have an effect. I remember walking around in my little white blouse and skirt and tourists standing there eating their ice cream cones and watching us like the zoo had opened.”
—Martha Shelley, U.S. author and social activist. As quoted in Making History, part 3, by Eric Marcus (1992)
“Yet shall he mount, and keep his distant way
Beyond the limits of a vulgar fate:
Beneath the Good how far—but far above the Great.”
—Thomas Gray (1716–1771)