Definition
Let F : J → C be a diagram in C. Formally, a diagram is nothing more than a functor from J to C. The change in terminology reflects the fact that we think of F as indexing a family of objects and morphisms in C. The category J is thought of as an "index category". One should consider this in analogy with the concept of an indexed family of objects in set theory. The primary difference is that here we have morphisms as well.
Let N be an object of C. A cone from N to F is a family of morphisms
for each object X of J such that for every morphism f : X → Y in J the following diagram commutes:
The (usually infinite) collection of all these triangles can be (partially) depicted in the shape of a cone with the apex N. The cone ψ is sometimes said to have vertex N and base F.
One can also define the dual notion of a cone from F to N (also called a co-cone) by reversing all the arrows above. Explicitly, a cone from F to N is a family of morphisms
for each object X of J such that for every morphism f : X → Y in J the following diagram commutes:
Read more about this topic: Cone (category Theory)
Famous quotes containing the word definition:
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)