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Concept
Cone (category theory)
Summary
In , a branch of mathematics, the cone of a functor is an abstract notion used to define the of that functor. Cones make other appearances in category theory as well.
Let F : J → C be a 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 and morphisms in C. The 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. Thus, for example, when J is a , it corresponds most closely to the idea of an indexed family in set theory. Another common and more interesting example takes J to be a . J can also be taken to be the empty category, leading to the simplest cones.
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 notion of a cone from F to N (also called a co-cone) by reversing all the arrows above. Explicitly, a co-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:
At first glance cones seem to be slightly abnormal constructions in category theory. They are maps from an object to a functor (or vice versa). In keeping with the spirit of category theory we would like to define them as morphisms or objects in some suitable category. In fact, we can do both.
Let J be a small category and let CJ be the of type J in C (this is nothing more than a ). Define the diagonal functor Δ : C → CJ as follows: Δ(N) : J → C is the constant functor to N for all N in C.
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