Cone (category theory)

In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well.

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. Thus, for example, when J is a discrete category, 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 span. 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

ψ

X

:

N

→

F

(

X

)

{\displaystyle \psi _{X}\colon N\to F(X)\,}

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 co-cone from F to N is a family of morphisms

ψ

X

:

F

(

X

)

→

N

{\displaystyle \psi _{X}\colon F(X)\to N\,}

for each object X of J, such that for every morphism f : X → Y in J the following diagram commutes:

Equivalent formulations

At first glance cones seem to be slightly abnormal constructions in category theory.