# The Unapologetic Mathematician

## Cones and cocones

There are a few auxiliary concepts we’ll need before the next major topic. Let’s start with two categories $\mathcal{J}$ and $\mathcal{C}$, and the category of functors $\mathcal{C}^\mathcal{J}$. If the following seems very complicated, consider $\mathcal{J}$ to be any particular toy category you’d like, so this category of functors is a diagram category.

Now, for every object $C\in\mathcal{C}$ there is a constant functor that sends every object of $\mathcal{J}$ to $C$, and every morphism to the identity on $C$. Actually, this assignment of the constant functor to an object of $C$ is a functor from $\mathcal{C}$ to $\mathcal{C}^\mathcal{J}$. Indeed, given a morphism $f:C\rightarrow D$ we get a natural transformation from the one constant functor to the other, whose component at each object of $\mathcal{J}$ is $f$. We call this the “diagonal” functor $\Delta:\mathcal{C}\rightarrow\mathcal{C}^\mathcal{J}$. That is, $\Delta(C)$ is the constant functor with value $C$.

Let $F:\mathcal{J}\rightarrow\mathcal{C}$ be some particular functor in $\mathcal{C}^\mathcal{J}$. A cone is an object in the comma category $(\Delta\downarrow F)$. Let’s unpack this definition. Since $\Delta$ has $\mathcal{C}$ as its source, and $F$ is a fixed object — the same thing as a functor from the category $\mathbf{1}$ to $\mathcal{C}^\mathcal{J}$, we know what objects of this category look like.

An object of the category $(\Delta\downarrow F)$ consists of an object $C$ of $\mathcal{C}$ and an arrow from $\Delta(C)$ to $F$. But an arrow in $\mathcal{C}^\mathcal{J}$ is a natural transformation of functors. That is, for each object $J$ of $\mathcal{J}$ we need an arrow in $\mathcal{C}$ from $\left[\Delta(C)\right](J)$ to $F(J)$. But $\left[\Delta(C)\right](J)=C$ for all objects $J$. So we just need an object $C\in\mathcal{C}$ and an arrow $C\rightarrow F(J)$ for each $J\in\mathcal{J}$.

Of course, there’s also naturality conditions to be concerned with. If $j:J_1\rightarrow J_2$ is an arrow in $\mathcal{J}$, and $c_1:C\rightarrow F(J_1)$ and $c_2:C\rightarrow F(J_2)$ are the required arrows from $C$, then naturality requires that $c_2=F(j)\circ c_1$. So we need a collection of arrows from $C$ to the objects in the image of $F$ that are compatible with the arrows from $\mathcal{J}$. Such a collection defines an object in the comma category $(\Delta\downarrow F)$ — a cone on $F$.

Cocones are defined similarly. A cocone on $F$ is an object in the comma category $(F\downarrow\Delta)$. That is, it’s an object $C\in\mathcal{C}$ and a collection of arrows from the objects in the image of $F$ that are compatible with the arrows from $\mathcal{J}$.

The description may seem a little odd, but try writing it out for some very simple categories $\mathcal{J}$. For example, let $\mathcal{J}$ be a set. Then try letting it be an ordinal, or another preorder. After you write down the definition of a cone and a cocone on some simple categories the general idea should seem to make more sense.

June 18, 2007 Posted by | Category theory | 4 Comments