The Unapologetic Mathematician

Mathematics for the interested outsider

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