Pointed category: why is it defined that way?

Before I start talking about what I want, I should point out that pointed category is pretty much the lowest generalization of abelian category, which is an important concept when thinking about algebra from a categorical perspective.

What is a pointed category? What should the term ‘pointed category’ refer to? Let’s start with a simpler case. What is a pointed set? A set with a distinguished point. What is a pointed space? A topological space with a distinguished point. What’s a pointed group? Never heard of it, but we could guess that it’s a group with a distinguished element. A group sorta already has a distinguished element, its identity element. Maps between pointed sets and pointed spaces must preserve the point. This is true for the identity elements in groups. It seems like groups are already pointed.

How should we express this concept of ‘pointed’ in categorical terms, in order to categorify it and figure out what a pointed category should be? In pointed sets and pointed spaces, they have a distinguished element. Elements of an object are expressed in categorical terms as a morphism from the terminal object. This comes from the situation in the category , where the terminal object is a singleton, and a function from a singleton carries no more information than the element of its codomain which it points to. The same is true in the category . In the category , the terminal object is a singleton also (usually this is called the trivial group, or the zero group). However for a group , there is only one map from the singleton into because group homomorphisms must preserve identity elements. This also makes the singleton the initial object in .

This leads to the following standard definition: Let be a category with a terminal object, denoted . A pointed object in is an object with an element . A map of pointed objects is a morphism such that the triangle commutes .

Great, now it should be pretty clear what a pointed category should be, right? A category with a distinguished object? Close, but no. A pointed category is actually a category which has a zero object, meaning an object which is both initial and terminal. We’ve seen an example of one of these already: . Why this though? Why doesn’t this one fit the trend? Something I’ve learned about categorification is that usually your first guess is missing something, some important point of nuance that you couldn’t see in the decategorified context. A classic example of this is the fact that strict -categories really don’t seem to be the right way to define -categories.

A set is just a bag of dots. So when you want to make it pointed, you just pick one, and you’re done. A category however is a bag of dots with arrows between them. The arrows are the most important part of category theory. Simply distinguishing an object doesn’t really seem aligned with the philosophy of category theory. The right thing to do is to distinguish an object and and make its distinction known to every other object in the category, ie by having a unique arrow to and from every other object. Something nice that happens with this definition is that every hom-set in a pointed category is a pointed set. Another nice thing is that in any category with a terminal object, the corresponding category of pointed objects with maps of pointed objects is a pointed category.

So when we jump up a level to categories, rather than just sticking a pin into your favorite object, you have to make it the centerpiece of your category in some sense. Let’s see what this concept would mean in the other cases of pointed things we know about. A group is a pointed thing naturally, not the pointed version of something, but its distinguished point is the centerpiece of the structure, in that multiplying any other element of the group by the identity does nothing. For a set, there is no relationship between the elements, so sticking a pin in some element is everything you can do. Pointed spaces seems like they should pose a more interesting case though. A topological space does provide a notion of connection between its elements. But what should we say? That the distinguished point is in an open set with every other point? This is automatically true once you distinguish a point: the whole space must be an open set.

Maybe the reason many people don’t expect the definition of pointed category is that pointed sets and pointed spaces can’t see centerpiece-ness, and groups embody it so naturally, that you don’t notice it. This is exactly what I would expect out of categorification though, to see something that is important that you can’t see in the decategorified context. Perhaps it should be called something like ‘centered’ rather than ‘pointed’.