Good Math, Bad Math

Things are a bit busy at work on my real job lately, and I don't have time to put together as detailed a post for today as I'd like. Frankly, looking at it, my cat theory post yesterday was half-baked at best; I should have held off until I could polish it a bit and make it more comprehensible.

So I'm going to avoid that error today. Since we've had an interesting discussion focusing on the number zero, I thought it would be fun to show what zero means in category theory.

There are two zeros it category theory: the zero object, and the zero arrow.

Zero Objects

The zero object is easier. Suppose we've got a category, C. Suppose that C has a terminal object - that is, an object t for which other object x in C has exactly one arrow f : x → t. And suppose that C also has an initial object: that is, an object i for which every object x in C has exactly one arrow, f : i → x. If C has both an initial object i, and a terminal object t, and i = t, then it's a zero object for the category. A category can actually have many zero objects. For example, in the category of groups, any trivial group (that is, a group which contains only one element) is a zero object in the category of groups. For an intuition of why this is called "zero" think of the number zero. It's a strange little number that sits dead in the middle of everything. It's the central point on the complex plane, it's the center of the number line. A zero object sits in the middle of a category: everything has exactly one arrow to it, and one arrow from it.

Zero Arrows

The idea of a zero arrow comes roughly from the basic concept of a zero function. A zero function is a function f where for all x, f(x) = 0. One of the properties of a zero function is that composing a zero function with any other function results in a zero function. (That is, if f is a zero function, f(g(x))) is also a zero function.)

A zero morphism in a category C is part of a set of arrows, called the zero family of morphisms of C, where composing any morphism with a member of the zero family results in a morphism in the zero family.

To be precise: suppose we have a category C, and for any pair b of objects a and b ∈ Obj(C), there is a morphism 0_a,b : a → .

The morphism 0_d,e must satisfy one important property:

For any two morphisms m : d → e, and n : f → g, the following diagram must commute:

To see why this means that any composition with a 0 arrow will give you a zero arrow, look at what happens when we start with an ID arrow. Let's make n : f → g an id arrow, by making f=g. We get the following diagram:

So - any zero arrow composed with an id arrow is a zero arrow. Now use induction to step up from ID arrows by looking at one arrow composed with an ID arrow and a zero arrow. You've still got a zero. Keep going - you'll keep getting zeros.

Zero arrows are actually very important buggers: the algebraic and topographical notions of a kernel can be defined in category theory using the zero morphisms.

One last little note: in general, it looks like the zero objects and the zero morphisms are unrelated. They're not. In fact, if you have a category with a zero object and a family of zero morphisms, then you can find the zero morphisms by using the zero object. For any objects a and b, 0_a,b = (a → 0) º (0 → b).