Zeros in Category Theory

Jul 26 2006 Published by under Category Theory, goodmath

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
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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
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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 0a,b : a → .
The morphism 0d,e must satisfy one important property:
For any two morphisms m : d → e, and n : f → g, the following diagram must commute:
zero.jpg
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:
zero-id.jpg
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, 0a,b = (a → 0) º (0 → b).

3 responses so far

  • Blake Stacey says:

    The zero arrow, invented by Mayan category-theoreticians around the year 700. . . .

  • Chad Groft says:

    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).

    More that that; it looks like, given a zero object 0, the family of arrows 0_{a,b} = (a → 0) º (0 → b) are automatically zero morphisms. Plugging these into the commutative square above, and using the fact that 0 is both initial and terminal, both directions along the square should be 0_{d,g}. Is that correct?

  • Zero says:

    Excellent. I'd love to see more posts about zero. It reminds me of my love for Xeno's Dichotomy. I'm off to check out the Zero Saga.
    Posted by: Gabe | July 21, 2006 03:46 PM
    ****************************************
    Size 101 So how big am I?
    IMO, there are two infinities, big & small.
    Infinity "big" is plus. Infinity "small" is minus.
    Isa 40:12 ¶ Who hath measured the waters in the hollow
    of his hand, and meted out heaven with the span, and comprehended
    the dust of the earth in a measure, and weighed the mountains in scales, and the hills in a balance?
    me..................a speck on earth
    earth...............a speck in the milky way
    milky way.......a speck in the universe
    universe..........a speck in God's pocket
    From my perspective:
    Anything smaller than me is small.
    Anything bigger than me is big.
    Zero

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