January 7, 2012 3 Comments
Category theory starts out with a lot of definitions. So let’s jump right in to it.
Let be a category and let . Now suppose that whenever we have , there is a unique morphism from to . Then we say that is an initial object of . Let’s see some examples.
- In the category (with sets as objects, and functions as morphisms), the empty set is an initial object. Think about it. Take any set . There is exactly one map .
- In the category – (vector spaces over a field and –linear transformations), the -dimensional vector space is an initial object.
- In the category (groups and group homomorphisms), the trivial group is an initial object.
Not every category has an initial object. The category of non-empty sets with functions as the morphisms is such an example.
Here’s an interesting fact! Initial objects are unique (up to isomorphism).
Suppose is a category, and are initial objects. Then by the initial-ness of , there is a unique map . By the initial-ness of , there is a unique map . What happens when we compose them, we get a map . But is initial, so there can only be one such map. The identity is such a map, so . Similarly, , which is exactly the criteria for an isomorphism. This is to say, the following diagram commutes:
Oh, I never said what it means for a diagram to commute. I mean that any way you can get from one place to another by following arrows, the composition of maps you followed produces the same thing. In this case, it says that and are the same, and the analogous statement for the identity morphism .
Can you guess what type of arrow the package I’m using can’t make?
Anyway, what does this mean? Let’s look in the examples we described. In the category , it means that the empty set is the only set that maps into every other set. In the category of vector spaces, it means that the -dimensional space is the only space that can map into every other space. In the category of groups, the trivial group is the only one that maps into every other group.
So there you go. Three theorems for the price of one. If we had more examples of initial objects in categories, we’d cover even more ground. None of these proofs would be hard to cover on their own. After all, the proof we just did would have sufficed in a specific category. But now you see the power of this more generalized technique. You can see the similarities between , the trivial group, and the -dimensional vector space.