Sorry for the delay since my last post (to those who actually read this…)
So I stumbled across a really nice way of looking at universal properties that is equivalent to specifying them as a terminal (or initial) object in a suitable comma category, but it has a much nicer “intuitive feel.”
Cones (and co-Cones)
Let C be some category. Let be a collection of objects of C indexed by some set , and let be a collection of morphisms in C (we do not require that there is a morphism for any two , we only allow for the possibility of there being one). We call this collection of objects and morphisms a diagram in C.
Let be a diagram in C. A cone on is a C-object and collection of morphisms , such that for all , . If we have two such “cones,” and on this diagram, we say such that the appropriate diagram commutes (try to figure it out! It’s a good idea to get an intuitive feel for these things). It follows pretty quickly then that we have a category of cones over , call it . A limit of the diagram is then just a terminal object in .
Similarly, a co-cone is just a cone with all the arrows reversed (i.e. an object together with maps for each ). A co-limit of such a co-cone is an initial object in the category of co-cones over the appropriate diagram.
Say we’re working in the category R-Mod for some ring with unity .
- Pullbacks: Let be three -modules, and consider the diagram . The limit of this diagram is then just the ordinary pullback (or fiber product), the module
- Products: Let be modules. Then the limit of the “diagram” consisting of just and and no morphisms between them is the product .
- co-Products: consider the same diagram used for the product. The co-limit of this is then the co-product of and , .
- Terminal objects: are just the limit of the “empty diagram.”
- Initial objects: are just the co-limit of the “empty diagram.”
and so on.
Having “Finite (co-) Limits”
Notice that all the above limits and co-limits were taken over a “finite” diagram. That is, there were only finitely many objects and morphisms in each diagram. Such (co-)limits are referred to as “finite” (co-)limits (I wonder why…). It turns out that it’s a highly desirable property for a category to “have all (finite) (co-)limits.” It took me a lonnnnnggg time to grok this.
Remember when we first started talking about universal properties? When you specify that an object satisfies a certain universal property, it is unique up to unique isomorphism if it actually exists. These objects don’t have to exist. The property of having, say, all finite limits or co-limits means that whenever you specify a universal property for an object with a finite diagram, that object actually exists. It’s a theorem (that I don’t currently know how to prove) that a category C with a terminal object and all pullbacks has all finite limits. Dually, if C has an initial object and all pushouts, it has all finite co-limits. Is this so unreasonable? Look at the list of examples again.
Back? Good. Suppose we’ve got all pullbacks and a terminal object, call it 1. Then the product is just the limit of the pullback diagram . The equalizer of two parallel maps is the pullback of . The kernel of (we’re still working with modules) is the pullback of . Get the picture?
A pretty good thing to try here would be to find out how these are equivalent to universal properties. So go try that. 🙂